Creative Physics Problems for Physics with Calculus: Mechanics 1 1442190957, 9781442190955

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Creative Physics Problems – Chris McMullen, Ph.D.

Creative Physics Problems For Physics with Calculus Vol. 1: Mechanics Chris McMullen, Ph.D.

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Volume 1: Mechanics

Creative Physics Problems for Physics with Calculus Volume 1: Mechanics Copyright (c) 2009 Chris McMullen All rights reserved. This includes the right to reproduce any portion of this book in any form. However, teachers who purchase one copy of this book, or borrow one physical copy from a library, may make and distribute photocopies of selected pages for instructional purposes for their own classes only. www.faculty.lsmsa.edu/CMcMullen Custom Books Textbooks / science / physics ISBN: 1442190957 EAN-13: 9781442190955

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Creative Physics Problems – Chris McMullen, Ph.D.

Contents 1 One-Dimensional Motion

4

2 Projectile Motion and Vectors

14

Review 1: Motion

25

3 Newton’s Laws of Motion

34

4 Uniform Circular Motion

48

Review 2: Newton’s Laws

54

5 Conservation of Energy

64

6 Systems of Objects

75

Review 3: Conservation of Energy and Momentum

86

7 Rotation

95

8 Fluids (One Problem)

114

Review 4: Rotation

115

Final Review: Mechanics

119

Answers to Selected Problems

109

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Volume 1: Mechanics

1 One-Dimensional Motion Note: An asterisk (*) designates a problem that does not have an answer at the back of the book. Note: Problems near the surface of the earth involve gravitational acceleration equal to 9.81 m/s 2. However, in some of the answers, this value was rounded to 10 m/s2 – usually, for problems where the numbers would then work out nicely without the need of a calculator. *1. Monkey Units I. Below are some conversions between Monkey Units and more common units. 1 banana = 8.34 inches 1 coconut = 9.42 cm 1 pound = 6.34 peels 1 eek = 0.79 seconds (A) How tall is a monkey in cm whose height is 5.2 bananas? (B) How much does a 150-lb. person weigh in peels? (C) How many coconuts are there in one banana? (D) What is the area of an 8.5”  11” sheet of paper in coconuts2? (E) Express 9.81 m/s2 in bananas/eek2.

 *2. Monkey Units II. What are the SI units of the monkey symbol defined as

202 g ?

3. Physics Makes You Run Around in Circles. Two monkeys jog around a circular track with a length of one kilometer. One monkey jogs clockwise at a rate of 5 m/s, while his sister jogs counterclockwise at a rate of 8 m/s. They begin from the same position. (A) How much time passes before they first meet? (B) How many laps has the sister completed when her brother completes his 15th lap? (C) After one minute, how much further has the sister traveled compared to her brother? (D) What are the components of the net displacement of the monkeys when they first meet? 4. Going Bananas! A monkey jogs around the track illustrated below. The bottom arc is semicircular with 50-m radius. The monkey completes the lap in 40 seconds. The monkey runs: (i) at a constant speed of 10 m/s from A to B along the semicircular arc (ii) at a different constant speed from B to A along the straight section.

A

B

4

Creative Physics Problems – Chris McMullen, Ph.D. (A) What is the total distance traveled by the monkey in trip (i)? (B) What is the net displacement of the monkey in trip (i)? (C) How fast did the monkey travel in trip (ii)? (D) What is the average speed of the monkey for the whole lap? (E) What is the magnitude of the average velocity of the monkey for the whole lap? *5. Monkey Motion. A monkey runs 120 m east at some constant speed, then runs 160 m north at a constant speed of 8 m/s. The whole trip takes one minute. (A) What is the net displacement of the monkey? (B) What is the average speed of the monkey? (C) How fast did the monkey run to the east? (D) What is the average velocity of the monkey? 6. Monkey Climbing. A monkey’s velocity is plotted as a function of time below.

10 8 6 4 2

A ● ●B

v (m/s) 0 -2 -4 -6 -8 -10

C ●

0

1

2

3

4

5

6

7

8

t (s) (A) Find the velocity of the monkey at point B. (B) Find the acceleration of the monkey at point C. (C) At what time is the velocity of the monkey equal to zero? (D) For what time interval(s) is the monkey moving in the negative x direction? (E) For what time interval(s) is the acceleration of the monkey negative? (F) What is the maximum speed of the monkey? (G) Find the total distance traveled by the monkey. (H) Find the net displacement of the monkey. (I) What is the average speed of the monkey? (J) What is the average velocity of the monkey? 7. Wild Monkeys I. The position of a monkey is illustrated below as a function of time.

x(m) 8 6 4 2 0 -2 -4 -6 -8 0

3

6

9

12

15

t (s) 5

18

21

24

27

30

Volume 1: Mechanics (A) What is the monkey’s net displacement? (B) What is the total distance traveled? (C) What is the monkey’s initial velocity? (D) What is the monkey’s final velocity? (E) Where is the monkey at s? (F) When is the monkey moving fastest? 8. Three Little Piggies. The 1st Little Piggy, who went to the market: 40 32 24 16 8

x(m) 0 -8 -16 -24 -32 -40 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20

t s 

Note: Clearly show all work. Explain how you obtain your answers. (A) What is the net displacement of the 1st little piggy? (B) What is the average speed of the 1st little piggy? (C) For how many seconds (total) is the 1st little piggy moving backwards? (D) For what time interval(s) is the 1st little piggy moving with uniform velocity? (E) What is the final velocity of the 1st little piggy? (F) What is the maximum speed of the 1st little piggy? The 2nd Little Piggy, who chased the sun: 10 8 6 4

 (m/s)

2 0 -2 -4 -6 -8

-10 0

3

6

9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60

t (s)

nd

(G) At what time(s) is the velocity of the 2 little piggy equal to zero? (H) At what time(s) is the acceleration of the 2nd little piggy equal to zero? (I) What is the maximum speed of the 2nd little piggy? 6

Creative Physics Problems – Chris McMullen, Ph.D. (J) What is the maximum acceleration of the 2nd little piggy? (K) What is the total distance traveled by the 2nd little piggy? (L) What is the average velocity of the 2nd little piggy? The 3rd Little Piggy, who went Weeeeeee! all the way home: 2.5 2 1.5 1 0.5

a(m/s2 ) 0 -0.5 -1 -1.5 -2 -2.5 0

2

4

6

8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

t (s) rd

Note: The initial velocity of the 3 little piggy is 10 m/s. (M) For what time interval(s) does the 3rd little piggy travel with negative acceleration? (N) For what time interval(s) does the 3rd little piggy travel backwards? (O) For what time interval(s) does the 3rd little piggy travel with uniform acceleration? (P) What is the velocity of the 3rd little piggy at t  8 s ? (Q) At what time(s) is the acceleration of the 3rd little piggy equal to zero? (R) What is the acceleration of the 3rd little piggy at t  25 s ? (S) Make a sketch of acceleration as a function of time for the 2nd little piggy. *9. Wild Monkeys. For each graph of a wild monkey illustrated below, sketch a curve of velocity as a function of time. Take care drawing significant features. 20 15 10 5

x(m)

0 -5 -10 -15 -20 0

3

6

9

12

15

t (s)

7

18

21

24

27

30

Volume 1: Mechanics

20 15 10 5

a(m/s2 )

0 -5

-10 -15 -20 0

3

6

9

12

15

18

21

24

27

30

t (s) *10. Little Piggy. Sketch the time-dependence of displacement, velocity, and acceleration for a uniformly accelerated little piggy (in one dimension) with x0  0 , 0  0 , and a  0 . *11. Average Monkey. An average monkey runs from home to the banana store at a constant speed of 3 m/s. The average monkey then returns home at a constant speed of 5 m/s. (A) Conceptually explain why the average monkey’s average speed is not 4 m/s for the roundtrip. Should the average speed be greater or less than 4 m/s? Explain. (B) Compute the average monkey’s average speed for the roundtrip. 12. Monkey Skater I. A monkey on roller skates has a velocity given by

 (t )   (  t ) 2   where  and  are constants. In their appropriate SI units,  = ½ and  = 2. The monkey skates for 8 seconds with an initial velocity of -10 m/s. (A) What are the SI units of  and ? (B) Solve for . (C) What is the maximum speed of the monkey? (D) What is the acceleration of the monkey after skating for 4 seconds? (E) How far does the monkey travel before changing direction? (F) What is the average velocity of the monkey? (G) What is the average speed of the monkey? 13. Monkey Skater II. A monkey on roller skates has an acceleration given by

a(t )   t where  = 15 in SI units. The monkey skates for 4 s with an initial velocity of -40 m/s. (A) What are the SI units of ? (B) What is the final velocity of the monkey? (C) What is the average velocity of the monkey? *14. Lazy Monkey. A monkey is so lazy that instead of actually running around, he just imagines running around with the following velocity:  (t )  3(2  t ) 2 where SI units have been suppressed for 8

Creative Physics Problems – Chris McMullen, Ph.D. convenience. The monkey imagines his motion to begin at t  0 and end at t  6 seconds. (A) What acceleration does the monkey imagine having at t  4 seconds? (B) What is the net displacement for the monkey’s imagined trip? (C) At what time(s), if any, does the monkey imagine running 27 m/s? *15. Scaredy Monk. A monkey runs with a velocity of  (t ) 

t2 7 after a black cat sneaks up on  2 12

him and meows. SI units have been suppressed for convenience, but you may not take that luxury in your answers. (A) What is the net displacement of the monkey after 4 seconds? (B) What is the acceleration of the monkey at t  13 sec? 16. Monkey Wish. A monkey makes a wish (for more bananas, of course!) as she drops a banana into a wishing well. She hears the splash of the banana two and one-half seconds later. (Neglect the time it takes for the sound to travel to her ears.) (A) How deep is the well? (B) What was the final speed of the banana? 17. Monkey Jordan. Many monkeys claim that the great Monkey Jordan can jump with a hangtime of 2 seconds. How high would Monkey Jordan need to jump to have such a hangtime? Is this realistic? 18. Banana’s Revenge. A 30-kg banana throws a 20-kg monkey upward with a speed of 25 m/s from the roof of a 50-m tall building. (A) How long is the monkey in the air? (B) What is the final speed of the monkey? (C) What is the maximum height of the monkey in the air (relative to the ground)? 19. Ceiling. A ball is thrown straight upward at a speed of 40 m/s. It strikes the ceiling 3 seconds later. About how high is the ceiling? (Can this be a typical room?) *20. Stopping on a Dime. A monkey driving a bananamobile 40 m/s spots a baby monkey crossing the street 100 m ahead. The monkey uniformly decelerates, stopping just in time. For how much time does the monkey decelerate? *21. Jumping for Joy. A monkey leaps upward from a trampoline with an initial speed of 30 m/s. How high does the monkey leap? *22. Chutes and Ladders. A monkey slides down a 40-m long incline from rest, reaching the bottom in four seconds. What is his acceleration? *23. Banana Split. A monkey driving a bananamobile spots a banana split 200 m ahead on the highway. The monkey uniformly decelerates for 5 seconds, stopping just in time to avoid squashing the banana split. Find (A) the initial velocity and (B) acceleration of the bananamobile. 24. Wait a Minute! A knight rides 15 m/s on horseback to rescue a princess when he realizes that he forgot to brush his teeth in the morning. What is his stopping distance if he uniformly decelerates at a rate of 2.0 m/s2?

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Volume 1: Mechanics 25. Towering Knight. A knight jumps on a trampoline to rescue a princess from the highest tower of a castle. If his maximum height is 60 m, find his (A) speed and (B) jump time when his height is 40 m on his return down. 26. Stairway to Heaven. A princess runs upstairs to rescue a knight from a high tower. As she tires, she uniformly decelerates at a rate of 0.2 m/s2. Five seconds later, her speed is 3.6 m/s. During these five seconds, how many 20-cm tall stairs did she climb? 27. Court Jouster. The court jester is jousting with a pencil as his lance. He stands 200 m from his opponent. From rest, he uniformly accelerates at a rate of 2.5 m/s2 as he approaches his fierce opponent, who gallops at a constant rate of 10 m/s. (A) Where will they meet? (B) When will they meet? (C) What is the speed of the court jester when they meet? *28. Halfway Up. A chimpanzee throws a banana vertically upward at a speed of 30 m/s. How fast is the banana moving when it is halfway up? *29. Monkey Wrench. A monkey on the ground throws a 3-kg monkey wrench vertically upward. The wrench is moving 15 m/s when it reaches 50% of its maximum altitude. (A) How long is the wrench is in the air? (B) How high does the wrench rise? (C) How fast was the wrench thrown? *30. Hot Pursuit. A police orangutan is driving 30 m/s when he spots a speeding monkey ahead. In order to catch the speeding monkey, the police orangutan uniformly accelerates to 50 m/s over a distance of 200 m. (A) Find the acceleration of the police orangutan. (B) How much time elapses during this acceleration? (C) Once the police orangutan reaches 50 m/s, a monkey crosses the street, pushing her baby carriage. The police orangutan uniformly decelerates for 100 m, just barely stopping in time. For how much time does the police orangutan decelerate? *31. Stress Reduction. A monkey ties a firecracker to his physics textbook and launches the textbook straight upward with an initial speed of 40 m/s.1 The firecracker explodes six seconds after launch. (A) How high is the textbook above the ground when the firecracker explodes? (B) What is the velocity of the textbook just before the firecracker explodes? (C) What is the maximum height of the textbook above the ground? *32. Interception. You lean out of your second-story dorm window (10 meters above the ground) and throw a ball upward with an initial speed of 30 m/s. Unfortunately, while the ball is returning downward somebody in a room above leans out the window (35 meters above the ground) and intercepts it. (A) How much time elapsed after the ball was thrown before it was intercepted? (B) What was the maximum height of the ball above the ground? 33. Mighty Pen. To illustrate that the pen is mightier than the sword, a brave knight throws his sword high up into the air. He patiently waits for its return, holding a pen up. After the sword is 10 m above the ground on the way up, four seconds elapse before the sword is moving 15 m/s on the way down. (A) Where is the sword when it is moving 15 m/s on the way down? (B) What are (i) the average velocity 1

Professional stunt monkey. Do not attempt.

10

Creative Physics Problems – Chris McMullen, Ph.D. and (ii) the average speed of the sword for this part of the trip (from 10 m above the ground on the way up until it is moving 15 m/s on the way down)? 34. Kidnapped! You are kidnapped by little green men and transported to Planet Fyzx. Unfortunately, they give you physics lab equipment to make measurements and complete a quiz. (A) You drop your watch from the top of a 100-m high cliff. (Time flies!) Your watch strikes the ground 5 seconds later. What is the acceleration of gravity near the surface of Planet Fyzx? (Use this value in any subsequent calculations.) (B) Then you throw a pig downward with an initial speed of 20 m/s. (Pigs fly!) What is the velocity of the pig just before it strikes the ground? (C) Now you throw a cow upward, and it returns to you 6 seconds later. (Cows fly!) Find the (i) height, (ii) velocity, and (iii) acceleration of the cow at the top of its trajectory. 35. Dunk a Skunk. A skunk is thrown straight upward from a bridge. The skunk is moving at a rate of 50 m/s just before it splashes into the lake below 8 seconds after being thrown. (A) How high is the bridge above the water? (B) What is the initial speed of the skunk? (C) What is the average velocity of the skunk for the whole stinking trip? (D) What is the average speed of the skunk for the whole stinking trip? 36. Monkey Chase. A monkey steals his uncle’s banana and runs at a constant rate of 10 m/s. Two seconds later his uncle chases the monkey, accelerating at a constant rate of 5 m/s2. (A) When does the uncle catch his nephew? (B) How far do the monkeys run? (C) What is the final speed of the uncle? *37. Colliding Coconuts. A monkey at the top of a 100-m cliff drops a coconut at the same time that a monkey on the ground below throws a coconut vertically upward with a speed of 40 m/s. (A) When will the coconuts collide? (B) At what height will the coconuts collide? *38. Chimpestry. A chimpanzee throws an acid-filled flask vertically upward at a rate of 40 m/s. Three seconds later the chimpanzee throws a base-filled test tube vertically upward at a rate of 50 m/s. Find the height and time of the collision between the acid-filled flask and the base-filled test tube. 39. Stop, Thief! A monkey steals the king’s crown and makes a mad dash for it. A royal guard chases the monkey. The monkey has a 20-m head-start as he runs 2 m/s when he sees the royal guard pursuing at a constant rate of 5 m/s. The monkey then uniformly accelerates. What must be the acceleration of the monkey if the monkey is to escape with the king’s crown? *40. The Elusive ‘A.’ You are standing on the floor, trying to jump into the air to grab a letter ‘A’ that is suspended from above by a cord. When you reach up without jumping you can grab an object that is 2 meters above the ground. The letter ‘A’ is 2.5 meters above the ground when the cord begins to unwind, at which time the ‘A’ begins to travel upward at a constant rate of 0.5 m/s. Your reaction time is 0.3 seconds. So, after the ‘A’ has moved upward for 0.3 seconds, you jump high into the air – hoping to earn your ‘A.’ (A) What is the minimum speed with which you can jump and still catch the ‘A’? (Assume the ceiling to be infinitely high.) (B) How high above the ground is the ‘A’ when you catch it if you jump with this minimum speed?

11

Volume 1: Mechanics *41. Garbage Physics. Frustrated with physics, you throw your notebook into a trash can at the top of a hill. It tips over and rolls from rest with a uniform acceleration of 0.5 m/s2. Then you realize that your homework was in your notebook, which you still need to turn in. So you chase the stinking homework assignment at a constant rate of 4 m/s after the trash can has rolled a distance x0 down the incline. What is the minimum value x0 can be if you are to catch the trash can before it lands in a pile of manure at the bottom of this very high hill?

 4 , a ( 2 )  ?, υ ? t3 (C)  (t )  2t  3, a(4)  ?, x(5)  ? (D)

3 2 42. Monkeymatics I. (A) x(t )  3t  2t  4, 0  ?,

over the interval from t = 1 sec to 4 sec.

a(3)  ? (B) x(t ) 

a(t )  t , 0  2,  (4)  ?, x(4)  ? (E) x(t )  2t 5 / 2  8t 3 / 2 : At what time(s) is the velocity 3 2 zero? (F)  (t )  2  : What is the minimum velocity? t t *43. Homing Pigeon I. A homing pigeon takes off to deliver a homework assignment to a physics student. Its acceleration is

a  3t 2 where SI units have been suppressed. Derive the following equations:

at 2 x  x0  0t  12 at   0  3 3     0 4 *44. Monkey Proof. A monkey challenges you to prove that you’re smarter than the average monkey with a physics proof: Show that if the acceleration has the form, a(t )   t , where  is a constant, then the average velocity is not the arithmetic mean of the initial and final velocities (as it is in the case of uniform acceleration), but is instead given by the following “weighted” average:  

30  2 . 5

*45. Homing Pigeon II. A homing pigeon takes off to deliver a homework assignment to a physics student. Its acceleration has the form

a  bt c where b and c are constants and c  -1 or -2. Derive the following equations: 12

Creative Physics Problems – Chris McMullen, Ph.D.

x  x0  v0 t 

at 2 (c  1)(c  2)

at c 1 v (c  1)  v v 0 c2 v  v0 

v  (c  1)v0 v  v0   a( x  x0 )(c  2) c 1

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Volume 1: Mechanics

2 Projectile Motion and Vectors 1. The Triangle. For the labeled angle in each triangle below, express the sine, cosine, and tangent of the angle as ratios. Clearly show all work. (A)

(B)

(C)

(D)

1 km





3m

 2m

6 mm

10 cm

4 km

7 cm



4 mm

2. Trig Practice. Find all the unknown sides and angles in each of the following right triangles. Clearly show all calculations. Draw and completely label diagrams, too. (A)

(B)

(C)

(D)

35º 3m 6 mm

25º

4 km

7 cm 55º

2m

3. The Banana Triangle. For each triangle below, solve for the indicated quantities. As always, clearly show all work.

12

 60º

L

x

b

13





2

2

 c

3

14

Creative Physics Problems – Chris McMullen, Ph.D. 4. Monkeymatics II. Solve for the unknown quantity in each of the following problems. Clearly show all work. (A) sin   0.4 (B) 3 cos(  30º )  1 (C) sin 60º  x (D) 2 x tan(70º )  3 (E) 3 sin(2 )  4 sin(2 )  0.4 (F) 2 sin   3 cos 5. Monkey Tails. A monkey pirate etches pictures of monkeys on large rocks. The monkeys’ tails are



vectors pointing to buried bananas. (A) One of the tail vectors is A , with magnitude 300 m and



direction 35º. What are the x- and y-components of this vector? (B) Another of the tail vectors is B , with an x-component of 40 m and y-component of 200 m. What are the magnitude and direction of this





vector? (C) To find buried bananas, you walk along A , then walk along B . This is equivalent to      walking along C , where C  A  B . Find the magnitude and direction of C . 6. Monkey Madness. A monkey is going bananas with monkey vectors. (A) The monkey finds that he can draw two different monkey vectors with a y-component of -40 m and a magnitude of 50 m. Explain why this is possible and find all possible directions for this monkey vector. (B) The monkey draws a monkey vector along the positive x-axis. Then he draws two more monkey vectors. All three monkey vectors have equal magnitude, yet the resultant is zero. Explain how this is possible and find the directions of all three monkey vectors.



7. Monkey Vectors I. Monkey vector A has a magnitude of 350 bananas and a direction of 63º.









Monkey vector B has a magnitude of 200 bananas and a direction of 345º. Monkey vector C  B  A .  Find the magnitude and direction of C . 8. Royal Vectors I. The King Vector is 100 crowns at an angle of 140º. The Queen Vector is 150 crowns at an angle of 15º. The King Vector is the resultant of the Queen Vector and the Princess Vector. What are the magnitude and direction of the Princess Vector? 9. Gold Strike! Two villagers pull on a bag of gold as shown below. If F1 = 50 N and F2 = 70 N, what is the net force (magnitude and direction) exerted on the bag of gold?

25º 40º

10. Family Jewels. A monkey steals a diamond necklace. In order to circumnavigate a pond, the monkey travels 200 m at an angle of 70º counterclockwise from the y-axis, 100 m at an angle of 35º counterclockwise from the –x axis, and then 300 m at an angle of 150º clockwise from the x-axis. A parrot in pursuit flies over the pond (in a straight line) and catches the monkey just as he completes the 15

Volume 1: Mechanics journey described above. (A) How far and in which direction did the parrot fly? (B) What are the (x,y) coordinates of the monkey’s final position? 11. Gravitational Pull. In the diagram below, the Star Wreck Monkeyprize is being pulled by two tractor beams. Tractor beam A is 50 kN, and tractor beam B is 70 kN.

y

 A 25 40

Monkeyprize

x

 B (A) Draw a sketch to illustrate the net force exerted on the Monkeyprize. Clearly label the magnitude and direction of this net force. (B) Find the magnitude and direction of the net force exerted on the Monkeyprize. *12. Monkey Math. Use the equations below to answer the following questions:

 s  (6 m, 8 m)  r  (5 m,  2 m)







(A) Find the magnitude and direction of r . (B) Find the magnitude and direction of 4s  5r . (C) Find   the magnitude and direction of 2s  3r . *13. Monkeystick. A monkey wishes to place a monkeystick in a 36 cm  24 cm  12 cm trunk. What is the maximum length of monkeystick that will fit into the trunk?

 *14.

Flower Vector.

The monkey vector

has a magnitude of 4 N and a direction of



 30º. The banana vector has a magnitude of 8 N and a direction of 270º. The flower vector defined by the following equation:



 =

 –

 (A) Find the magnitude of the flower vector

 . (B) Find the direction of the flower vector 16

.

is

Creative Physics Problems – Chris McMullen, Ph.D.



 *15. Monkey Addition I. Consider the monkey vector and banana vector The magnitude of each vector is given in meters per second.

illustrated below.

y

 

50 3.9 m/s

5.2 m/s 70

 The flower vector

x

is defined by the following equation:





=3



– 4 

Find the magnitude and direction of the flower vector

.

*16. Pentagon. The 4 vectors illustrated below have equal magnitudes of 5 m with their tips lying at 4 corners of a regular pentagon. Find the magnitude and direction of the resultant.

y

x

17

Volume 1: Mechanics *17. Motorboats. Three monkeys ride motorboats to the other side of a river in the directions shown below. Each motorboat would travel the same speed if the water were calm, but there is a river current directed downward.

A

B

C

river current (A) Which monkey reaches the other side in the least time? Explain. (B) Which monkey travels fastest? Explain. (C) Which monkey travels the shortest distance? Explain. *18. Spin the Monkey. A bored physics student blindfolds a monkey, spins the monkey around several times, and then lets go. The monkey proceeds to run, but not in a straight line, since he is very dizzy. Thus, his motion is two-dimensional. The x - and y -coordinates of his position are plotted as functions of time in the graphs below.

x(m)

25

y(m)

25

20

20

15

15

10

10

5

5

0

0

-5

-5

-10

-10

-15 0

5

10

15

20

25

30

35

40

t (s)

-15 0

5

10

15

20

25

30

35

40

t (s)

(A) What is the net displacement of the monkey? (B) What is the speed of the monkey at t  25 seconds? (C) What is the direction of the monkey’s velocity at t  25 seconds? (D) When is the monkey is Quadrant III? (E) Sketch the monkey’s velocity and acceleration. (F) Find the monkey’s average speed for the trip. 18

Creative Physics Problems – Chris McMullen, Ph.D. 19. Monkeyin’ Around. The position of a wild monkey is given by

 r(t )  (t )ˆi  (t 2   )ˆj where  = 3 m/s,  = 2 m/s2, and  = –4 m. (A) How far from the origin is the monkey at t = 4 seconds? (B) What is the net displacement (magnitude and direction) of the monkey from t = 2 to t = 4 seconds? (C) What is the velocity (magnitude and direction) of the monkey at t = 3 seconds? (D) What is the acceleration (magnitude and direction) of the monkey at t = 5 seconds? 20. Royal Vectors II. His Highness has defined the vectors below.

 K ( x)  3x 2 ˆi  4ˆj  3 sin(2 x)kˆ   dK Q( x )  dx x   P( x)   K ( x)dx x 0

 R ( x)  (3x 2  5,220º )

    in terms of ˆi and ˆj . (B) Express the magnitude and direction 2    of P2  in spherical coordinates. (C) Express P  2Q in terms of Cartesian unit vectors. (D) Express     the magnitude and direction of 3Q   2P  in spherical coordinates. 2 2 where x is in radians. (A) Express R



Note: The comma in the vector R separates the magnitude from the direction (the angle is not a 4digit number). 21. Silly Physics. A monkey and his uncle love to dangle from tree limbs and discuss silly physics



concepts. The monkey defines the silliness vector S( ) to have a magnitude of

 direction of  . His uncle defines the ludicrous vector L( ) according to:

3 sec and a

  d  L( )  2S( )  3 S( ) d

      . (B) Find the magnitude and direction of L  . Note: Of 6 6

(A) Find the x- and y-components of S

course, you can’t just plug the magnitudes into the vector equation!

19

Volume 1: Mechanics





*22. Monkey Addition II. Consider the monkey vector and banana vector illustrated below. The magnitude of each vector is expressed as a function of t , where SI units have been suppressed for convenience.

y

 

60 t2  2

3t 60

x

 The flower vector

is defined by the following equation:





(t )

=

–

3

d 2 dt



 at t  4 seconds.

Find the magnitude and direction of the flower vector *23.

Banana Vectors.



A banana vector B(t ) has a magnitude of 3t and angle of 300º



counterclockwise from the x-axis. A coconut vector C(t ) has an x-component of 2t  15 and a y2



component of  4t . SI units have been suppressed for convenience. (A) Express B(t ) in terms of ˆi

 dC and ˆj . (B) Evaluate the magnitude and direction of when t = 3 seconds. (C) Evaluate the dt   magnitude and direction of the quantity 2B(2)  C(2) . *24. Toilet Bowl Physics I. When you flush the toilet, the contents travel down a curved pipe with an



2 acceleration given by a(t )  ( t   )ˆi   t ˆj , where  = 2,  = 3, and  = 4 where SI units have been



suppressed. The initial velocity is v 0  10ˆj . (A) What are the SI units of , , and ? (B) Find the

20

Creative Physics Problems – Chris McMullen, Ph.D. magnitude and direction of the velocity at t = 2 seconds. (C) What is the net displacement of the contents after 4 seconds? *25. Toilet Bowl Physics II. When you flush the toilet, the contents travel down a curved pipe with a



velocity given by v(t )  (t 2   )ˆi   t ˆj , where  = 2,  = 3, and  = 4 when velocity and time are expressed in SI units. (A) What are the SI units of , , and ? (B) Find the magnitude and direction of the acceleration at t = 2 seconds. (C) Find the average velocity for the first 4 seconds.





*26. Flatulence. The magnitude and direction of the stench vector S and reek vector R are

S  2t 2  5t

 S  160º 

dS dt  R  310º R





where S and R each have units of PU’s when t is expressed in seconds. The flatulence vector F is    related to the stench vector and reek vector by R  4S  3F . Find the magnitude and direction of the



flatulence vector F at t = 2 seconds. *27. Monkey Vectors II. The Monkey Vector, Banana Vector, Coconut Vector, and Uncle Vector are defined as follows:

Monkey Vector : M (t )  2t  4ˆi  (6t )ˆj Banana Vector : B(t )  (4t 2 )ˆi  (3t  5)ˆj dB dt Uncle Vector : U(t ) has magnitude 3t and direction 140º

Coconut Vector : C(t ) 

counterclockwise from the x - axis where t is time. Units are suppressed for convenience. (A) What are the magnitude and direction of the Monkey Vector when t = 2 seconds? (B) Evaluate the x- and y-components of the Uncle Vector at t = 4 seconds. (C) What are the magnitude and direction of the quantity when t = 3 seconds? *28. Monkey Help. A monkey needs to express the vector (2,2 3,4 3 ) in spherical coordinates. Lend a monkey a hand. *29. Cannonball. A monkey launches a cannonball with an initial speed of 100 m/s at a 30º angle. What is the speed of the cannonball at the top of its trajectory?

21

Volume 1: Mechanics *30. Monkey Sail. A monkey sails in a boat where the ocean current contributes a velocity 50 m/s at 160º and the wind contributes a velocity 30 m/s at 220º to the boat. What are the speed and direction of the boat relative to the ground? 31. Sick of Physics. Sick of studying, a physics student vomits from a window. The vomit has an initial speed of 20 m/s at an angle of 65º above the horizontal. The vomit lands 50 m away from the building. As usual, assume that air resistance effects are negligible. (A) How high is the window above the ground? (B) What is the maximum height of the vomit? (C) What is the speed of the vomit at the top of its trajectory? (D) What is the acceleration of the vomit at the top of its trajectory? (E) What is the magnitude of the net displacement of the vomit? (F) What is the direction of the net displacement of the vomit? 32. Knights in Smoldering Armor. A fire-breathing dragon spits knights in smoldering armor out from the roof of a 200-m tall castle. (A) What will be the final velocity (implying both magnitude and direction in the answer – in contrast to final speed) of a knight spat out with a velocity of 30 m/s at an angle of 25º above the horizontal? (B) Repeat for a knight spat out with a velocity of 30 m/s at an angle 25º below the horizontal? (C) Repeat for a knight spat out with a horizontal velocity of 30 m/s. (D) Which knight strikes the ground with the greatest speed? 33. Scaredy-Cat. A cat sharpens its nails on the furniture, then hides 15 m above the ground in a tree. When the cat’s companion returns home from a long day at work and discovers the damage, the companion goes outside and throws a shoe at a speed of 20 m/s directly at the cat. The shoe is thrown from a height of 2 m above the ground and a distance of 10 m from the tree. At the instant the shoe is thrown, the cat falls from the tree (hoping to avoid the oncoming shoe). (A) At what angle is the shoe thrown? (B) What is the vertical position of the shoe when the horizontal position of the shoe matches the horizontal position of the cat? (C) What is the vertical position of the cat at this time? (D) Does the shoe strike the cat? 34. Serenade. A young lad has prepared romantic physics poems in order to serenade his lover, but first he must get her attention by throwing an orange against her window. If he throws the orange at an angle of 50º, her window is a height of 10 m above the ground, he stands a distance of 20 m away from her wall, and he throws the orange from height of 2 m, how fast must he throw the orange in order for it to strike her window? 35. Bomb’s Revenge. A plane is flying 250 km/hr at a constant altitude of 20 km. How far away from its target should the bomb be when it releases a pilot? (Yes, you read that correctly.) 36. Football’s Revenge. A football kicks a quarterback with an initial speed of 50 m/s at an angle of 60º. (A) What is the maximum height of the quarterback? (B) What is the speed of the quarterback at the top of his trajectory? (C) How far away from the football does the quarterback land on the ground? (D) For how much time is the quarterback in the air? *37. Baseball’s Revenge. A baseball on the top of a 20-m high building throws an outfielder at a speed of 30 m/s horizontally toward a catcher who is standing 50 m away from the base of the building. (A) 22

Creative Physics Problems – Chris McMullen, Ph.D. How far must the catcher below run in order to catch the outfielder? (B) Must he run toward or away from the building? *38. Stress Relief. In order to relieve some physics stress, you make a piñata that looks like Sir Isaac Newton, suspend it from a tree 10 m above the ground, and throw apples at it. You throw one apple from the roof of a building at a rate of 20 m/s and at an angle 30º below the horizontal and it strikes Newton right in the belly. Good aim! The horizontal distance between the building and Newton is

40 3 m. (A) How tall is the building? (B) For how much time is the apple in the air? (C) Find the final speed of the apple. *39. Haunted Physics. Upon hearing you creatively express your physics frustrations, the ghost of Isaac Newton grabs you, takes you to the roof of a building, and throws you at a velocity of 40 m/s at an angle of 30º below the horizontal. You strike the ground below a distance 100 3 m away from the base of the building. How tall is the building? *40. Monkey Physics. A monkey stands at the edge of a 45-m tall cliff, where he throws a coconut at a rate of 40 m/s at an angle 35º below the horizontal. The coconut strikes his uncle, standing on the ground below, right on the top of his head. Nice throw! (A) How far was the monkey’s uncle standing from the base of the cliff? (B) How fast was the coconut moving when it struck the monkey’s uncle? (C) At what angle was the coconut moving, relative to the ground, when it struck the monkey’s uncle? (If you get this right, will you be a monkey’s uncle?) *41. Pirate Ships. One pirate ship heads 10 m/s east, while another approaches at 20 m/s west. (A) If the cannons launch the cannonballs at 80 m/s at 30º angles above the horizontal, what is the minimum distance the pirate ships can be apart when a cannonball can be launched and reach the enemy ship? (B) What will be the maximum height of the cannonball in the air? 42. Golf Ball’s Revenge. A sore golf ball tees up a frustrated golfer at the bottom of a hill with a steady 20º incline. The golf ball then swings a golf club, striking the golfer, who flies off with an initial speed of 30 m/s at an angle of 55º relative to the horizontal (so 35º relative to the incline). How far up the incline has the golfer traveled when he lands? *43. Monkey Ball I. A monkey on the ground throws a ball straight upward. After rising 10 m, it is moving with one-fourth of its initial speed. At the top of its trajectory, a monkey hanging from a tree hits the ball with a tennis racket. The ball then moves 20 m/s at an angle of 40º above the horizontal. (A) How high does the ball rise before being struck by the tennis racket? (B) After being struck by the tennis racket, what is the speed of the ball at the top of its trajectory? (C) What is the horizontal range of the ball? (D) For how much time is the ball in the air? *44. Banana Ball. A monkey kicks a banana ball with an initial velocity v at an angle above the horizontal. Derive an equation for its maximum height relative to its starting point in terms of , , and/or . No other symbols may appear in your final expression.

23

Volume 1: Mechanics *45. Monkey Ball II. A banana kicks a monkey ball with an initial velocity v at an angle above the horizontal on level ground. (A) Derive an equation for its horizontal range in terms of , , and/or . No other symbols may appear in your final expression. (B) Determine the angle that maximizes the range for a given . *46. Rock Throwing. Three monkeys stand atop a cliff and simultaneously throw rocks with the same initial speed. Monkey A throws his rock horizontally. Monkey B throws his rock 30º above the horizontal. Monkey C throws his rock 30º below the horizontal. Which strikes the ground with the greatest final speed? *47. Cannonball’s Revenge. A smoldering cannonball places a pirate inside a cannon on the rear of a ship that is sailing southbound at a rate of 25 m/s. The cannonball lights the cannon, launching the pirate southbound with an initial speed of 40 m/s. What angle of the cannon produces maximum range of the pirate relative to the ocean? *48. Monkey Position. The acceleration of a wild 20-kg monkey has components

a x  4 m/s2

, a y  8 m/s2

The initial velocity of the monkey is

vx 0  10 m/s , v y 0  0 Assuming the acceleration to be uniform, find the magnitude and direction of: (A) the acceleration of the monkey; (B) The net force exerted on the monkey; and (C) The net displacement of the monkey at t  3s. *49. Stray Cat. A stray cat has interrupted the jousting tournament. Several knights chase the cat through the arena. The cat’s acceleration is given by a(t )  (3t )ˆi  (t  2)ˆj , where SI units have been suppressed. When t = 2 seconds, the cat’s velocity is 5 m/s at an angle of 40º. (A) What is the cat’s initial velocity (magnitude and direction)? (B) What is the cat’s minimum acceleration? (C) What is the net displacement of the cat after running for four seconds? 2

*50. Monkey Relatives. A monkey’s son just marries his brother’s friend’s daughter. The monkey’s son and daughter head east on the roof of a train at a constant rate of 20 m/s. As the train passes the monkey’s brother, the monkey’s son throws a coconut straight up with an initial speed of 10 m/s relative to the monkey’s son. Find each of the following relative to (i) the monkey’s son and (ii) the monkey’s brother: (A) the initial velocity (magnitude and direction) of the coconut; (B) the time the coconut is in the air; (C) the horizontal distance traveled by the coconut; (D) the velocity of the coconut at the top of its trajectory; (E) the acceleration of the coconut at the top of its trajectory. *51. Gravity’s Revenge. Gravity suddenly changes from being approximately constant near earth’s 2 surface to having the form, a  ( t )ˆi  (t   )ˆj , where  = 2,  = 3, and  = 1 in their appropriate SI

units. An object has an initial velocity of (30 m/s) ˆi . (A) What are the units of , , and ? (B) Find the magnitude and direction of the acceleration when t = 2 s. (C) Find the magnitude and direction of the velocity when t = 2 s. (D) Find the average velocity of the object from t = 0 to t = 5 s. 24

Creative Physics Problems – Chris McMullen, Ph.D.

Review 1: Motion A General Physics Review by Chimp Dickens Dizzy and Wizzy are two monkeys stuck to one and the same tail. Two hundred meters in length, their tail leaves quite a trail. (Applies to all problems!) 1. Tug of War. Dizzy and Wizzy have a unique way of playing tug-of-war with their little brother, Bizzy. They tie their tails to their uncle, then they all pull in the directions illustrated below. Dizzy exerts a force of 200 N, Wizzy exerts a force of 150 N, and Bizzy exerts a force of 120 N. y

Dizzy 50º

x Bizzy

25º Wizzy

(A) Find the magnitude of the net force exerted on their uncle. (B) Find the direction of the net force exerted on their uncle. 2. Drinking Well. Wizzy launches Dizzy straight upward with a cannon at a speed of 25 m/s and Dizzy splashes in the well below 8 seconds later. (A) How deep is the well? (B) What is Dizzy’s final speed? (C) What is Dizzy’s average velocity for these 8 seconds? (D) What is Dizzy’s average speed for these 8 seconds?

25

Volume 1: Mechanics 3. Helicopter Rescue I. Dizzy lowers Wizzy from a helicopter to save their little sister, Lizzy. A plot of Wizzy’s velocity is shown below.

v (m/s)

10 8 6 4 2 0 -2 -4 -6 -8 -10 0

2

4

6

8

10

12

14

16

t (s) (A) Find Wizzy’s acceleration when t = 14 seconds. (B) At what time is Wizzy’s acceleration equal to zero? (C) Find Wizzy’s net displacement. *4. Helicopter Rescue II. Dizzy lowers Wizzy from a helicopter to save their little sister, Lizzy. Wizzy’s

15 7 and   in appropriate SI 8 8 units. (A) What are the SI units of  and  ? (B) Find Monkbob’s acceleration at t  4 s. (C) Find velocity is given by:  (t )  t 5 / 2  t 3 / 2

, 0  t  4 s , where  

Monkbob’s average speed. 5. Banana Split. While standing back-to-back, Dizzy spots a banana in the east, while Wizzy spots a banana in the west. Dizzy runs at a constant velocity of 5 m/s east, while Wizzy uniformly accelerates to the west from rest with an acceleration of 2 m/s2. (A) How much time elapses before their 200 m tail begins to stretch? (B) How far does Wizzy travel during this time? 6. Elevator. In order to climb a 30-m tall cliff, Wizzy launches Dizzy with a cannon at an angle of 70º and speed of 50 m/s. The cannon is 40 m away from the cliff. Then Wizzy simply climbs up the tail.

30 m 70º

40 m

26

Creative Physics Problems – Chris McMullen, Ph.D. (A) For how much time is Dizzy in the air? (B) How far does Dizzy travel horizontally? (C) What is Dizzy’s speed at the top of the trajectory? (D) Find Dizzy’s maximum height. (E) Find the magnitude and direction of Dizzy’s final velocity.

*7. Evil McMonkey. Evil McMonkey has just robbed the Gotchimp City Bank. (A) With his springloaded shoes, Evil McMonkey leaps vertically upward with an initial speed of 40 m/s. What is his maximum height in the air? (B) Evil McMonkey lands in the driver’s seat of his convertible getaway car, which is moving 30 m/s. Evil McMonkey uniformly accelerates to 50 m/s as he drives 50 m. What is the car’s acceleration? (C) For how much time does the car accelerate? *8. The Monkmobile I. As the Evil McMonkey speeds away at a constant speed of 50 m/s, Batmonk and Gibbon pursue him in the Monkmobile. The Monkmobile initially trails Evil McMonkey by 200 m. With an initial speed of 10 m/s, the Monkmobile accelerates with a uniform acceleration of 5 m/s2. (A) How much time elapses before the Monkmobile catches Evil McMonkey? (B) How fast is the Monkmobile moving when it catches Evil McMonkey? *9. The Monkmobile II. Batmonk and Gibbon are pursuing Evil McMonkey in their Monkmobile. Batmonk swerves, taking the Monkmobile up a ramp to cross a canyon. The Monkmobile leaves the ramp with an initial speed  0 at an angle of 30º above the horizontal. The canyon measures 100 3 m across and the far end of the canyon is 25 m below the top of the ramp. ramp

 υ0

30º 25 m safety

100 3 m deep canyon What initial speed  0 is needed for the Monkmobile to cross the canyon safely? *10. Smokescreen I. Just as the Monkmobile reaches Evil McMonkey, Evil McMonkey releases a toxic gas, creating a smokescreen. Thus, the chase continues. The position of Evil McMonkey is illustrated below as a function of time. 27

Volume 1: Mechanics

x(m)

25 20 15 10 5 0 -5 -10 -15 0

5

10

15

20

25

30

35

40

t (s)

(A) What is Evil McMonkey’s net displacement? (B) What is Evil McMonkey’s velocity at t  30 s ? The Monkmobile’s acceleration is illustrated below as a function of time. The Monkmobile’s initial velocity is 30 m/s.

a(m/s2 )

25 20 15 10 5 0 -5 -10 -15 0

5

10

15

20

25

30

35

40

t (s)

(C) What is the Monkmobile’s velocity at t  20 s ? (D) What is the Monkmobile’s acceleration at t  20 s ? *11. Smokescreen II. Just as the Monkmobile reaches Evil McMonkey, Evil McMonkey releases a toxic gas, creating a smokescreen. The chase continues. The position of Evil McMonkey, x1 (t ) , and acceleration of the Monkmobile, a2 (t ) , are:

x1 (t )  8t  t 3 a2 (t )  12t where SI units are suppressed. The Monkmobile’s initial velocity is 8 m/s and Evil McMonkey has a 64-m headstart. (A) What is Evil McMonkey’s velocity at t  2 seconds? (B) What is the Monkmobile’s net displacement at t  2 seconds? (C) At what time does the Monkmobile catch Evil McMonkey? *12. Riddlemur. The Riddlemur rescues Evil McMonkey by lowering a ladder from a helicopter. The Riddlemur leaves behind the following riddles for Batmonk and Gibbon to ponder. 28

Creative Physics Problems – Chris McMullen, Ph.D. Riddle A: Velocity is to position as acceleration is to ____________________? Also, explain, conceptually, the difference between velocity and acceleration. Riddle B: Velocity has ____________________, whereas speed does not. Also, explain, conceptually, what speed and velocity are. Riddle C: How is it possible to have a 100 m total distance traveled, yet no net displacement? Also, explain the distinction between total distance traveled and net displacement. Riddle D: Drive 30 m/s to work, but 20 m/s back home. Why is the average speed not 25 m/s? Also, should it be greater or less than 25 m/s? Explain why. *13. Catwomonk I. Catwomonk apprehends Evil McMonkey and the Riddlemur in their (no longer) secret hideout. She lassoes each separately, then pulls Evil McMonkey with a force of 200 N at an angle of 120º and the Riddlemur with a force of 100 N at an angle of 240º. (A) Draw a diagram, clearly illustrating the resultant of these two forces. (B) Find the magnitude and direction of the net force exerted by Catwomonk. *14. Catwomonk II. Catwomonk is tracking Evil McMonkey on the radar monitor of her monkicycle.

 15 3 t 2 ˆ j , while the Catwomonk is following vector M , 2  and direction of 120º. Catwomonk needs to travel along vector I in 

Evil McMonkey is following vector E  10 t ˆi  which has a magnitude of

5t4 2

order to intercept Evil McMonkey, where

 t   dM I (t )    Edt dt 0 

Find the magnitude and direction of the interceptor vector I (2) .

29

Volume 1: Mechanics

*15. Bacon Racin’. A little piggy’s velocity is illustrated below as a function of time.

 (m/s) 8 6 4 2 0 -2 -4 -6 -8 0

3

6

9

12

15

18

21

24

27

30

t (s) (A) For what time interval(s) is the little piggy moving in the negative x-direction? (B) For what time interval(s) is the little piggy’s acceleration positive? (C) What is the little piggy’s maximum speed? (D) What is the little piggy’s acceleration at t  12 s ? (E) What is the little piggy’s net displacement for the whole trip? (F) What is the little piggy’s average speed for the whole trip? *16. Bacon Huntin’. A big bad wolf walks from a straw house to a wooden house, traveling 200 m at an angle of 120º. The big bad wolf then walks from the wooden house to a brick house, traveling 100 m at an angle of 240º. (A) Find the magnitude of the big bad wolf’s net displacement. (B) Find the direction of the big bad wolf’s net displacement. *17. Bacon Chasin’. A little piggy running from a big bad wolf runs in a circle with 100-m diameter. The little piggy uniformly accelerates from rest, completing one lap in one minute. (A) How fast is the little piggy running when he completes his first lap? (B) Find the total distance traveled, the net displacement, the average speed, and the average velocity of the piggy after completing one lap. Be clear which is which. *18. Jumpin’ Bacon. A little piggy is standing on the roof of a brick house 10 m above a big bad wolf on the ground below. The little piggy leaps straight upward, traveling through the air for 2 seconds before landing on the big bad wolf. (A) How fast is the little piggy traveling just before landing on the big bad wolf? (B) What is the little piggy’s maximum height above the ground? *19. Fallin’ Bacon. A 50-kg little piggy pig jumps off a 100-m tall cliff with a parachute, traveling downward with a constant speed of 5 m/s. Ten seconds later, a 200-kg big bad wolf jumps off the same 100-m tall cliff (from rest) without a parachute. How high is the little piggy above the ground at the time of the collision?

30

Creative Physics Problems – Chris McMullen, Ph.D.

Who lives in a coconut up in a tree? Monkbob Squaretail! Who’s furry, brown, and physics-crazy? Monkbob Squaretail! *20. Driving School. Monkbob uniformly accelerates from rest at a rate of 3 m/s 2 for 150 m, travels for 10 seconds with constant speed, and then uniformly decelerates until coming to rest 20 seconds later. (A) All together, how far does Monkbob travel? (B) What is Monkbob’s average velocity for the whole trip? *21. Putrid Pear. The Putrid Pear restaurant features a throw-to for customers on the run. Monkbob, sitting in a tree, throws a banana vertically upward. A chimpanzee catches the banana on the ground below five seconds later. The banana is moving 30 m/s when the chimpanzee catches it. (A) How fast did Monkbob throw the banana? (B) How high above the ground is Monkbob? 22. Bumblebee Hunting. Monkbob, who is on roller skates, has a velocity given by  (t )   sin( t ) where  = 4 and    / 2 in SI units. Monkbob skates for 6 s. (A) What are the SI units of and ? (B) What is the acceleration of Monkbob at t  2 sec? (C) What is the net displacement of Monkbob? *23. Friendship. Monkbob and his best friend, Patchimp, love to dangle from tree limbs and discuss  silly physics concepts. Monkbob defines the silliness vector S to have a magnitude of 40 smiles and a



direction of 60 and the ludicrous vector L to have a magnitude of 40 smiles and a direction of  30









. Patchimp defines the zany vector Z according to: Z  2S  3L . Find the magnitude and direction



of Z . *24. Chase. Monkbob steals Slothward’s flute and runs at a constant speed of 10 m/s. Three seconds later, Slothward uniformly accelerates from rest at a rate of 4 m/s2. How far does Monkbob run before he is caught? 25. Slothward. Monkbob steals Slothward’s flute and uniformly accelerates from rest at a rate of 4 m/s2. Moments later, Slothward chases Monkbob at a constant speed of 10 m/s. How many seconds can Slothward afford to wait and still catch Monkbob?

A Connecticut Monkey in King Arthur’s Court by Monk Twain A monkey in a twenty-first century Connecticut zoo awakens one morning in legendary medieval Camelot under the rule of King Arthur and the Knights of the Round Table. The monkey’s understanding of physics makes him a valuable asset to the kingdom.

31

Volume 1: Mechanics *26. Damsel in Distress. The lovely Guinevere is trapped in the highest tower of the darkest castle. The monkey estimates that Sir Lancelot could run around a lake to get there by traveling 600 m at an angle of 115º counterclockwise from east and then 400 m at an angle of 255º counterclockwise from east. However, the monkey points out that Sir Lancelot could rescue Guinevere sooner by swimming across the lake. (A) How far must Sir Lancelot swim to save Guinevere? (B) In what direction should Sir Lancelot swim? *27. Jousting Match. The monkey is amused when the court jester uses a banana in a jousting tournament. The court jester heads toward the knight at a constant speed of 20 m/s, while the knight uniformly accelerates from rest at a rate of 10 m/s2 toward the court jester. They meet 4 seconds later. (A) How far were the knight and court jester apart initially? (B) What is the final speed of the knight when they meet? *28. Heavy Artillery. The monkey offers a class on projectile motion to improve the aim of trebuchet launches. The trebuchet – an improved variation of the catapult – is useful for hurling large stones at castles. The monkey’s example involves a stone that is hurled at a speed of 25 m/s an angle of 50º above the horizontal. (A) What is the speed of the stone at the top of its trajectory? (B) What is the acceleration of the stone at the top of its trajectory? (C) What is the maximum height of the stone above the ground?

Five Little Monkeys *Five little monkeys, who were good buddies, sat at a table engaged in their studies. (A) “What does a vector have that a scalar does not?” (B) “What does acceleration mean, conceptually?” (C) “What is the distinction between average speed and average velocity?” (D) “How do you interpret the number 9.8 m/s2, conceptually?” (E) “What do the slope and area of a velocity graph represent?” *Four little monkeys raced their cars, all the while playing guitars! (A) Indicate the time(s) when each monkey is at rest. Explain your reasoning. (B) Determine the maximum speed of each monkey. Explain and show work.

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Creative Physics Problems – Chris McMullen, Ph.D.

16 12 8 4 0 -4 -8 -12 -16

16 12 8 4 0 -4 -8 -12 -16

16 12 8 4 0 -4 -8 -12 -16

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

16 12 8 4 0 -4 -8 -12 -16

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

*Three little monkeys hunted for treasure; though they found none, they always took pleasure. The first little monkey digs 6 m at an angle of 330º. The second little monkey digs m at an angle of 120º. The third little monkey digs along the resultant of these two vectors. (A) Draw and label a sketch to illustrate the resultant. (B) Find the magnitude of the third little monkey’s vector. (C) Find the direction of the third little monkey’s vector. *Two little monkeys rode in trains – they better slow down, or there will be pains. The two trains are heading toward one another with speeds of 25 m/s. When the trains are 1 km apart, both trains begin to decelerate uniformly with the same deceleration. (A) What deceleration is needed in order to avert disaster? (B) For how much time do the trains decelerate? *One little monkey threw a banana-cream pie as hard as he could straight up into the sky. The banana-cream pie rose to a maximum height of 80 m. How fast was the banana-cream pie thrown?

33

Volume 1: Mechanics

3 Newton’s Laws of Motion *1. Physics Dummies. For each of the following questions, first identify which of Newton’s laws is (are) involved in the problem, then use the law(s) (or definitions) to answer the question. (A) A crash test dummy wearing a seatbelt drives into a brick wall. What is the first force the dummy feels? (B) A crash test dummy is standing up with his shoes glued to the roof of a car as the car rounds a turn to the left. Describe the dummy’s lean. (C) A crash test dummy drives at a constant speed, yet he is accelerating. How can this be? (D) You place a large block of metal on a dummy’s head. Then you place a block of wood on the block of metal. Then you hammer a nail into the block of wood. How is it that the block of metal serves as a cushion? (E) A crash test dummy standing on the street is hit by a bulldozer traveling 20 m/s. Compare the forces exerted on the dummy and the bulldozer. (F) A robotic spaceship dummy is fixing the exterior of a spaceship with a wrench when he loses his tie rope. How can he return to the spaceship? (G) A dummy jumps off the top of a tall building. Identify the action and reaction force pair involving the dummy. (H) Two robotic dummies with equal mass each hold one end of a rope passing over a pulley suspended from the ceiling. One dummy is 1 m above the other dummy. Both dummies are above the floor. What happens when the lower dummy shortens the rope between them by 1 m? 2. Lunar Dummy. A dummy deemed to be 180-lbs. on earth is transported to the moon to partake in some free fall experiments. What are the dummy’s mass and weight on the earth and moon (4 answers in all)? 3. Astronautic Dummy. A dummy deemed to be 123-kg on earth is transported to another planet where he takes 4 seconds to strike the ground when dropped from a height of 100 m. What are the dummy’s mass and weight on this planet? *4. Textbook Physics. A 25-g bullet traveling 300 m/s shoots into a physics textbook that is 3.5 cm wide, and exits the textbook traveling 40 m/s. Poor defenseless textbook! (A) What was the average deceleration of the bullet while it was penetrating through the textbook? (B) What was the average resistive force exerted on the bullet while it was penetrating through the textbook? *5. Terminal Speed. A 200-kg skydiver parachutes from an airplane. What is the acceleration of the skydiver when the force of air resistance is (A) zero, (B) 900 N, (C) 2500 N, (D) 1960 N. (E) Give the direction of the speed and acceleration for each case. (F) What is the force of air resistance when the skydiver has reached his terminal velocity? (G) When the skydiver’s speed is least, is the acceleration least or greatest? 6. 00. A physics spy, 00, rescues a physics student from distractions as illustrated below. The 1200kg helicopter is pushed upward with a force of 20 kN. The spy weighs 800 N, and the mass of the student is 60 kg.

34

Creative Physics Problems – Chris McMullen, Ph.D.

helicopter

00 student

(A) Find the acceleration of the helicopter. (B) Find the tension in each rope. *7. Kidnapped. You are kidnapped by little green monkeys who transport you to Planet Mnqy in a UFO. While you are being kidnapped, you hang onto a rope connected to their UFO, as illustrated below. Assume that your weight is 200-N.

Find the tension in the cord if: (A) The UFO accelerates upward at 15 m/s2; (B) the UFO accelerates downward at a rate of 4 m/s2; (C) the UFO travels upward with a constant speed of 5 m/s. 8. Special Delivery. Three boxes of precious physics textbooks (m1 = 500 kg, m2 = 1000 kg, m3 = 1500



kg) are connected by a cord. The leading block is pulled by a force P = 2000 N at a 40º angle as shown below. Neglecting friction, find (A) the acceleration of each box and (B) the tension in each cord.

 P 40º

35

Volume 1: Mechanics 9. Treasure Chest. A 20-kg monkey is connected to a 40-kg treasure chest as shown below. The coefficient of kinetic friction between the chest and the ground is 0.25. The block is presently moving to the right. (A) Find the acceleration of the chest of bananas. (B) Find the acceleration of the monkey. (C) Find the tension in the cable.

*10. Lucky Charms. Lemur and Chimp found a box of lucky charms on their expedition in North Aperica. Unfortunately, both tripped and fell off a cliff when they were rigging a pulley system to get their treasure home. However, they are still hanging by the skin of their teeth, as illustrated below. The coefficient of friction between the box and ground is 7 60 . The block is presently moving to the left. What is the acceleration of the system?

60 kg

30 kg 10 kg *11. Banana’s Revenge. Two boxes of monkeys connected by a cord are pulled by a banana with a force of 800 3 N as illustrated below. The coefficient of friction between the boxes of monkeys and the horizontal is 1

3.

800 3 N 30º

40 3 kg

60 3 kg

(A) Find the acceleration of the system. (B) What is the tension in the connecting cord? 36

Creative Physics Problems – Chris McMullen, Ph.D. *12. Sweet Physics Thoughts. Two smiling students love physics so much that on Saturday, in the middle of Winter Break, they go outside to play with a 60 3 -kg box of physics textbooks. The smiling students pull on the box with the forces illustrated below. The coefficient of friction between the box of physics textbooks and the horizontal is

1 2 3

.

800 N

200 3 N 60º

30º

60 3 kg

(A) Merrily find the acceleration of the box of physics textbooks. (B) Clearly explain why a beginning student might look at this diagram and think, naïvely, that Newton’s third law is being violated. Once you have identified the problem, resolve it – clearly explaining what Newton’s third law does say about what might naïvely be confused. 13. Minimal Effort. A student finally finished completing her homework assignment. It only took 10,000,000 pages (double-sided). She fills a large crate with her homework, and attaches a rope to the center of one side. Now she is prepared to drag the crate to class by pulling along the rope with force

 P directed at an angle  above the horizontal. See the diagram below.  P 

The coefficient of kinetic friction between the sidewalk and the crate is 0.4. (A) If she wants to pull the crate with constant velocity with minimum effort (i.e. minimum P), at what angle  should she pull on the rope? (B) Explain, conceptually, why the answer isn’t 0º? *14. Hanging Around. The monkey of mass m hanging from the rope below is in static equilibrium.

2

1

m 37

Volume 1: Mechanics Derive equations for the tension in each cord in terms of m , 1 ,  2 , and/or g . *15. Soothsloth. A sloth realizes it’s going to be a bad day when he looks at his calendar. Not only is it Monkday the 13th, but the calendar suddenly falls. The sloth reacts quickly, supporting the 130-g calendar with a force as illustrated below. The coefficient of friction between the calendar and the wall is 3 3 . What force must the sloth exert in order to prevent the calendar from falling?

30º

16. Play Time. A monkey sits in a large saucer and slides down an incline of angle  with respect to the horizontal. The coefficient of friction between the monkey and the ground is . Solve for the acceleration of the monkey in terms of , , and/or g. 17. Neanderthal Escalator. A Neanderthal built an escalator with a 200-kg passenger box, a very sturdy vine, a pulley, and a large rock as illustrated below. The coefficient of static friction between the box and the ground is 0.6. The passenger box is presently moving up the incline. pulley

stone passenger box 25º

What mass of the stone is needed to accelerate a 100-kg passenger (sitting in the passenger box) up the incline?

38

Creative Physics Problems – Chris McMullen, Ph.D. 18. Trig Torment. A 750-N Neanderthal slides down (even though there is a push up the incline, as illustrated below – how?) a dinosaur’s back at a constant rate of 30 m/s as illustrated below. The coefficient of kinetic friction between the dinosaur and the Neanderthal is 0.3. What is P? dinosaur’s back

Neanderthal 25º 35º 19. To Steal the Treasure. Lemur and Chimp find a 200-kg treasure chest filled with gold-plated bananas. The chest rests on an incline as illustrated below. The coefficient of friction between the chest and incline is 0.6. The incline makes a 35º angle with the horizontal. A cord parallel to the incline prevents the box from moving.

200 kg

35º (A) What is the tension in the cord? (B) After Lemur cuts the cord, what is the acceleration of the treasure chest? (C) What is the speed of the treasure chest after sliding 30 m down the incline?

39

Volume 1: Mechanics *20. Down Escalator. A 40-kg monkey is connected to a 20-kg box of bananas as illustrated below. The coefficient of friction between the box of bananas and the incline is 0.5.

pulley

30º Find the two possible accelerations of the system. Explain why two are possible. *21. Pirate Physics. A 500-kg treasure chest rests on the deck of a pirate ship. The coefficients of friction between the treasure chest and the deck are 0.36 and 0.39. One is the coefficient of static friction, while the other is the coefficient of kinetic friction. The ship is sailing through rough weather. The crew is concerned that the treasure chest may slide and disturb the valuable contents. The crew is quarreling over the maximum angle that the deck can make with the horizontal before the treasure chest will begin to slide. Do the crew a favor and figure this out for them. *22. The Adventures of Lemur and Chimp. As illustrated on the following page, brave explorers Lemur and Chimp find themselves in a sticky situation on their quest for the Golden Banana. The coefficients of friction between Lemur and the incline are 0.6 and 0.7. In the position shown, 90-kg Lemur and 40-N Chimp are moving 5 m/s (downward) and Lemur is 80 m from the cliff.

40

Creative Physics Problems – Chris McMullen, Ph.D. Chimp, I told you that we shouldn’t have skipped our physics class to search for the Golden Banana!

80 m

pulley

20º

50 m

Relax, Lemur. Friction is on our side.

150 m

crocodile-infested river (A) Which coefficient is which, and which is needed to solve this problem? (B) Find the acceleration of Lemur and Chimp. (C) Find the tension in the rope. (D) What is the fate of Lemur and Chimp? *23. Sacrificial Physics. A monkey pulls two boxes of bananas up an incline, as illustrated below. When he reaches the top, he will dump them into a volcano, hoping his great sacrifice will reap great rewards on his final exam. The coefficient of friction between the boxes of textbooks and the incline is 1 / 3 .

41

Volume 1: Mechanics

1500 N

60 kg 40 kg 30º (A) Find the acceleration of the boxes of physics textbooks. (B) Find the tension in the connecting cord. (C) If the connecting cord breaks, what will be the acceleration of each box? *24. Going Bananas! A 20-kg monkey is connected to a 40-kg monkey as illustrated below. The coefficient of friction is to the right.

3 / 2 between either monkey and the surface. The system is presently moving

20 kg

40 kg

60º

60º

(A) Find the acceleration of the monkeys. Also, indicate the direction. (B) Find the tension in the connecting cord. (C) If the cord snaps, which monkey will have greater acceleration? Explain. 25. To Bring Home Glory. Lemur and Chimp find two more treasure chests: one 50-kg chest filled with copper-plated watermelons and a 100-kg chest filled with silver-plated coconuts. The coefficient of static friction between the chests is 0.4, but the ground is frictionless. As illustrated below, Lemur and Chimp exert an 800-N horizontal force on the chest of watermelons.

800 N

50 kg 100 kg

(A) What is the acceleration of the top chest? (B) What is the tension in the cord? *26. To Measure Acceleration: Lemur and Chimp drive home on a level road. Lemur ties a coconut to a string and suspends it from the ceiling of the car. When the velocity of the car is constant, the pendulum is vertical. When the car accelerates, the pendulum makes an angle  with the vertical. 42

Creative Physics Problems – Chris McMullen, Ph.D. Thus, this pendulum functions as an accelerometer. Derive an equation for the acceleration of the car in terms of  and/or g . *27. To Defy Gravity. A little green monkey pushes the treasure chest of coconuts of mass m1 with



force P , yet you (of mass m2 ) do not fall. The ground is frictionless, but the coefficient of friction between you and the chest is  .

m2

 P

m1

(A) Derive an equation for the acceleration in terms of m1 , m2 , for P in terms of m1 , m2 ,

 , and/or g . (B) Derive an equation

 , and/or g .

28. Defying Gravity. A monkey of mass m sits on the frictionless wedge of mass M in the diagram below, yet the monkey is stationary relative to the wedge. How can this be? Because the wedge is



being pushed with a horizontal force P . Derive an equation for P in terms of m, M, g, and/or .

*29. Banana Cavern. The entrance to Banana Cavern is blocked by a 200-kg rock. Lemur and Chimp summon their strength to push it out of the way. The net force exerted on the rock is:

 F(t )  t 2 ˆi  (t   )ˆj , where   3 ,   5 , and   15 in their appropriate SI units. The initial velocity of the rock is zero. (A) What are the SI units of  ,  , and  ? (B) What are the magnitude and direction of the initial acceleration? (C) At what time is the direction of the acceleration equal to 180º? (D) What is the magnitude of the momentum at t  3 seconds? *30. Momentum. While Lemur and Chimp were sleeping by a campfire, an orangutan ran off with their food pack. Fortunately, there is a homing beacon inside the food pack. The momentum of the 10-kg



food pack is: p(t )  t 2ˆi  t   ˆj . where   3 ,   5 , and   15 in their appropriate SI units. 43

Volume 1: Mechanics (A) What are the SI units of  ,  , and  ? (B) What are the magnitude and direction of the initial velocity? (C) What is the direction of the net force exerted on the food pack at t  2 seconds? (D) What is the magnitude of the net displacement for the first 2 seconds?



*31. Monkey Position. The acceleration of a wild 20-kg monkey is a(t )  tˆi  ˆj where   3 and



  4 in SI units. The initial velocity of the monkey is v 0 (t )  ˆi where   5 in SI units. (A) Find the SI units of these Greek symbols. Find the magnitude and direction of: (B) the initial acceleration of the monkey. (C) The net force exerted on the monkey at t  3 s . (D) The momentum of the monkey at t  3 s . (E) The net displacement of the monkey at t  3 s . *32. Doggy Paddle. A 60-kg dog paddles its way around a pool. The magnitude and direction of the velocity are

v(t )  3t

  2t where  has units of radians and v has units of m/s when t is expressed in seconds. (A) Find the magnitude and direction of the dog’s momentum at t = 3 seconds. (B) Express the net force exerted on the dog in ˆi and ˆj notation. (C) At what time is the dog moving in the negative y-direction? *33. Scaredy Monk. A 169-kg monkey runs with a momentum of p(t ) 

t3  3t after a black cat 13

sneaks up on him and meows. SI units have been suppressed for convenience, but you may not take that luxury in your answers. (A) What net force is exerted on the monkey at t  13 sec? (B) How fast is the monkey moving at t  13 sec? (C) What is the net displacement of the monkey after 13 sec? (D) When is the monkey’s acceleration equal to zero? (E) When does the monkey change direction? *34. Physics is a Slice of . The momentum of a 700-g slice of banana-cream pie is: 3 p(t )  t 2 ˆi  ( t )½ ˆj

where  = 3 and  = 5 in their appropriate SI units. (A) What are the SI units of  and ? (B) What is the initial velocity of the slice of pie? (C) How far does the slice of pie travel during the first 4 seconds? (D) What is the direction of the net displacement of the pie for the first 4 seconds? (E) What is the net force exerted on the pie when t = 2 seconds? (F) At what time(s) does the net force exerted on the pie have a direction of 330º? *35. I Scream for Physics. The magnitudes and directions of two forces acting on a 600-g banana split are:

44

Creative Physics Problems – Chris McMullen, Ph.D.

F1  3t

1  110º  2  240º

F2  2t

where the initial velocity of the banana split is 3ˆi  4ˆj . SI units have been suppressed for convenience. (A) What are the magnitude and direction of the net force exerted on the banana split when t = 3 seconds? (B) What is the initial momentum of the banana split? (C) What is the acceleration of the banana split when t = 3 seconds? (D) What is the velocity of the banana split when t = 3 seconds? (E) What is the average velocity of the banana split for the first 3 seconds? *36. Physics is a Stroll in the Dark. The position of a 2-kg flashlight is r(t )  r0 cos(t )ˆi  r0 sin(t )ˆj , where r0 = 20 m and  = 0.4 rad/sec. (A) What are the Cartesian coordinates of the flashlight’s initial position? (B) What are the magnitude and direction of the flashlight’s initial velocity? (C) What are the magnitude and direction of the momentum of the flashlight when t = 3 seconds? (D) What are the magnitude and direction of the acceleration of the flashlight when t = 3 seconds? (E) What are the magnitude and direction of the net force exerted on the flashlight when t = 3 seconds? (F) At what time is flashlight accelerated along 135º? *37. Elevator Monkey. A 50-kg monkey stands on a scale while riding a 1000-kg elevator. (A) What does the scale read (i) as the elevator travels upward at a constant rate of 4 m/s and (ii) as the elevator travels downward at a constant rate of 3 m/s2? (C) As the elevator travels with uniform acceleration, the monkey weighs 800 N. What are the magnitude and direction of the acceleration? (D) In part (C), the monkey drops a quarter from a height of 2m above the elevator floor. How long does it take to reach the floor? (E) The cable snaps when the elevator is 40 m above a giant spring while the elevator is at rest. What does the scale read as the elevator falls? (F) What is the speed of the elevator just before striking the spring? (G) If the spring is compressed a maximum of 5 m, what is the spring constant? 38. Green Cheese. All monkeys know that the moon is not made of green cheese. It’s really made of banana-coconut-cream pie. Mmmm delicious! (A) Use astronomical data to compute gravitational acceleration near the moon’s surface. (B) What force does the earth exert on the moon? (C) What force does the moon exert on the earth? 39. Monkey Earth. Monkeys call our planet Monkey Earth (instead of Mother Earth). (A) Look up the mass and radius of the earth to compute the earth’s gravitational acceleration near the surface of the earth. (B) At what altitude is the earth’s gravity equal to half of what it is at the surface? 40. Law of Attraction. (A) A 30-kg monkey stands 5 m away from a 20-kg banana. According to Newton, what is the gravitational force of attraction between the monkey and the banana? (B) Two twin monkeys have identical mass and stand 10 m apart. What would the mass of each monkey need to be to make the gravitational force of attraction equal to one Newton? *41. Into the Light. As you begin to do your homework, a bright light suddenly appears in the classroom – first a tiny blinding ball of light, then it expands into a much dimmer rectangle. Hoping to 45

Volume 1: Mechanics avoid taking the quiz, you walk through this rectangle… and wind up surrounded by little green monkeys. Wouldn’t you know it? They make you take a physics quiz! Planet Ban, as they call it, which is a trillion light-years from earth, has a radius of 2 10 m . Gravitational acceleration near the surface of Ban is 4 m/s2. Ban’s moon, Ana, has a mass that is onethird of Ban’s mass, and an orbital radius that is ten times Ban’s radius. (A) What is Ban’s mass? (B) What force does Ban exert on Ana? 5

42. Planet Mnqy I. Planet Mnqy has five times the mass of the earth and three times the diameter. You may not look up the mass and radius of the earth for this problem. (A) Calculate gravitational acceleration near the surface of Mnqy using only the value for the gravitational acceleration near the surface of the earth. (B) A monkey weighs 100 N near the surface of Mnqy. Find the mass and weight of the monkey when he stands atop a mountain with an altitude equal to 10% of Mnqy’s radius. *43. Planet Mnqy II. Five minutes after starting this problem, you are kidnapped by little green monkeys who transport you to Planet Mnqy in a giant flying banana. (Well, you can always hope.) 9  1020 kg 2  105 m

Mass of Mnqy: Radius of Mnqy:

You kiss the ground when you realize that Planet Mnqy is made of banana! (A) If you throw a rock into the air (yes, there is air, and, yes, you may neglect it, except as needed for biological purposes), what will be the rock’s acceleration? (B) If you weigh 400 N on earth, what are your mass and weight on planet Mnqy? (C) Would it be easier for Supermonk to leap a tall building on the earth or planet Mnqy? Would it be easier for Supermonk to stop a locomotive on the earth or planet Mnqy? Explain in terms of physics concepts. 44. Magic Bananas. If a monkey puts a bunch of banana’s near the earth, they fall toward the earth; if he puts the bananas near the moon, they fall toward the moon. However, there is a “magical” point where the bananas will neither fall toward the earth nor the moon. Where is this “magical” location? 45. Giant Leap for Monkeykind. A spherical moon with uniform density, radius R0 , and mass M is made of banana. Monkeys mining the banana cut out a spherical cavity of radius R0 / 2 such that one end of the cavity touches the moon’s center and the opposite end touches the moon’s surface. The monkeys take the mined banana back to their home planet. (A) What is the moon’s mass now? (B) What force would be exerted on a test monkey of mass m located a distance 3R0 from the moon’s center on a line joining the moon’s and cavity’s centers?

46

Creative Physics Problems – Chris McMullen, Ph.D.

R0

3R0 ,0 x

47

Volume 1: Mechanics

4 Uniform Circular Motion 1. Father Time. (A) What is the angular speed of the minute-hand of a clock in rad/sec? (B) How many radians does the hour-hand sweep out in 7 hours? 2. Round-N-Round. We spend our lives going in circles around the sun. (A) What is the angular speed of the earth in its orbit around the sun? (B) What is the tangential speed of the earth in its orbit around the sun? (C) What are the magnitude and direction of the centripetal acceleration of the earth? (D) What are the magnitude and direction of the centripetal force acting on the earth? *3. Monkimedes. A 20-kg monkey who discovers buoyancy while taking a bath suddenly runs around the streets of Sillinews shouting Eek! Eek! The monkey runs with constant speed in a circle with a diameter of 400 m, completing ¾ of a revolution in 5 minutes. (A) Find the period. (B) Find the frequency. (C) Find the monkey’s angular speed. (D) Find the monkey’s speed. (E) Find the total distance the monkey travels. (F) Find the monkey’s net displacement. (G) Find the acceleration of the monkey. (H) Find the net force acting on the monkey. (I) The average velocity of the monkey. (J) What is the minimum coefficient of friction needed between the monkey’s shoes and the ground for the monkey to run as described without slipping? 4. Banana Dance. A monkey ties a 200-g banana to its tail and spins it in a horizontal circle with 3-m diameter at a constant rate of 50 rev/min. (A) What is the angular speed of the banana? (B) What is the tangential speed of the banana? (C) What angle does the monkey’s tail make with the horizontal? (D) How long is the monkey’s tail? (E) What is the acceleration of the banana? (F) What kind of acceleration is this? (G) What is the direction of the acceleration? (H) What is the tension in the monkey’s tail? *5. Time Flies! A monkey spins a 500-g alarm clock by its power cord in a horizontal circle with 4-m diameter. Each minute the alarm clock completes 15 revolutions. (A) Find the period. (B) Find the frequency. (C) Find the angular speed. (D) Find the tangential speed. (E) Find the acceleration. (F) Find the tension. (G) Find the angle that the cord makes with the vertical. *6. Spit-the-Can. A couple of monkeys play spit-the-can as follows: One monkey ties an 8-kg spit can to a rope and spins it in a horizontal circle of radius of the moving target. The period of the spit can is

3 m, while the other monkey spits, hoping to hit

6 seconds. (A) What angle does the rope make with 3

the vertical? (B) What is the tension in the rope? (C) How long is the rope?

48

Creative Physics Problems – Chris McMullen, Ph.D. *7. Superfly. When a 500-g frog attempts to eat Superfly, Superfly grabs the frog by its tongue and

3 m. The period of the frog’s uniform circular 6 motion is seconds. (A) What angle does the frog’s tongue make with the vertical? (B) What is the 3 spins the frog in a horizontal circle with a radius of

tension in the frog’s tongue? (C) How long is the frog’s tongue? *8. “Eye” See You! A monkey takes the 500-g glass eye out of his socket and connects it to a string. The monkey then spins the glass eye in a horizontal circle with an angular speed of

3 rad/sec. The 2

string makes an angle of 60º with the vertical. (A) Find the period. (B) Find the tension in the string. (C) Find the acceleration of the eye. (D) How long is the string? (E) By what factor would the angular speed need to increase/decrease in order for the string to make an angle of 30º with the vertical? 9. Centrifuge. You can ‘defy’ gravity on an amusement park centrifuge, which consists of a large cylindrical room with horizontal sides. You stand against the wall. First, the centrifuge rotates, then the floor disappears! Yet, you cling to the wall, rather than slide down the wall (unless you get soaking wet on a water ride just before entering the centrifuge). Derive an equation for the period T of rotation of the centrifuge needed to prevent you from sliding down the wall in terms of the radius R of the cylinder, the coefficient of static friction s between you and the wall, and/or g. 10. Swinging Around. An amusement park ride consists of a 20-m diameter horizontal disk high above the ground. Several swings are suspended from the disk near its edge with 15-m long chains. As the disk spins, the swings extend outward at an angle of 50º from the vertical. How fast are the passengers moving? *11.  tan t . A mutant monkey has wheels where his feet should be. The monkey is driving down Monk Street when he rounds a turn to the left. The diameter of the turn is 80 m. The coefficient of static friction between the monkey’s wheels and the pavement is 0.5. What maximum speed can the monkey have if he wishes to round the turn safely? 12. Racecar Physics. A circular racetrack has a diameter of 200 m (from the top view, of course) and constant banking angle of 20º (as viewed below, with the racecar headed out of the page). (A) What is the maximum constant speed a car can have if there is no friction? (B) Repeat (A) if the coefficient of static friction is 0.4.

49

Volume 1: Mechanics 13. To Spin a Tail. Inside the cave, Lemur and Chimp meet a giant ogre. The ogre picks up 20-kg Lemur by his 2-m tail and spins him around in a horizontal circle with constant angular speed. As illustrated below, Lemur’s tail makes an angle of 60º with the vertical.

60º

2m

Lemur (A) Draw a free-body diagram for Lemur. Label relevant coordinates on your diagram. (B) Sum the relevant components of the forces acting on Lemur. (C) What is the tension in Lemur’s tail? (D) What is Lemur’s acceleration? (E) What is the name for this kind of acceleration? (F) What is the direction of this acceleration? (G) With what speed is Lemur moving? *14. Physics Drives You Crazy! A 50-kg monkey drives a 200-kg bananamobile with a constant speed of 20 m/s along a semicircular hill with a radius of 80 m, as shown below.

80 m 30º

(A) What is the angular speed of the bananamobile? (B) What is the normal force acting on the bananamobile in the position shown? (C) What is the fastest speed that the car could have in this problem and not lose contact with the ground at the top of the hill? *15. Physics ‘R’ Us. Little Isaac Newton the 12th buys a monster truck in the local physics store. The 500-g toy monster truck drives along a vertical loop with 10-m diameter with constant speed – without falling away from the track! The period is  seconds.

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Creative Physics Problems – Chris McMullen, Ph.D.

90º

0º 225º 270º (A) Explain why frictional effects can be neglected, and why the rotational effect of the wheels can also be safely neglected. In particular, this would not be the case for a sliding block or a rolling ball. (B) What is the speed of the monster truck? (C) Solve for the normal force for each of the positions shown in the diagram above. (D) What is the least speed the monster truck can have if the normal force is to be positive at every point on the loop? *16. A Bucket O’ Fun. A monkey whirls a bucket of water with combined mass radius R with constant angular speed  , as shown below.

R

m in a vertical circle of



Express your answers using m ,  , R ,  , and/or g only. (A) What is the speed of the bucket? (B) What is the period of the bucket? (C) What is the frequency of the bucket? (D) What is the acceleration of the bucket? (E) Draw a free-body diagram for the bucket in the position shown. (F) Sum the components of the forces. (G) What is the force of the monkey’s pull in the position shown? (H) What is the slowest speed that the bucket could have without spilling any water at the top of the arc? 17. Geosynchronous Satellite. What must be (A) the speed and (B) altitude of a geosynchronous satellite in a circular orbit around the earth (above the equator)? *18. Supermonkey. Supermonkey flies to the top of a mountain, where the altitude is 100 km above the earth’s sea level. He throws a banana horizontally. He throws it so fast that he catches it! (A) How fast did he throw the banana? (B) For how much time did Supermonkey have to wait for the banana to return?

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Volume 1: Mechanics 19. Banana Planet. Lemur and Chimp live on Banana planet, which orbits Banana star in a circular orbit. Lemur weighs 200 N on Banana planet, while Chimp weighs 350 N on Banana planet. Below are astronomical data for Banana planet. Radius of Banana Planet: 1.45  105 m Mass of Banana Planet: 9.24  1023 kg Distance between Banana Planet and Banana Star: 6.59  109 m Mass of Banana Star: 3.72  1026 kg One ‘Day’ on Banana Planet: 8.23 hours One ‘Year’ on Banana Planet: 720 days (A) What is the value of gravitational acceleration near the surface of Banana planet? (B) What is Chimp’s mass? (C) How much does Lemur weigh on the earth? (D) What force does Banana star exert on Banana planet? (E) What is the angular speed of Banana planet in its orbit around Banana star? (F) What is the tangential speed of Banana planet in its orbit around Banana star? (G) What is the acceleration of Banana planet? (H) What is the net force exerted on Banana planet? (I) What would be the period of a banana satellite orbiting Banana planet at a constant altitude of 200 km above the surface of Banana planet? *20. Planet 6. Below are astronomical data for Planet 6 and its moon: Mass of 6: 6  1023 kg Mass of 6’s moon: 9  1021 kg Radius of 6: 4  106 m Radius of 6’s moon: 2  105 m 10 Orbital radius of 6: 8  10 m Orbital radius of 6’s moon: 4  107 m (A) If you stand on the surface of 6 and throw a rock into the air, what will be its acceleration? (B) What force does 6 exert on its moon? (C) What is the speed of 6’s moon in its circular orbit around 6? (D) What is the period of 6’s moon in its circular orbit around 6? (E) A common myth is that 6’s moon is made of green banana. What can you calculate to resolve this myth? Calculate this quantity. *21. Monkey Laws. The little green monkeys are thinking of changing the laws of physics. In particular, they wish to modify Newton’s law of universal gravitation to have the form:

FH

m12 m23 r4

without modifying Newton’s 2nd law. (A) What are the SI units of H ? (B) Derive an equation for the speed of a satellite of mass m2 orbiting a planet of mass m1 in a circular orbit of radius r in terms of m1 , m2 , r , and/or H only. (C) Derive an equation for the period of a satellite of mass m2 orbiting a planet of mass m1 in a circular orbit of radius r in terms of m1 , m2 , r , and/or H only.

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Creative Physics Problems – Chris McMullen, Ph.D.

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Volume 1: Mechanics

Review 2: Newton’s Laws *A student named ______________________ was taking a test, After staying up all night studying without a moment’s rest. The teacher says to complete five pages, then walks out of the class. Enter a monkey with the nerve to ask, “Who cares whether you pass?” He throws one banana horizontally and drops another from rest at the same time. Heights being equal, which banana lands first? As always, explain, or it’s a crime.

He drops ten-kilogram and twenty-kilogram bananas from rest at the same time. Heights being equal, which banana lands first? As always, explain, or it’s a crime.

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Creative Physics Problems – Chris McMullen, Ph.D.

The monkey throws a bunch of bananas horizontally with as much force as he can. Why does the monkey fall backward when he lets go? Surely, that was not his plan.

The monkey rides a skateboard over a block, where the wheel gets caught. The skateboard suddenly stops. Explain why it is that the monkey does not.

The monkey jumps high in the air, landing on a banana – fresh off the tree. What is the ratio of the forces exerted between the banana and monkey?

A monkey pushes a box of bananas – first at rest, then while in motion. Explain which is greater – the coefficient of static or kinetic friction.

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Volume 1: Mechanics

A scale reads forty kilograms when a monkey stands on it and flips the switch. What are the mass and weight of the monkey? Make it clear which is which.

“Now listen here, little monkey,” says a student. “We’re taking a test today. We can’t concentrate with all your distractions. Will you please go away?” “Oh! You’re taking a test?” asked the monkey. “Why didn’t you say? I’ll be back in a moment. I have just the thing to save the day!” Wouldn’t you know it? The monkey lands safely on an air mattress. When he returns, he is pulling a box by a cord with a look of success. Supposing 3 meters per second per second is the box’s acceleration, And the pull is a hundred Newtons, what’s the coefficient of friction?

20 kg

30º

Calculate the normal force, and it would also be great To explain why the normal force is not simply the weight.

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Creative Physics Problems – Chris McMullen, Ph.D.

The monkey leaps horizontally out of the window at twenty meters per second. The window is forty-five meters above the ground. Where does he land?

45 m

Wouldn’t you know it? The monkey lands safely on an air mattress. He returns in a helicopter, lifting a box by a cord with a look of success. Accelerating four meters per second squared with a distinctive flair, What lifting force pushes the helicopter straight up through the air?

200 kg

helicopter

50 kg

box

In case you might otherwise find yourself bored, Also find the tension in the connecting cord. Out of the box pop Chimp One and Chimp Two, Each wearing tennis shoes from Kalamazoo. The funny thing about these chimps is that neither is a foot tall. The chimps incline a track at thirty degrees, but that is not all. Chimp One shoves Chimp Two up the inclined track, Waiting at the bottom for Chimp Two to come back. 57

Volume 1: Mechanics

Explain why including the shove in the free-body diagram would be incorrect. If the shove does not influence the acceleration, then what does it affect?

30º If one-half square-root three is the coefficient of friction, Then going up the incline what is Chimp Two’s acceleration? Out of the window, forty-five meters high, Chimp Two throws Chimp One Eighty meters per second, thirty degrees above the horizontal – how fun! Find Chimp One’s height, speed, and acceleration at the trajectory’s top. Also find where on the ground (or air mattress) Chimp One will stop.

45 m

The monkey grabs Chimp One by the tail, extends his arm horizontally, whirling Chimp One in a horizontal circle. With what angular speed does Chimp One swing?

extendedarm

tail

30º

60 3 cm

58

140 3 cm

Creative Physics Problems – Chris McMullen, Ph.D.

Chimp One returns, and is now connected to Chimp Two by a cord – see below. If the coefficient of friction equals 1 / 3 , with what acceleration will they go?

20 kg

40 kg

60º Chimp One sets Chimp Two on a table, then ties his body to a pole with a cable. What minimum horizontal force must Chimp One apply to accelerate the table?

ChimpTwo table

cable

ChimpOne Chimp One’s mass = 20 kg Chimp Two’s mass = 40 kg Table’s mass = 60 kg Coefficient of friction between table and Chimp Two = 0.25 But the ground is frictionless If perchance you are able, Find the tension in the cable. Chimp One and Chimp Two skate ten meters per second in a circle, graciously, Completing twelve revolutions per minute, until falling down rather humorously. What is his angular speed? What acceleration does he need? What is the circle’s diameter? What’s the circumference, for that matter? What is the period of revolution? What is the frequency of the motion?

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Volume 1: Mechanics

What is the net force exerted on Chimp Two? What coefficient of friction is needed, too? “You silly monkey,” says a student, “you’ve made matters worse. You were distracting enough, but these tiny chimps are a curse!” “But,” replies the monkey, “we’ve demonstrated every problem on the test. And I closed the door and put glue in the lock. So don’t act so stressed.”

Spookier than a skeleton’s chattering teeth, feared by the bravest of souls, eager to haunt any foolish enough to cast their eyes upon it, Halloween Physics now has you under its spell! Muhahahaha! Do you, simple mortal, dare to accept this challenge? Can you combat the evil curse and mighty demons, armed only with a pencil, calculator, and Newton’s laws? Well then, enter the realm of Evil Physics and see if you have what it takes to rescue 100 souls from the depths of darkness! Good luck – for you shall need it! *1. Eyes of Newton. Lower the drawbridge to cross the moat of blood by gazing into the eyes of Newton, removed from their sockets and dangling in the air, while answering the following bloodcurdling riddles. Explain your answers with Newton’s laws. (A) When the Cyclops fires his gun, filled with gooey slime pellets, why does he experience a recoil force? (B) Why is the slime pellet accelerated more than the Cyclops? (C) Why does the coffin slide in the hearse when the driver slams on the brakes? *2. Staircase of Doom. After climbing 50 meters up a 45º staircase, the stairs flatten, you fall, and you slide back down. The coefficient of kinetic friction between you and the staircase is 0.33. How much time elapses before you return to the bottom of the staircase? *3. Vampire Coffins. As you reach the bottom of the stairs, a trap door opens and you fall into the dark, dreary basement, where two coffins are connected by a cord as shown below. Neglect friction. m1 = 400 kg, m2 = 800 kg. Find the acceleration of the coffins.

20º

60

Creative Physics Problems – Chris McMullen, Ph.D. *4. Screeching Bat. You enter the torture chamber. Awaiting you is none other than the Grim Reaper. He ties a 100-g bat to a blood-soaked vine and whirls the screeching bat around in a horizontal circle at a rate of 0.2 rev/sec. (A) What is the period of the bat’s rotation? (B) What is the frequency of the bat’s rotation? (C) What is the tangential speed of the bat? (D) What is the bat’s acceleration? (E) What is the tension in the vine? *5. Planet Zqmby. Finally, you are simply overwhelmed and faint. When you awaken, you find that you have been captured and transported to planet Zqmby.* As you lie on the cold, slimy floor, you read the bloody hieroglyphs that those poor souls before you have written upon the walls, and realize that you can escape from your perils if you can solve these two last physics problems. The hieroglyphs also reveal that the mass of planet Zqmby is 1.107  1024 kg and its radius is 3.33  106 m. (A) What is the gravitational acceleration near the surface of planet Zqmby? (B) At what height above the surface of planet Zqmby would a satellite have a speed of 999 m/s? Congratulations, brave soul, for you have accepted the difficult challenges presented to you and faced them head-on with determination, perseverance, and your all-powerful knowledge of physics. Although you have suffered many scars, you should be proud of your valiant effort. Now, there is time for you to revel in your accomplishments. But wait – there are more questions! Is there no mercy? *6. Screeching Bat Revisited. Reconsider the screeching bat problem (#4). What angle does the vine make with the horizontal? *7. Vampire Coffins Revisited. Reconsider the vampire coffins problem (#3) with the following alterations. You no longer know the angle of the incline – call it . Include the effects of friction (k = 0.2). What must  be if the acceleration of the coffins is 2.0 m/s2 up the incline? (Assume that the system begins from rest.) [Before you utter your complaints, be grateful that the instructor did not turn off the lights halfway through the exam and howl like a werewolf!]

*8. Torture Chamber. Solve these riddles to save anguished souls from unimaginable physics torture. (A) You ride in a haunted train that is heading north. You hold a rotten apple directly over an X on the floor. You let go of the rotten apple as the train decelerates. Relative to the X on the floor, explain where the rotten apple lands and why. (B) (You are no longer on a train.) You throw a rotten apple with a velocity of 30 m/s at an angle of 60º above the horizontal. What are the speed and acceleration of the apple at the top of its trajectory? (C) A giant 800-kg ogre running 10 m/s collides head on with little 80kg you (at rest prior to the collision). Compare the forces that you and the ogre exert on each other. Also, compare your change in velocity to that of the ogre. Explain your answers. (D) You simultaneously shoot a bullet horizontally and drop a bullet from rest from the same height. Both strike the level ground below. Which bullet strikes the ground first? Explain your reasoning. (E) A one-eyed alien 61

Volume 1: Mechanics weighs 900 N on the surface of the earth. What are the mass and weight of the alien on the moon? (F) A ghoul gives a coffin a shove on a frictionless horizontal surface. Does the shove influence the coffin’s motion after the ghoul is no longer in contact with the coffin? Explain. Describe the coffin’s velocity and acceleration after the ghoul is no longer in contact with the coffin. Also, what happens to the ghoul, and why? 9. Bloody Physics. You scream in horror as you slide down an incline covered with bloody physics equations. The angle between the incline and the horizontal is 30º. The coefficient of kinetic friction between you and the bloody incline is

3 . 4

30º (A) What is your acceleration if you slide down the incline? (B) What is your acceleration if you instead slide up the incline? *10. Coffin Ride. You are connected to a coffin by a blood-soaked vine as illustrated below, but there is good news: The surfaces are covered in green slime, making them effectively frictionless. The system is initially at rest.

pulley

160 kg

80 kg

30º (A) What is your acceleration down the incline? (B) What is the tension in the blood-soaked vine? *11. Headless Horseman. A headless horseman ties his 8-kg head to a blood-soaked vine and spins it in a horizontal circle with a radius of

3 m. The frequency of his head is

3 Hz. (A) What angle does 6

the blood-soaked vine make with the vertical? (B) What is the tension in the blood-soaked vine? (C) How long is the blood-soaked vine?

62

Creative Physics Problems – Chris McMullen, Ph.D. *12. Wheel of Torture. You ride a Ferris wheel with 100-m diameter that is spinning with a frequency of 0.1 Hz. Unfortunately, the ghoulish passengers became sick and now you’re covered with slimy vomit. Your mass is 50 kg. [Realize that the seat has multiple surfaces (e.g. a back and a bottom) pointing in different directions.]

y A

C

x

B (A) What are the magnitude and direction of the force that the seat exerts on you when you are at the top of the Ferris wheel (position A)? (B) What are the magnitude and direction of the force that the seat exerts on you when you are at the bottom of the Ferris wheel (position B)? (C) What are the magnitude and direction of the force that the seat exerts on you when you are halfway up the Ferris wheel (position C)? 13. The End. The Grim Reaper is pulling two coffins connected with a blood-soaked vine as illustrated below. The coefficient of friction between the coffins and the ground is ¼. The system is moving to the right.

400 2 N

20 kg

80 kg

45º

(A) What is the acceleration of the system? (B) What is the tension in the connecting cord?

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Volume 1: Mechanics

5 Conservation of Energy *1. Will Work for Bananas. A 400-g box of bananas rests on the floor. The coefficient of friction between the box of bananas and the floor is 0.6. (A) A monkey pushes the box of bananas horizontally with a force of 5 N for 7 seconds. How much work did the monkey do? (B) A monkey pulls the box of bananas at an angle of 40º above the horizontal with a constant force of 5 N until the speed of the box is 10 m/s. How much work did the friction force do? (C) A monkey lifts the box of bananas 1.5 m in the air. How much work did the monkey do against gravity? (D) A monkey pulls the box of bananas at an angle of 80º above the horizontal with a constant force of 5 N for 7 seconds. How much work did the friction force do? (E) A monkey pulls the box of bananas at an angle of 20º above the horizontal with a constant force of 5 N. The monkey delivers 2.4 hp to the box of bananas. What is the average speed of the bananas? 2. Fun, Fun, Fun! Fun, being the antithesis of work, is naturally defined as the negative of the work:

Fun  W A 50-kg chimpanzee swings from a chandelier, slides down a 35º banister, lands on a skateboard, and crashes into a cabinet of fine china. Now that’s fun! The chimpanzee slides down the 20-m long banister with an initial rate of 10 m/s. The coefficient of kinetic friction between the chimpanzee and the banister is 0.3. (A) How much fun is provided by friction as the chimpanzee slides down the banister? (B) How much fun is provided by the normal force as the chimpanzee slides down the banister? (C) How much fun does the chimpanzee have while sliding down the banister? (D) What is the average power delivered to the chimpanzee by gravity? *3. Precious Joules I. A monkey with a 700-g box of precious banana gems stands at the bottom of a 20º incline. (A) The monkey, standing at the bottom of the incline, shoves the box upward – i.e. he gives the box an initial speed, but he remains at the bottom. The box runs out of speed after traveling 30 m along the incline. How much work does gravity do on the box while the box travels up the incline assuming the incline to be frictionless? (B) In the previous exercise, what average power does gravity deliver to the box of bananas on its way back down? *4.

Precious Joules II.

A monkey pushes a 4-kg box of precious banana gems with a force

 F( x)  (x 2  x) ˆi , where  = 3 and  = -5 in their appropriate SI units. The banana travels from (-1

m, 4 m) to (2 m, 4 m) along a straight line. The initial speed of the banana is 5 m/s. (A) What are the SI units of  and ? (B) How much work did the monkey do? (C) What is the final speed of the banana? (D) Is the work path-independent? *5. Play Time. The more you work, the less you play. The more you play, the less you work. Thus, a monkey hypothesizes that Play Work  1000 J . Knowledge is power. Therefore, a monkey predicts 64

Creative Physics Problems – Chris McMullen, Ph.D. that Ignorance Power  1000 W . A monkey exerts a force of 200 N to pull a 28-kg box of bananas

3 m/s a distance of 12 m along the ground, as illustrated below. The coefficient of friction between the box and ground is 1 / 3 . with an initial velocity of

30º

(A) How much does the normal force play with the box of bananas? (B) How much does friction play with the box of bananas? (C) How much does gravity play with the box of bananas? (D) How much does the box of bananas play? *This is analogous to, “How much work is done on the box of bananas?”+ (E) How much ignorance does the monkey deliver to the box of bananas? [What force is associated with the monkey?]



6. Monkey Product. A monkey pulls a wagon full of bananas with a constant force F  5ˆi  4ˆj along a



displacement s  32ˆi  16ˆj . where SI units have been suppressed for convenience. (A) Compute the scalar product of the force and displacement vectors (using the dot product between unit vectors – do not use any angles or cosines in part (A)) to find the work done by the monkey in pulling the wagon. (B) Find the magnitude and direction of the force exerted by the monkey. (C) Find the magnitude and direction of the net displacement of the wagon. (D) Find the angle between the force and the net displacement from the directions found in (B) and (C). (E) Verify that Fscos agrees with the scalar product computed in (A).

1 . (A) A puppy playfully tugs on the tail of a Work   kitten with a force F  3ˆi  4ˆj , while the displacement of the kitten is s  5ˆi  2ˆj (where SI units have *7. Puppy Love. Play is the inverse of work: Play 

been suppressed). How much does the puppy play? (B) A 15-kg puppy drags a 5-kg wedding dress 40 m along a 30º muddy incline. How much does gravity play with the dress? (C) A 20-kg puppy runs along horizontal ground with an acceleration of 5 m/s2 for 8 seconds. How much does the normal force play? (D) A 10-kg puppy rides on a 20-kg sled on horizontal ground for 50 m. The coefficient of friction between the ground and the sled is ½. How much does friction play? 8. More Fun! SI units are suppressed. (A) A puppy playfully tugs on the tail of a kitten with a force

  F  3ˆi  4ˆj , while the displacement of the kitten is s  5ˆi  2ˆj . How much fun does the kitten have  playing with the puppy? (B) A kitten nudges a mouse with a force F  3( x  4) 2 ˆi from x  0 to x  6 . How much fun does the kitten provide to the mouse? (C) The amount of fun a kitten has playing with a ball of yarn is Fun  3(t  2) 2 . How much power does the kitten deliver to the ball of yarn at t = 6 seconds? What is the average power delivered to the ball of yarn during the first 6 seconds?

65

Volume 1: Mechanics *9. Hot Pursuit. Monk – James Monk, aka secret agent 00 – is pursuing his nemesis, Osamonk Been



Rotten, in a high-speed car chase. (A) 00’s sporty car is powered by a force F  5x 2 yˆi  10 x 2 y 3ˆj as his car travels along the curve y  4x 2 over the interval 0  x  2 , where the force is expressed in kN and position is expressed in km. How much work is done by the car’s engine? (B) Osamonk Been Rotten’s monster truck delivers a power P  6t 2  4t  2 over the interval 0  t  8 , where the power delivered is expressed in kW and time is expressed in SI units. How much work is done by the truck’s engine (assuming the initial work to be zero)? *10. Blue Collar Monkey. A monkey with a blue collar pulls an 8-kg box of bananas a distance of 6 m along a 60º incline with a force of 120 N, as illustrated below. The coefficient of friction between the box and incline is 0.5.

60º

(A) How much work does the monkey do? (B) How much work does gravity do? (C) How much nonconservative work is done? (D) Starting from rest, what is the average power delivered by the monkey’s pull? *11. White Collar Monkey. A monkey with a white collar sits at a desk and contemplates the work done and power delivered by various hypothetical forces. How much work is done by: (A) a force

   x  F  3ˆi  2ˆj that displaces an object s  2ˆi  ˆj ? (B) a force F  2 sin  that displaces an object  12  from x  2 to x  10 ? (C) How much power is delivered at t  4 if W  4t 5 / 2 ? SI units have been suppressed for convenience.

*12. Monkey Units. What are the SI units of

1  4sP    2g  m 

2/3

? [Variables appear in italics, whereas units

do not. In this way, for example, it is easy to distinguish between work (W) and Watts (W).] 13. Banana Peels. A miniature 200-g monkey skis 150 m down a 25º frictionless incline on banana peels. The initial speed of the monkey is 20 m/s. (A) How much does the potential energy of the monkey change? (B) What is initial kinetic energy of the monkey? (C) What is the final kinetic energy of 66

Creative Physics Problems – Chris McMullen, Ph.D. the monkey? (D) What is the final speed of the monkey? (E) What is the work done by gravity? (F) What is the work done by the normal force? (G) How much power does gravity deliver to the monkey? *14. Monko, Inc I. A monkey exerts a force of 50 N to pull a 10-kg box of bananas with a constant speed of 2 m/s a distance of 20 m along the ground, as illustrated below.

60º (A) Find the work done by gravity. (B) Find the work done by the monkey’s pull. (C) Find the average power delivered by the monkey’s pull. (D) Find the work done by friction. *15. Monko, Inc II. Monkeys work hard to earn their bananas at Monko, Inc. (A) A monkey walks up a





60º incline. (i) When computing the work done by gravity, what is the angle between F and s ? (ii)





When computing the work done by the normal force, what is the angle between F and s ? (iii) When





computing the work done by friction, what is the angle between F and s ? (iv) When computing the work done by gravity, what is the force? (v) When computing the work done on the monkey, what is the



force? B) A monkey exerts a force F  2ˆi  2 3ˆj on a box of bananas, displacing the box of bananas



along s  24ˆi  8 3ˆj with an average speed of 4 m/s. (i) How much work does the monkey do? (ii)





What is the magnitude of F ? (iii) What is the direction of s ? (iv) What average power does the monkey deliver to the box of bananas? *16. Iceboarding. A 500-kg polar bear is iceboarding down the side of an enormous glacier at an angle of 35º with the horizontal. He travels 500 meters. (A) How much did his potential energy change? (B) How much work did gravity do? (C) How much power was delivered to the polar bear? (D) How much kinetic energy did the polar bear gain? (E) What was the final speed of the polar bear? (F) What was the final momentum of the polar bear? (G) What would the final speed of the polar bear been if there were friction ( = 0.2)? *17. Bananamobile. A 50-kg bananamobile is traveling 30 m/s down the highway. How much work is required to bring the bananamobile to a stop? *18. Double the Power. A motor does work W  t to accelerate a 200-kg bananamobile from rest for a duration of five seconds, where   500 in SI units. (A) How much horsepower does the motor deliver to the bananamobile? (B) The bananamobile, presently moving 50 m/s, exerts a braking force 3

given by F ( x)  x , where  is a constant. If the car comes to rest in 200 m, what is  ? 2

*19. Monkey Torment. All of the monkeys chase you with physics notes. Ahhhh! They catch you and torment you with physics problems. Consider the following equations: 67

Volume 1: Mechanics

 F1  4ˆi  7ˆj  F2  (2 x 2  3 x)ˆi W3 (t )  2t 2  5t  4 U 4 ( x) 

9 4  2 4 x x





(A) How much work is done by F1 in moving an object along the displacement s  12ˆi  5ˆj ? (B) What is



the potential energy associated with F2 ? (C) Evaluate the power for W3 at 3 seconds. (D) Locate the positions of stable equilibrium for a particle with potential energy U 4 . *20. Political Monkey. When a monkey is asked whether or not he is conservative, he responds that he



is as conservative as the force F  2 x 2 yˆi  xy 2ˆj . (A) Find the work done by the force from (0,0) to (3,2) along the straight line that connects the two endpoints. (B) Compare with the work done by the force from (0,0) to (3,2) along the path (0,0)  (3,0)  (3,2) . (C) Is the force conservative? *21. Monkeymatical Physics. 2 Monkeys + 1 Banana = 1 Fight. (A) Find the partial derivative of

3x 2 y  7 x 3 y  4(2  y) 2

with

respect

x.

to

(B)

Find

the

partial

derivative

of

2

x y  3xy 2  2 . (D) Is the force y x  2 ˆ 2ˆ (E) Is the force F  4 x yi  4 xy j conservative? (F) Given

3x 2 y  7 x 3 y  4(2  y) 2 with respect to y . (C) Find the gradient of

 F  4 xy 2ˆi  4 x 2 yˆj conservative? U ( x, y)  x  y 2 , find the SI units of  and  . (G) Given U ( x, y)  3x  2 y 2 , find the magnitude  and direction of F . (H) Given U (r )  r 3  3r 2  16r  9 , find the equilibrium positions and  2 characterize each as stable, neutral, or unstable. (I) Given F(t )  8t  16t  32 , find the impulse from  t  0 to t  30 ms . (J) Determine whether or not F  2ˆi  4 x 2ˆj is a conservative force by computing

the work done from (0,0) to (2,2) along the path (0,0)  (2,0)  (2,2) and comparing with the work done from (0,0) to (2,2) along the path (0,0)  (2,0)  (2,2) . (K) Determine whether or not

 F  4 x 2ˆi  2ˆj is a conservative force by computing the work done from (0,0) to (2,2) along the path (0,0)  (2,0)  (2,2) and comparing with the work done from (0,0) to (2,2) along the path (0,0)  (2,0)  (2,2) .

*22. Pink Collar Worker. A monkey wearing a pink collar is doing research on the physics of tennis



serves. She hypothesizes that the force delivered by the tennis racket is F  θˆ kr 2 sin(2 ) , where k is a constant and r and  are polar coordinates that describe the initial position of the tennis racket. Either prove that this force is conservative or show that it is not.

68

Creative Physics Problems – Chris McMullen, Ph.D. 23. Banana Path. A monkey runs along the banana path: 3/ 2

4  y ( x)   x   9 

,

4 8 x 9 9

where SI units have been suppressed. Find (A) the total distance traveled and (B) the net displacement (this is a vector, so you must specify both the magnitude and the direction) by formally integrating the appropriate differential element. (C) Express the ratio of the magnitude of the net displacement to the total distance traveled as a percentage. 24. Monkey Rushmore. A 600-g banana is thrown upward at some angle at a rate of 30 m/s off the top of a 100-m high hill carved in the shape of a monkey. The maximum height of the banana is 140 m above the ground. (A) Use energy methods to find the speed of the banana at the top of its trajectory. (B) At what angle is the banana thrown? 25. Banana Sled I. A miniature 200-g monkey slides 150 m down a 25º incline in a 400-g sled. The initial speed of the sled is 20 m/s. The coefficient of friction between the sled and the incline is 0.3. (A) How much energy is lost to friction? (B) Use energy methods to find the final speed of the monkey. 26. Banana Sled II. A miniature 200-g monkey slides 150 m down a 25º incline in a 400-g sled. The initial speed of the sled is 20 m/s. If the monkey loses 10% of its energy to friction, what is its final speed? 27. Pendulemur I. A 5-kg lemur swings back-and-forth on a 30-m vine suspended from the branch of a banana tree. The vine makes a maximum angle of 30º with the vertical. (A) What happens to the potential and kinetic energy as the lemur swings from 30º to 0º? What happens to the total energy? Describe the energy transformation. (B) Find the speed of the lemur at the bottom of its arc. 28. Pendulemur II. A lemur of mass m swings back-and-forth on a vine of length L suspended from the branch of a banana tree. The vine makes a maximum angle of  with the vertical. Take the height of the lemur to be zero when the vine is vertical. Also, take initial to be maximum angle, and final to be vertical. (A) Derive equations for the initial and final height of the lemur. (B) Derive equations for the initial and final velocity of the lemur. (C) Derive equations for the initial and final acceleration of the lemur. (D) Derive equations for the initial and final tension in the vine. (E) If   30º , what is the angle of the vine when the lemur’s potential energy is half its maximum value? (It’s not 15º! Why not?) 29. Simple Harmonic Banana. A 400-g banana oscillates with amplitude 2.5 cm at the end of a horizontal spring whose constant is 1.0 N/cm. Find the acceleration and speed of the banana when its displacement from the equilibrium position is (a) 2.5 cm, (b) 0 cm, (c) 0.5 cm. *30. Icecoaster. Penguinland has just opened its newest attraction, the Icecoaster. After waiting in line for three hours on the frozen tundra penguins seat themselves in a roller coaster car, restrain themselves with safety harnesses, and prepare for a 30 second ride of thrilling terror. Polar bears push 69

Volume 1: Mechanics the 400-kg roller coaster car 20 meters against a giant spring (k = 6000 N/m) and let go. As illustrated below, the roller coaster then zooms forward, completes the loop-the-loop, and is finally slowed by snow at the end. B 30 m A

C

(A) What is the speed of the roller coaster at point A? (B) What is the speed of the roller coaster at point B? (C) What is the speed of the roller coaster at point C? (D) When the roller coaster reaches point C it plunges into snow and comes to a halt in 200 meters. What is the average force that slows the car? (E) What is the normal force at point B? 31. Monkey Coaster I. In the frictionless roller coaster illustrated below, a 20-kg monkey passes through the 35-m diameter loop once and then compresses a spring 8 m from its equilibrium position (after which he repeats his journey in reverse). In the position shown, the monkey is 60 m above the ground and moving 20 m/s.

A

B (A) What is the speed of the monkey at point A ? (B) What is the normal force at point A ? (C) What is the spring constant? *32. Monkey Coaster II. In the roller coaster illustrated below, a 6-kg monkey is launched to the right by a spring with constant 200 N/m that is compressed 3 m from equilibrium. The diameter of the loop is 10-m. Neglect friction.

70

Creative Physics Problems – Chris McMullen, Ph.D.

B

(A) How much energy is stored in the compressed spring? (B) How high up the hill does the monkey get (before falling back down)? (C) What is the normal force at point B ? (D) What percentage of the monkey’s initial energy can the monkey afford to lose and still complete the loop without falling away from the track? 33. Free Fall I. Monkeys wait in line for hours to ride on Free Fall at Monkey Mountain. While the monkeys are biting their nails at Point A, the car suddenly drops. The combined mass of the car and monkeys is 400 kg. The ride is frictionless between Points A and B , but there is friction between Points B and C . The ride comes to rest at Point C in the illustration below.

A

Car

60 m 40 m 20 m

B

C

(A) What is the speed of the monkeys at Point B ? (B) How much nonconservative work is done between Points B and C ? (C) What is the coefficient of friction between the car and surface between Points B and C ? 34. Free Fall II. Monkeys wait in line for hours to ride on Free Fall at Monkey Mountain. While the monkeys are biting their nails at Point A, the car suddenly drops (see illustration above). The ride



applies a nonconservative braking force between Points B and C given by F( x)  x 2 ˆi . The combined mass of the car and monkeys is 400 kg. (A) What is the maximum speed of the monkeys? (B) What value of  is required in order to bring the car to a stop at Point C? 35. Monkey Slide I. A 2-kg monkey slides down a 30º frictionless incline with an initial speed of 5 m/s. The monkey slides 20 m before reaching a spring with a spring constant of 22 N/m. 71

Volume 1: Mechanics

30º (A) How fast is the monkey sliding just before reaching the spring? compression of the spring from equilibrium?

(B) What is the maximum

36. Monkey Slide II. Enjoy a ride on the monkey slide! Consider the illustration below. The 60-kg monkey steps onto the 70º escalator at Point A . The monkey rides up the escalator at a constant rate of 20 m/s. When the monkey reaches the top, she slides down at an initial rate of 20 m/s. The coefficient of friction between the monkey and the 50º incline is 0.2. After leaving the 50º incline, the remainder of the track is frictionless. After traveling along the horizontal, the monkey collides with a 20-kg banana. The collision between the monkey and the banana is completely inelastic. After the collision, the monkey travels through the loop. The monkey then rises to some maximum height at Point F .

B F E 100 m 40 m

A

70º

50º

C

D

(A) The escalator exerts two forces on the monkey: normal force, plus a pull along the incline. Find the force of this pull. (B) How much work does the escalator do to transport the monkey from A to B ? [You can deduce which force(s) to use.] (C) How much power does the escalator deliver to the monkey? (D) What is the momentum of the monkey as he rides up the escalator? (E) What is the kinetic energy of the monkey as he rides up the escalator? (F) How much does the potential energy of the monkey change from A to B ? (G) How much nonconservative work is done as the monkey slides down the incline (from B to C )? (H) What is the speed of the monkey at Point C ? (I) What is the speed of the monkey at Point D ? Reminder: The banana enters the problem between C and D . Note: You can deduce when, if ever, the banana leaves the problem. (J) Find the normal force exerted on the monkey at Point E . (K) How high does the monkey rise up the incline (i.e. what is the height of Point F )? 72

Creative Physics Problems – Chris McMullen, Ph.D. *37. Here Kitty Kitty! A monkey stands on the roof of a 10-m tall house and throws an adorable kitty cat upward (but not straight upward) with an initial speed of 5 m/s as part of his science fair project to determine if cats always do land on their feet. The cat does, in fact, land on its feet after losing 36% of its energy (relative to the ground) to air resistance. What is the cat’s speed just before landing on the ground? *38. Ferocious Lion. It turns out that the kitty cat in the previous problem has a godfather who is a ferocious lion. The lion places the 15-kg monkey atop a vertical spring with constant 8 N/cm, compresses the spring 2 m from equilibrium, and releases the spring from rest. How high does the monkey travel (above equilibrium) if he loses 25% of his spring potential energy to air resistance? 39. Eskimonkey. An eskimonkey of mass igloo of radius R as illustrated below.

m slides from rest from the top of a frictionless hemispherical

R



(A) What is the initial potential energy of the eskimonkey relative to the ground? (B) What is the initial kinetic energy of the eskimonkey? (C) What is the potential energy of the eskimonkey relative to the ground when the eskimonkey is at angle  ? (D) What is the speed of the eskimonkey when the eskimonkey is at angle  ? (E) What force does the igloo exert on the eskimonkey when the eskimonkey is at angle  ? (F) What is the speed of the eskimonkey when the normal force equals zero? (G) At what angle  c does the monkey lose contact with the igloo? *40. Good Morning, Physics! As illustrated below, a 10 -kg monkey sleeps on a 30 -kg bed. The spring constant is 30 N/m. The spring is presently stretched 20 -m from equilibrium. This elaborate alarm clock cuts the cord at 3:14 a.m., thereby sending the bed and monkey into simple harmonic oscillation. What a way to start the day! The coefficient of friction between the floor and the bed is 0.2.

Zzzzzzzzz!

73

Volume 1: Mechanics (A) What will be the speed of the bed when it returns to equilibrium? (B) What is the speed of the bed after traveling 10 m to the right of its initial position? (C) What must be the coefficient of friction between the bed and the monkey in order to prevent the monkey from sliding on the bed during oscillation? *41. Double the Oscillation. A spring with a constant of 60 N/m and natural length of 12 m is attached to a wall on the left, while another spring with a constant of 40 N/m and natural length of 18 m is attached to a wall on the right. As illustrated below, a monkey with a mass of 20-kg is stuck between the two springs. The two walls are 40 m apart. (Assume the monkey to have negligible thickness.)

40 m (A) When the system is in equilibrium, how far is the monkey from the left wall? (B) If the monkey is released from rest from a position 12 m from the left wall, what will be his speed when he is 16 m from the left wall?

74

Creative Physics Problems – Chris McMullen, Ph.D.

6 Systems of Objects *1. Center of Monkeys I. Find the location of the center of mass of the following system of three monkeys: a 100-kg monkey located at (10 m, 0) ; a 150-kg monkey located at (4 m, 6 m) ; and a 200kg monkey located at (2 m,  8 m) . 2. Center of Monkeys II. A banana is often located in the center of monkeys. Find the center of mass for each of the following mass distributions: (A) A 30-kg monkey is 20-m away from a 90-kg monkey. Find the center of mass of the monkeys. (B) Find the center of mass of the following monkeys: a 20-kg monkey at (0,0) ; a 40-kg monkey at (5 m,10 m) ; and a 60-kg monkey at (15 m, 5 m) . (C) The following freshly baked banana cookies have uniform density: A rectangular cookie of mass 4m has coordinates (0,0), (2L,0), (0, L), (2L, L) ; a circular cookie of mass 6m has diameter 2 L and center located at (4 L,2 L) ; and a long, thin cookie of mass 5m has endpoints (2L,2L), (0,3L) . (D) A banana-cream pie of radius R and uniform mass density is centered about the origin. There is a circular

R  ,0  . 2 

hole of radius R / 2 centered about 

(E) A monkey balances a 40-kg wooden mallet with uniform density on a fulcrum as illustrated below. The monkey then saws the mallet into two pieces at its center of gravity. How much does each end weigh? (The answer is not 192 N. Why?) [Note: This problem continues onto the next page.]

4 cm 20 cm

18 cm 75

Volume 1: Mechanics Note: The handle and the head are not the two ends. The two ends are the two pieces of the mallet that remain after sawing – 18 cm of the handle being one end, and 2 cm of handle connected to the head being the other end. (F) A 50-kg monkey stands at one end of a 10-m long plank. A fulcrum is placed underneath the plank, 3 m away from the monkey. The system is in static equilibrium – that is, the fulcrum is positioned beneath the center of mass of the system. What is the mass of the plank? *3. Invisible Monk. An 80-kg chimpanzee is at (4 m,2 m) and a 20-kg lemur is at (12 m,3 m) . In addition, there is an invisible 40-kg monkey somewhere. The center of mass of the whole system (i.e. the chimpanzee, lemur, and invisible monkey) is at (2 m,1 m) . Where is the invisible monkey? That is, find his x- and y-coordinates. *4. Hippy Longtalking. Hippy Longtalking wishes to lift a 20-meter long, 200-kg platform with a 150-kg man standing 4 meters from one end. If she lifts the platform in the center, it will tip over. Where should she lift the platform such that it won’t tip over? *5. Center of Monkeys III. The top view of the building below is filled with monkeys with approximate uniform density. Find the location of the center of mass of the monkeys.

y

2m

4m 2m

x 4m

76

Creative Physics Problems – Chris McMullen, Ph.D. *6. Banana-Cream pie I. The banana-cream pie illustrated below has uniform density. Find the location of its center of mass.

y

2m

2m

4m

2m

x 4m *7. X Marks the Spot. Consider the map of Monkey Island drawn below. A treasure chest of golden bananas is buried beneath the center of mass of Monkey Island. Monkey Island has uniform density. Monkey Island is the entire shaded region (rectangle and circles). Where is the treasure.

y

10 km

50 km 20 km

x 50 km 77

Volume 1: Mechanics *8. Banana Acres. Consider the map of Banana Acres drawn below. A treasure chest of golden bananas is buried beneath the center of mass of Banana Acres. Banana Acres has uniform density. Banana Acres is the entire shaded region (square minus circle). Where is the treasure?

y

R 5R

2R

x 5R *9. Center of Attention. The shaded object illustrated below has uniform mass density  . If you stand in the center of mass of this object, all of the monkeys will howl at you. Where is the center of mass?

y

7D 5D 3D

x

78

Creative Physics Problems – Chris McMullen, Ph.D. 10. Non-Uniform Monkeystick. A monkeystick with endpoints (0,0) and (L,0) has non-uniform density   axb where a and b are constants. Show that xCM 

b 1 L. b2

11. Buried Treasure. As illustrated below, Monkey Isle is bounded by the parabola y  x 2 and the line y  4 . Monkey Isle has uniform density   2 . SI units have been suppressed. A monkey pirate buried a treasure in the center of mass of Monkey Isle. (A) Which of the three forms is appropriate for dm ? (B) What are suitable integration limits? (C) Which of the integrals must be performed first? (D) Write a symbolic equation to find the mass of Monkey Isle. Make a series of clear substitutions to express your integrand in terms of x and y (with no other variables). Include the limits of your integrals. (E) Integrate over the first variable – determined by your answer to (C). Clearly show your work. (F) Now integrate over the second variable. Clearly show all steps. Circle your final answer for the mass of Monkey Isle. (G) Write a symbolic equation for xCM . (H) Carry out steps (D) thru (F) to find

xCM , clearly showing all work. Circle your final answer for xCM . (I) Carry out steps (D) thru (F) to find yCM , clearly showing all work. Circle your final answer for yCM . *12. Taking Cover. Osamonk Been Rotten’s evil lair is about to be destroyed in an explosion. 00 knows that a vault is located in the lair’s center of mass – a good place to take cover. The lair is shaped like a solid semicircle with 50-m radius; the straight section runs along the x-axis and the rest extends in the positive y direction. The lair has non-uniform mass density    r , where  is a constant. At what coordinates will 00 find the vault? 13. Balancing Act I. A monkeystick has endpoints (L,0) and (2 L,0) and non-uniform mass density

  x 2 , where  is a constant. A monkey must balance this monkeystick on his fingertip in order to score well in this event. (A) What are the SI units of  ? (B) Solve for  in terms of m and L . (C) Express the center of mass xCM in terms of L . 14. Balancing Act II. The shaded semicircular ring illustrated below has inner radius R0 , outer radius

2R0 , and non-uniform mass density  (r )  r , where  is a constant. A monkey must balance this object on his fingertip in order to score well in this event. (A) Solve for  in terms of m and R0 . (B)  Express the center of mass vector rCM in terms of R0 and Cartesian unit vectors.

79

Volume 1: Mechanics

y

x

15. Banana Creations. Find the center of mass of these delicious creations. (A) An infinitesimally thin banana of uniform mass per unit length  in the shape of ¼ of a circle has endpoints R,0 and 0, R  . (B) A solid hemispherical banana cake with radius R and uniform mass distribution has its circular bottom lying in the xy plane and extends into the positive z-direction. *16. Center of Monkeys IV. A banana is often located in the center of monkeys. Find the center of mass for each of the following mass distributions: (A) A rod of density 1  a x has endpoints Another rod of density 2  by 3 has endpoints ( L,0), ( L,3L) . Note: (0,0), (2L,0) . a  3, b  4, L  2 in SI units. (B) A solid hemispherical banana cake with radius R and uniform mass distribution has its circular bottom lying in the xy plane and extends into the positive z-direction. A banana pie with non-uniform density   a x has the shape of a trapezoid, as illustrated below. Note: a  2 in SI units.

(0,2 m) (2 m,1 m)

*17. Double the Math. A monkey bakes a banana cream pie shaped like a triangle (as illustrated below) 2 with non-uniform density   x , where  is a constant.

80

Creative Physics Problems – Chris McMullen, Ph.D.

y ( L,0)

x

( L,0)

(L,0)

(A) Express the mass of the pie in terms of  and L . (B) Find the y-component of the center of mass of the banana cream pie. The only unknown appearing in your final answer should be L . *18. Banana-Cream Pie II. A monkey bakes a semicircular banana cream pie with non-uniform density

  r 2 , where  is a constant. y

R0

x

(A) Express the mass of the pie in terms of  and R0 . (B) Find the center of mass of the banana cream pie. The only unknown appearing in your final answer should be R0 . *19. Square Pie. The square pie illustrated below has non-uniform mass density  ( x, y)  xy , where SI units have been suppressed.

81

Volume 1: Mechanics

y

L

L

x



Express the center of mass vector rCM for the square in terms of L and Cartesian unit vectors. *20. Triangular Division. A monkey locates the center of mass of an equilateral triangle and then cuts the triangle into two pieces, as illustrated below. To the monkey’s surprise, the two pieces have different mass. Draw and label a coordinate system with a clear choice of origin.

CM

not drawn to scale

(A) Determine the coordinates of the triangle’s center of mass. As always, you must clearly show all work, beginning with valid starting equations. (B) Determine the ratio of the mass of the bottom piece to the mass of the top piece. 21. Bouncing Coconut. A monkey drops a 50-g rubber-coated coconut from a height of 10 m. The coconut is in contact with the ground for 30 ms. The maximum height of the rebound is 8 m. (A) What is the momentum of the coconut just before striking the ground? (B) What is the momentum of the coconut just after striking the ground? (C) What is the average force exerted on the coconut during impact? (D) How much energy does the coconut lose during the collision? 22. Walking the Plank. A 50-kg monkey stands at one end of a 150-kg plank. The 20-m long plank is on ice. There is friction between the monkey and plank, but not between the plank and ice. The monkey and plank are initially at rest. The monkey walks forward at a constant rate of 2 m/s relative to the ice until the monkey walks off the plank. (A) As the monkey walks along the plank, list the forces acting on the system. Which are internal, and which are external? What is the net external force? Is momentum conserved for the system? (B) What is the final velocity (include direction) of the plank? (C) How much 82

Creative Physics Problems – Chris McMullen, Ph.D. kinetic energy has the monkey-plank system lost or gained (compared to the initial kinetic energy of the system)? 23. Monkey Mayhem I. It’s a jungle out there when monkeys drive bananamobiles on frictionless ice. (A) A 50-kg bananamobile traveling 8 m/s to the east collides head-on with a 100-kg bananamobile traveling 12 m/s to the west. The collision is completely inelastic. What is the final velocity of the bananamobiles? (B) A 600-kg bananamobile traveling 3 m/s to the east collides head-on with a 150-kg bananamobile traveling 9 m/s to the north. The collision is completely inelastic. What is the final velocity of the bananamobiles? 24. Monkey Mayhem II. A 600-kg bananamobile traveling 90 m/s to the north collides head-on with a 400-kg bananamobile traveling 50 m/s to the south. The collision is elastic. Find the final velocities of the bananamobiles. *25. Sumo Wrestling Orangutans I. This sumo wrestling match is held on horizontal frictionless ice. A 20-kg sumo wrestling orangutan traveling 3 m/s to the north collides head-on with a 10-kg sumo wrestling orangutan traveling 8 m/s to the south. The orangutans bounce off each other after the collision, and the 20-kg sumo wrestling orangutan travels 4 m/s to the south after the collision. (A) Find the final speed of the 10-kg sumo wrestling orangutan. (B) Which way is the 10-kg orangutan moving after the collision? (C) Is the collision elastic? (D) If the sumo wrestling orangutans are in contact with each other for 200 ms, what is the average force they exert on one another during the collision? *26. Sumo Wrestling Orangutans II. This sumo wrestling match is held on horizontal frictionless ice. A 20-kg sumo wrestling orangutan traveling 3 m/s to the north collides head-on in an elastic collision with a 10-kg sumo wrestling orangutan traveling 9 m/s to the south. (A) Find the final velocity (including direction) of each sumo wrestling orangutan. (B) What percentage of the kinetic energy does the system lose if the collision is instead completely inelastic? *27. Sumo Wrestling Orangutans III. This sumo wrestling match is held on horizontal, frictionless ice. Consider the illustration below. The orangutan on the left with mass 2m is moving toward the other orangutans with speed 150 . The other orangutans are initially at rest. Assume all collisions to be head-on, elastic collisions.

150

rest

rest

3m

15m

2m

(A) Express the final velocities of all three orangutans in terms of  0 . Include direction. (B) How many collisions are there? Explain. (C) Suppose instead that the collisions are all completely inelastic. What percentage of the initial kinetic energy is lost after all of the collisions?

83

Volume 1: Mechanics *28. Sumo Wrestling Orangutans IV. This sumo wrestling match is held on horizontal, frictionless ice. One orangutan with mass 4m is moving with speed 3 0 . A second orangutan with mass 3m and is moving with speed 2 0 . The orangutans are moving in the directions shown prior to the collision. They collide in a completely inelastic collision. Determine the magnitude and direction of the final velocity of each orangutan.

3 0

2 0

60º

60º

4m

3m

29. Look Here. You find some monkeys playing marbles with coconuts. Upon closer inspection, you realize that the coconuts are painted like eyeballs. A coconut of mass 4m traveling 3v0 to the north collides head-on with a coconut of mass 8m traveling 2v0 to the south. The collision is elastic. Find the final velocities (include direction). *30. Lunch at Last! A 40-kg coyote running 7 m/s west catches a 20-kg roadrunner jogging 2 m/s west. They stick together after the collision and fall off a 100-m cliff. Find the speed of the coyote and roadrunner just before striking the ground below. *31. Stranded I. A 48-kg monkey is at rest on horizontal frictionless ice. Fortunately, he is holding a 12kg watermelon. (A) If he throws the watermelon 8 m/s to the south, what will be the velocity (i.e. speed and direction) of the monkey? (B) Now that the monkey is moving, he decides that he would prefer to be at rest. The monkey has two 2-kg shoes that he can part with. How fast, and in what direction, must the monkey throw his shoes in order to come to a complete stop? (C) If the watermelon was in contact with the monkey’s hand for 400 ms as he was throwing it in part (A), what was the average force of the monkey’s throw? *32. Stranded II. A 20-kg monkey is at rest on horizontal frictionless ice. Fortunately, he is holding a 15-kg watermelon. (A) If he throws the watermelon 4 m/s to the south, what will be the velocity (i.e. speed and direction) of the monkey? (B) The monkey continues to move with the final velocity computed in (A) when he collides head-on with a 10-kg monkey traveling 9 m/s to the south. If the collision is elastic, what are the final velocities of each monkey? *33. Iceslide. A 200-kg polar bear slides down an iceslide. When he reaches the bottom, he slides horizontally at a speed of 5 m/s until bumping into a 100-kg polar bear sliding horizontally at a speed of 8 m/s in the opposite direction. (A) What is the final speed of each polar bear if the collision is inelastic? (B) Repeat if the collision is perfectly elastic. (C) Repeat if the collision is inelastic, but the 2 nd polar bear 84

Creative Physics Problems – Chris McMullen, Ph.D. is traveling 90º to the first polar bear. (D) Consider (B) again. If the polar bears are in contact with each other for 50 ms, what is the average force they exert on one another during the collision? 34. Monkey Pile. A 40-kg monkey is running with velocity 5 m/s at an angle of 0º, a 60-kg monkey is running with velocity 3 m/s at an angle of 40º, and an 80-kg monkey is running with velocity 2 m/s at an angle of 200º. The three monkeys collide at the same moment in a perfectly inelastic collision. (A) Find the final velocity of the trio. (B) How much energy is lost/gained during the collision? (C) What impulse does the 40-kg monkey experience during the collision? 35. Rescue Aperation. A male ape of mass m in distress stands on an island in a crocodile-infested river. A female ape of mass 2m swings down on a vine, catches the male ape at the bottom of the arc, and just barely reaches the other side with the male ape in her arms. The initial angle is  0 , and the final angle is  .

L 2m

0  safety m

(A) Derive an equation for  . (B) What is  if  0 

 2

? (C) What is  0 if  

 6

?

*36. Double the Tension. A 20-kg monkey and 30-kg monkey swing from 10-m long ropes as illustrated below. Each monkey begins from rest from the position shown. (The 20-kg monkey starts a little earlier in order to time their collision to occur at the bottom of their arcs.) The collision is completely inelastic.

60º

45º

30 kg 20 kg How high do the monkeys rise after the collision, and which way do they swing?

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Review 3: Conservation of Energy and Momentum *1. No Work. A 20-kg monkey was supposed to slide 10 m down a 45º incline from rest. The coefficient of friction would have been 2 / 2 . However, the monkeys are on strike. If the monkeys had not gone on strike: (A) How much work would have been done by the normal force? (B) How much work would have been done by friction? (C) How much power would have been provided by gravity? Overworked and Underfed!

Shame, Shame, Inhumane!

Don’t be a donkey. Support monkeys.

Will work for bananas.

*2. The Scabs. A monkey was scheduled to ride the roller coaster illustrated below. Since the monkeys are on strike, a 50-kg donkey signed up to take the monkey’s place. The donkey compressed the 40 N/m spring 8 m from equilibrium, then releases from rest. The track is frictionless, except between points C and D.

80 m

A C

60 m

B

D 20 m

(A) What is the speed of the donkey at point A? (B) What is the speed of the donkey at point B? (C) What coefficient of friction is needed between points C and D for the donkey to come to rest at point D? 86

Creative Physics Problems – Chris McMullen, Ph.D. *3. The Brawl. The disgruntled 20-kg monkey sees the 50-kg donkey walking 3 m/s to the east. The monkey runs 6 m/s to the east, colliding with the donkey head-on. (A) Find the final velocities (that implies direction, too) of the monkey and donkey if the collision is elastic. (B) What average force does the monkey exert on the donkey during the collision if the duration of the collision is 500 ms? *4. Pin the Tail. Instead of pinning a tail on the donkey, the donkey wants you to pin a tail onto the center of mass of a rod with endpoints (0, L) and (0,2L) and non-uniform density   y where  is a constant. (A) Solve for the mass of the rod in terms of  . (B) Solve for the center of mass of the rod. The only unknown in your final answer should be L . *5. Back to Work. The starving monkeys finally return to work. Excited, a 20-kg monkey runs for 8 seconds with a velocity   3t 2 (with SI units suppressed). (A) What is the final momentum of this monkey? (B) What is the acceleration of the monkey at t  3 sec ? (C) What is the net force acting on the monkey at t  5 sec ? (D) How far does the monkey travel? (E) What is the average power delivered to the monkey?

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6. Banana Jump. As illustrated below, one monkey uses a vertical spring with spring constant 1200 N/m to launch another monkey into the air in order to reach bananas suspended above. If the spring is compressed too little, the 4-kg monkey will not reach the bananas. However, if the spring is compressed too much, the monkey will overshoot the bananas, and the judges will deduct points. At equilibrium, the bananas are 2 m above the monkey’s reach. How far x should the spring be compressed in order for the monkey to barely reach the bananas? (The gray color shows the spring at equilibrium.)

Not Drawn to Scale

2m

Equilibrium Compressed

x

7. Stop the Box. Two monkeys are separated by horizontal frictionless ice (dotted line below). One monkey slides a 40-kg box with a velocity of 50 m/s to the left. The second monkey slides a 25-kg box to the right with the hope of stopping the 40-kg box.

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Creative Physics Problems – Chris McMullen, Ph.D.

10

50 m/s

25 kg

40 kg

Frictionless Ice

(A) If the collision is elastic what should be the initial velocity of the 25-kg box in order to stop the 40-kg box? (B) If the collision is instead completely inelastic, what should be the initial velocity of the 25-kg box in order to stop the 40-kg box? 8. Monkey Golf. As illustrated below, a monkey hits a golf cube2 with a putter, hoping that the golf cube will make it through the 1-m diameter loop and fall into the hole. The solid line represents frictionless metal. The dotted line represents carpet. The coefficient of friction between the golf cube and the carpet is ¼. (The gray boxes are drawn to help you visualize the ideal path of the golf cube.) Not Drawn to Scale

20 m Hole

Carpet

Frictionless Metal

(A) What is the minimum initial speed 0 of the golf cube needed to score a hole-in-one? (B) What is the minimum initial speed 0 of the golf cube needed to complete the loop without falling away from the track? 9. Monkey Power. As illustrated below, a monkey turns a crank rapidly to pull a 40-kg load of bananas up the incline at a constant speed of 5 m/s. The coefficient of kinetic friction between the box of bananas and the incline is

1 2 3

. (Neglect friction between the cord and the pulley.) How much

horsepower does the monkey deliver to the load of bananas?

2

Fortunately, a golf cube does not roll. (The problem would be more involved for a golf ball.)

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Bananas

30º Crank

10. Coconut Cream Pie. A monkey bakes a coconut cream pie in the shape of an equilateral triangle with one edge on the x-axis, symmetric about the y-axis, with edge length L . Find the center of mass of the coconut cream pie if (A) the pie is uniform and (B) if the mass density is   y .

Monkey Coaster Specifications: Consider the monkey coaster illustrated below. The escalator pulls the monkey car up the initial incline. At the top of the initial incline, the monkey car detaches from the escalator and slides down the curved track with an initial speed of 20 m/s. The monkey car collides with a banana car in a completely inelastic collision. The monkey car and banana car then travel through the loop-the-loop. When the monkey car and banana car reach the horizontal section of track at the end, a coconut car is launched via a spring mechanism. The coconut car collides with the other cars in an elastic collision, which brings the monkey car and banana car to a halt. (The monkey car and banana car remain stuck together at the end of the ride.)

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Creative Physics Problems – Chris McMullen, Ph.D.

escalator pulls the monkey car 75 2 m along the incline at a constant speed of 20 m/s monkey car 60 kg 45º

start here

monkey car detaches from the escalator at the top with an initial speed of 20 m/s

** track is frictionless after leaving the escalator ** 28-m diameter banana car (initiallyloop-theat rest) 15 m

spring is compressed 4 m from equilibrium to launch the coconut car to the left finish coconut car 40 m

80 kg

20 kg

loop completely inelastic collision between monkey car and banana car

91

final position of monkey and banana cars (end at rest after elastic collision with coconut car)

Volume 1: Mechanics *11. How much work must the escalator do to against gravity in order to lift the monkey car to the top of the incline? *12. How much power must the escalator deliver against gravity in order to lift the monkey car to the top of the incline? *13. What are the momentum and kinetic energy of the monkey car while the escalator pulls it up the incline? *14. What is the speed of the monkey car just before colliding with the banana car? *15. What is the speed of the monkey car just after colliding with the banana car? *16. How fast are the monkey and banana cars moving at the top of the loop? *17. What normal force do the monkey and banana cars experience at the top of the loop? *18. How fast are the monkey and banana cars moving just before colliding with the coconut car? *19. How fast must the coconut be moving just before colliding with the monkey and banana cars such that the monkey and banana cars come to rest just after the collision? *20. What spring constant is needed to give the coconut car the needed velocity? *21. What are the magnitude and direction of the velocity of the coconut car just after the collision with the monkey and banana cars?

Do You Suffer from a Bad Case of Physicsitis? *Diagnosis: Symptoms of physicsitis are demonstrated with the following examples. A student pounds his head against a brick wall with a frequency of 5 Hz. (A) What is the period of the student’s head-pounding? (B) How many times does the student pound his head against the wall in one minute? An 80-kg student dreams about the ghost of Isaac Newton chasing her as she accelerates at a constant rate of 5 m/s2 for 6 seconds along horizontal ground. How much work does gravity do? A student who thinks he turned into a monkey swings back and forth on a vine tied to a tree branch. Where is the potential energy greatest? Where is the kinetic energy greatest? Where is the total energy greatest? Be clear.

92

Creative Physics Problems – Chris McMullen, Ph.D. An 80-kg student dreams about the ghost of Isaac Newton chasing her as she slides 20 m down a 30º incline. (A) In the equation for the work done by normal force, what is  ? (B) In the equation for the work done by friction, what is  ? (C) In the equation for the work done by gravity, what is  ? A student connects a monkey to a horizontal spring on frictionless ice. The other end of the spring is attached to a wall. The student compresses the spring and lets go from rest. Where is the potential energy greatest? Where is the kinetic energy greatest? Where is the total energy greatest? Be clear. *Treatment: If you exhibit symptoms of physicsitis, seek professional medical help immediately. A student releases a 5-kg physics textbook from rest from the top of a 50-m tall building. The textbook is directly above a stuffed monkey lying on the ground below. The textbook loses 250 J (relative to the ground) of energy during the trip due to air resistance. What is the speed of the physics textbook just before squashing the stuffed monkey? *Side Effects: 90% of the students who take banana pills make a full recovery; 10% go bananas. A 90-kg student who thinks he turned into a monkey swings back and forth on a 10-m long vine suspended from a tree branch. The vine makes a maximum angle of 60º with the vertical.

10 m

60º

(A) How much does the height of the student change as  varies from 60º to 0º? (B) How fast is the student moving at the bottom of the arc? *Regression: Discontinued usage of banana pills may lead to more pronounced symptoms. A student dresses up in a monkey costume and visits an amusement park. The student screams Newton’s laws while riding the roller coaster illustrated below. The initial speed of the roller coaster is 50 m/s. Neglect friction.

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B

50 m/s

45 m

(A) How fast is the roller coaster moving at point B? (B) How high is the roller coaster when it comes to rest? *Rehabilitation: Patients with severe cases of physicsitis may be sent to the Monkey Farm. A 60-kg student climbs 20 m vertically upward on monkey bars in 40 seconds. (A) How much work is done against gravity? (B) What average power acts against gravity?



Monkeys wearing hats with I’s and J’s on them drag a student with a collective force of F  8ˆi  2ˆj



along a displacement s  3ˆi  6ˆj . How much work do the monkeys do? A monkey connects one end of a horizontal spring to a wall and connects a 40-kg student to the free end. The spring is compressed 2 m from equilibrium and released from rest. When the spring returns to equilibrium, the student is moving 4 m/s. Neglect friction. (A) What is the spring constant? (B) What is the acceleration of the student when the spring is compressed 2 m from equilibrium?

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Creative Physics Problems – Chris McMullen, Ph.D.

7 Rotation 1. Tired. A monkey puts his younger brother in a 70-cm diameter tire and gives it a big shove forward. It topples over 15 seconds later, after completing 60 revolutions. (A) How far does the tire travel? (B) What is the initial angular speed of the tire? 2. Physics Makes You Dizzy I. A monkey ties a rope to your leg, drags you outside, and sets you on a skateboard. The monkey spins you in an 8-m diameter circle. Starting from rest, you rotate: (i) with uniform angular acceleration from t  0 to t  5 sec (ii) at a constant rate of 0.3 rev/sec from t  5 sec to t  25 sec (iii) with uniform deceleration for 15 rev from t  25 sec until coming to rest (A) What is your angular acceleration during Part (i)? (B) What is your acceleration at t  4 sec? (C) How many revolutions do you complete during Part (i)? (D) How far do you travel during Part (ii)? (E) What is your acceleration during Part (ii)? (F) What is your tangential acceleration during Part (iii)? (G) For how much time do you decelerate during Part (iii)? (H) What is your average angular speed for the entire trip? *3. Physics Makes you Dizzy II. As you begin to read this problem, a monkey ties a rope to your leg, drags you outside, and sets you on a skateboard. The monkey spins you in a

60



-m diameter circle.

Starting from rest, you rotate with uniform angular acceleration until your angular speed is

2 rad/s . 3

You travel 300 m. (A) How many revolutions do you complete? (B) For how much time does the monkey spin you in a circle? (C) What is your tangential acceleration? (D) What is your acceleration at t  5 sec? *4. Head-Spinning. A monkey shows symptoms of being under the influence of too much physics. Specifically, his head begins spinning with an initial angular speed of 5 rad/s with an angular acceleration of  rad/s2. His head completes 36 revolutions during this uniform angular acceleration. (A) What is the final angular speed of the monkey’s head? (B) For how much time does the monkey’s head rotate? *5. Physicsitis. A monkey shows symptoms of physicsitis. Specifically, his stomach begins churning (from rest) with a uniform angular acceleration of 10 rad/s2. His stomach completes 90 revolutions. (A) What is the final angular speed of the monkey’s stomach? (B) For how much time does the monkey’s stomach rotate?

95

Volume 1: Mechanics *6. Wheel of Torture. The Wheel is given a final spin. The Wheel begins spinning at a rate of 0.4 rev/sec and lands on Bankrupt 7.4 seconds later. The Wheel has a 3-m diameter. (A) What was the average angular deceleration of the Wheel? (B) How many revolutions did the Wheel complete? (C) What was the initial speed of the pointer, located at the edge of the Wheel? (D) What was the initial centripetal acceleration of the pointer? *7. Monkey-Go-Round I. A monkey puts her baby on the edge of a 4-m diameter merry-go-round, initially at rest. The baby holds on tight. The monkey grabs the edge of the merry-go-round and runs around in a circle, providing uniform angular acceleration to the merry-go-round. The monkey runs for 8 seconds, reaching a final speed of 4 m/s. (A) What is the baby’s tangential acceleration? (B) What is the baby’s angular acceleration? (C) What is the baby’s final angular speed? (D) How far does the baby travel? (E) How many revolutions does the baby complete? (F) What is the baby’s final centripetal acceleration? *8. Monkeyball. A monkey throws a monkeyball up into the air with an initial angular speed of  rad/s. The monkeyball completes 4 revolutions during its flight. (A) What is the angular speed of the monkeyball at the top of its trajectory? (B) How high does the monkeyball rise? *9. Supermonk. Faster than a speeding photon,3 more powerful than a black hole,4 capable of leaping across the universe in a single bound,5 Supermonk hears the plea of desperate physics students across the world: There just isn’t enough time in the day to complete their homework assignments. Supermonk begins running (and, when necessary, swimming) to the east at a super-high speed in order to slow the earth’s rotation. The earth uniformly decelerates until there are 36 hours in a ‘day.’ Supermonk runs continuously for 720 hours to achieve this amazing feat. (A) What is the earth’s uniform angular deceleration? (B) How many revolutions does the earth complete while Supermonk runs? *10. Monkey Torque. A monkey suspends 35 g from the 80-cm mark of a 100-g uniform meterstick. Where must the fulcrum be placed in order to achieve static equilibrium? (Draw and label a diagram to illustrate your answer.) *11. Apples and Oranges. A monkey places an apple at the 10-cm mark of a uniform meterstick, and an orange at the 80-cm mark. The fulcrum is underneath the 50-cm mark, and the system is in static equilibrium. The combined mass of the apple and orange is 300 g. What is the mass of the apple? 12. Happy Monksgiving! In the illustration below, one 40-kg monkey hangs from one end of a 100-kg uniform rod that is 12 m long, another 60-kg monkey stands above the pivot (center of rotation), and a 20-kg turkey lies 1 m from the opposite end. Also, the standing monkey pulls on a rope as shown with a force of 200 N.

with an uncertainty of 300,000,000 m/s. from a distance of 100,000 lightyears or more. 5 through a wormhole. 3 4

96

Creative Physics Problems – Chris McMullen, Ph.D. Draw picture of turkey here.

60º pivot

5m 3m 8m (A) Find the torque exerted by the dangling monkey. (Relative signs are crucial.) (B) Find the torque exerted by the standing monkey’s weight. (C) Find the torque exerted by the tension in the rope (pulled by the standing monkey). (D) Find the torque exerted by the turkey. (E) Which torque is missing? Find it. (F) What is the net torque exerted on the rod? 13. Rotational Equilibrium. The uniform boom drawn below is 20-m in length and weighs 600 N. The pivot is located 5 m from the left end. Also, F1 = 2400 N, F2 = 1800 N, F3 = 3600 N, and F4 = 1200 N.

 F2

 P

5m 52º pivot

 F1

52º 42º

 F4

 F3

10 m

7.5 m

     (A) Find the torques exerted by F1 , F2 , F3 , and F4 . (B) What must be the magnitude of P for the system to be in rotational equilibrium? *14. Monkeyin’ Around. As illustrated below, a 60-kg monkey hangs from one end of a 12-m long rod, while an 80-kg monkey stands 4-m from the opposite end. A fulcrum rests beneath the center of the rod.

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30º fulcrum

(A) What torque is exerted by the hanging monkey? (B) What torque is exerted by the standing monkey? (C) Where, precisely, should a 40-kg box of bananas be placed in order to achieve static equilibrium? 15. Uniform Boom. In the diagram below, the uniform boom weighs 300 N and is 20 m long. The load weighs 3500 N and is connected 18 m from the hinge. The system is in static equilibrium. When No-Mo Yo-Yo cuts the support cable, the load lands on a physics textbook – unless, of course, the Powder Puff Pearls save the day.

10 m 40º 8m

(A) What is the tension in the support cable while the system is in equilibrium? (B) Find the magnitude and direction of the force supplied by the hingepin while the system is in equilibrium. *16. Crane. In the diagram below, the crane weighs 700 N and is 20 m long. The load weighs 6000 N and is connected 16 m from the hinge.

30º 16 m

98

Creative Physics Problems – Chris McMullen, Ph.D. (A) What is the tension in the support cable? (B) Find the horizontal and vertical components of the force supplied by the hingepin. *17. Monkey Engineering, Inc. A monkey applies his knowledge of physics in order to store his precious bananas. The uniform boom has a mass of 3 kg and is 12 m long; its lower end is connected to a hingepin. The boom is supported by a tie rope that connects to a wall. The tie rope can sustain a maximum tension of 30 N.

4m 30º

30º

(A) What maximum load can the boom support without snapping the tie rope? (B) Find the maximum horizontal and vertical components of the force exerted on the hingepin. (C) Suppose that the bananas have a mass corresponding to the maximum load and the tie rope is cut. What will be the initial angular acceleration of the system? 18. Vector Product. The Powder Puff Pearls save the day by lassoing 40-kg No-Mo Yo-Yo and then flying



around. The distance between No-Mo Yo-Yo and the axis of rotation is described by r  3ˆi  8ˆj  24kˆ .



The force Possum exerts on No-Mo Yo-Yo is F  16ˆi  8ˆj  2kˆ . SI units have been suppressed for convenience. (A) Express the torque that the Possum exerts on the monkey in terms of Cartesian unit





vectors. (B) What is the angle between r and F ?





 





*19. Monkey Vectors. Given A  3ˆi  3ˆj  2kˆ and B  3ˆi  ˆj  2 3kˆ , find: (A) A  B , (B) A  B ,





and (C) the angle between A and B . *20. Monkey Torques. Find the torques exerted on the following monkeys:

     (B) r  kˆ , F  kˆ (C) r  kˆ , F  ˆi  ˆj , F  ˆj     (D) r  2ˆi  3ˆj  4kˆ , F  2ˆj  kˆ (F) r  ˆi  2kˆ , F  3ˆi  6kˆ 

(A) r  kˆ

*21. Cross Monkeys. What do you get when you cross monkeys with multiplication? 99

Volume 1: Mechanics (A) Find kˆ  ˆi . (B) Find ˆi  kˆ .

(C) Find ˆj  ˆj . (D) Find ˆi  (ˆi  ˆj) .     (E) Find A  B where A  2ˆi  3ˆj  4kˆ and B  2ˆj  kˆ .     (F) Find A  B where A  ˆi  2kˆ and B  3ˆi  6kˆ .       (G) Find ( A  B)  C where A  2ˆi  ˆj , B  ˆj  2kˆ , and C  ˆi  kˆ .       (H) Find A  (B  C) where A  2ˆi  ˆj , B  ˆj  2kˆ , and C  ˆi  kˆ .





*22. Monkey Puzzle. A monkey knows that A  ˆi  2ˆj and C  2ˆi  kˆ , and wishes to find a third











vector B where C  A  B . When the monkey proceeds to find B , something ‘funny’ happens. (A) Exactly, what is ‘funny’? (B) Why does it happen? 23. Monkey of Inertia I. Find the moment of inertia of each object below about the indicated axis. Assume each object to have uniform density unless otherwise noted. (A) A dumbbell consists of a monkey of mass 5m and a monkey of mass 3m connected by a light, inextensible rod of length L . The dumbbell rotates about an axis perpendicular to the rod that passes through the center of mass of the system. Treat the monkeys as point-like objects. (B) Repeat the previous problem if the dumbbell rotates about an axis perpendicular to the rod that passes through the monkey of mass 3m . (C) Repeat the previous problem if the dumbbell rotates about an axis perpendicular to the rod that passes through the monkey of mass 5m . (D) Three monkeys are connected by light, inextensible rods as illustrated below. The system rotates about the z -axis. Treat the monkeys as point-like objects.

y

6m

2L

7m

4m 5L

x (E) Repeat the previous problem if the system rotates about the x -axis. (F) Repeat the previous problem if the system rotates about an axis parallel to the z -axis that passes through the center of mass of the system. (G) A solid disc of radius R in the xy plane centered about the origin has a circular hole cut out of it of radius

R R  centered about the point  ,0  . The system rotates about the 2 2 

z -axis. *24. Monkey of Inertia II. A monkey connects three small (point-like) 5-kg oranges together with three 4-m long lightweight (relatively massless) rigid rods to form the equilateral triangle illustrated below. 100

Creative Physics Problems – Chris McMullen, Ph.D. (A) Find the moment of inertia of this system about axis 1. (B) Find the moment of inertia of this system about axis 2.

axis 1

L axis 2

L 3

*25. Monkeystick. Find the moment of inertia of a uniform monkeystick about an axis perpendicular to the rod and passing through the rod a distance that is one-third of its length from one end. *26. Monkey Shapes. A dumbbell consists of a solid 10-kg sphere of 30-cm radius and a hollow 15-kg sphere of 20-cm radius joined by a 5-kg rod that is 80 cm long. (A) Where is the center of mass of the system? (B) A fulcrum is placed beneath the midpoint of the rod. What is the moment of inertia of the system about the axis of rotation? (C) What is the initial angular acceleration of the system? *27. Smartbell. A monkey uses recycled physics textbooks to manufacture the smartbell illustrated below. The smartbell consists of an infinitesimally thin, hollow sphere on the left and a solid sphere on the right connected by a uniform rod. The hollow sphere has a mass of 15 kg and radius of 10 m. The solid sphere has a mass of 20 kg and a radius of 5 m. The rod has a mass of 6 kg and length of 40 m. (Assume that holes were not drilled into the spheres to connect the rod.)

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10 m Find the moment of inertia of the smartbell. *28. Monkeysticks. Find the moment of inertia of (A) a monkeystick with endpoints (0,0), ( L,0) of uniform linear mass density  about the axis x  L / 4 and (B) a monkeystick with endpoints (0,0), ( L,0) of non-uniform linear mass density   3x about the axis x  0 . 29. No-Mo Baby Toys. Why did No-Mo Yo-Yo grow up to be such an evil monkey? Find the moment of inertia about the axis of rotation of each of his baby toys described below. Each toy has uniform mass density unless otherwise stated. (A) A thin ring of radius R in the xy plane centered about the origin

R  ,0  . (B) A solid disc of 2  R radius R in the xy plane centered about the origin has a circular hole cut out of it of radius 2 R  centered about the point  ,0  . The system rotates about the z -axis. (C) A thin spherical shell of 2  radius R rotates about its center. (D) A rectangular plate of width W and length L rotates about an axis perpendicular to the plate that passes through its center of mass. (E) Two rods of length L have endpoints 0,0, L,0 and 0,0, 0, L  and linear mass densities 1 ( x)  2 x and 2 ( y)  y 2 , rotates about an axis parallel to the y -axis that passes through the point 

respectively. The system rotates about an axis perpendicular to the rods that passes through the origin. *30. Monkey of Inertia III. Consider the uniform monkey tail illustrated below; it has the shape of a semicircular arc of radius R0 . Express the moment of inertia of the monkey tail about the axis y 

R0 4

in terms of the mass m of the monkey tail and the radius R0 only; no other symbols may appear in your final expression.

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y

x *31. Non-Uniform Monkeystick. A nonuniform monkeystick has density  ( x)  ax where a and b are constants. The system rotates about the axis shown below. Derive an equation for the moment of inertia in terms of the mass of the rod m , its length L , and the constant b only. b

axis *32. Bananaball. A spherical monkeyball has non-uniform density   r , where  is a constant. Derive an equation for the moment of inertia of the monkeyball about an axis through its center in terms of its mass m and radius R only. No other variables may appear in your final expression. *33. Bananacubes. Find the moment of inertia for each of the following uniform bananacubes. (A) A solid cube of edge length L rotates about an axis through one edge. (B) A solid cube of edge length L rotates about an axis through its center and perpendicular to one side. (C) A hollow cube of edge length L rotates about an axis through one edge. *34. Physics Is a Slice of . The slice of banana-cream pie illustrated below has radius R and nonuniform density  (r , )  r 2 . The y-axis is the axis of rotation. Find the moment of inertia.

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y

 6

x

*35. Bananacone. Find the moment of inertia of a solid cone of uniform banana of mass m , base radius R , and half-angle of 30º about its symmetry axis. *36. Monkey Ring. A thin ring of radius R and mass m has non-uniform linear mass density  ( )   cos2  . The ring is centered about the origin and lies in the xy plane. (A) What are the SI units of  ? (B) Express  in terms of the total mass of the ring. (C) Where is the center of mass of the ring? (D) What is the moment of inertia of the ring about the z -axis? (E) What is the moment of inertia



of the ring about the x -axis? (F) What is the angular acceleration of the ring if a force F  3mgˆi is applied at the point (0, R) ? *37. Physics Warps Your Mind. The ghost of Isaac Newton reaches inside your ear and yanks out your brain! He then stretches your brain into a long straight line, and finally folds it into a thin rectangle. Isaac then rotates your rectangular brain as illustrated below. Now your brain is an antenna capable of receiving physics signals with improved clarity. Think of your brain as four separate rods – not a rectangular plate. The vertical rods have uniform linear mass density, while the horizontal rods have nonuniform linear mass densities  ( x)  x , where  is a constant. The horizontal rods have twice as much mass as the vertical rods. The total mass of your brain is m .

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y axis 3L

 ( x )  x 2L

 ( x )  x x 4L (A) What is  ? (B) Find the mass of each rod. (C) Find the moment of inertia of each rod about the axis shown. (D) Express the total moment of inertia of your brain in terms of m and L (but not  ). *38. Banana Pizza. A monkey orders a pizza with his favorite topping – slices of banana, of course! Consider the triangular slice of pizza illustrated below, which has non-uniform density   x 2 y . Express the moment of inertia of the pizza about the line x  L in terms of the mass m of the pizza and the distance L only; no other symbols may appear in your final expression.

y xL ( L, L)

(L,0)

x

39. Monkey Dunking. As illustrated below, a 75-kg monkey is suspended from a rope that passes over a 20-kg pulley and connects to a 50-kg box of bananas on horizontal ground. The coefficient of friction between the box of bananas and the ground is 0.4. The pulley has a diameter of 3 meters. The monkey is released from rest and falls 50 meters before landing in a barrel of physics notes below. Assume that there is no slipping between the cord and the pulley. (A) How many revolutions does the pulley complete as the monkey falls 50 meters? (B) What is the force of friction between the bananas and the 105

Volume 1: Mechanics ground? (C) Draw free-body diagrams for the monkey and box of bananas. Sum the forces for each object. Note that the two tensions are not equal due to friction between the cord and pulley. You will have two equations and three unknowns (T1 , T2 , aT ) . (D) What is the moment of inertia of the pulley? What shape are you assuming for the pulley? You can look up the moment of inertia of various standard geometries in the textbook. (E) Draw the pulley. Draw the forces that exert torques on the pulley. Sum the torques. This will give you a third equation with unknowns (T1 , T2 ,  ) . Two of these four unknowns are related. Now you can solve the three equations in three unknowns. (F) Find the acceleration of the monkey. (G) Find the angular acceleration of the pulley. (H) Find the speed of the monkey just before landing in the barrel. (I) Find the final angular speed of the pulley. (J) Find the tension in each section of cord.

50 m

*40. The Golden Banana I. As illustrated below, a 60-kg monkey is connected to the 40-kg Golden Banana by a rope that passes over a 20-kg pulley. The incline is frictionless. The rope rotates with the pulley without slipping.

40 kg

60 kg

60º (A) What is the monkey’s acceleration? (B) What are the tensions in the connecting cords?

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Creative Physics Problems – Chris McMullen, Ph.D. *41. Golden Banana II. Chimp is connected to the Golden Banana by a rope that passes over a pulley, as illustrated below. The coefficient of friction is 0.5. The mass of the pulley is m / 4 . The cord rotates with the pulley without slipping.

m 3

30º

60º

(A) Derive an equation for the acceleration of the system in terms of m and/or g only. No other variables may appear in your final answer. Simplify your final expression. (B) Derive equations for the tensions in the connecting cords in terms of m and/or g only. No other variables may appear in your final answer. Simplify your final expressions. *42. Escalator. A monkey of mass 2m is connected to a box of bananas of mass m as illustrated below. The coefficient of friction between the box of bananas and the incline is 1

3 . The system

starts from rest. The pulley is a solid disk with a mass of m 2 . The cord rotates with the pulley without slipping. The system starts from rest.

pulley

30º (A) Find the acceleration of the system. (B) Find the tension in each cord.

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Volume 1: Mechanics 43. Monkey Bowling I. The coconuts arrange ten monkeys like bowling pens. The coconuts then climb a curved hill until they are 100 meters above the ground, from which point they release an object from rest, hoping to score a strike. (A) (i) Find the final speed of the object if it is a solid sphere. (ii) How faster would the sphere reach the bottom if it were to slide down the hill in the absence of friction? (B) Find the final speed of the object if it is a cylindrical shell with inner radius equal to one-half the outer radius. (C) If the object is a barrel of monkeys and the final speed is 40 m/s, find the moment of inertia of the barrel of monkeys. *44. Monkey Bowling II. A monkey stands at the bottom of an incline of angle  with a hollow sphere. He bowls the hollow sphere up the incline with an initial speed  0 . Assume that the hollow sphere rolls without slipping for the entire trip.

L

0  502 (A) Show that the hollow sphere travels a distance L  up the incline before running out of 6 g sin  speed. (B) Show that if the hollow sphere instead slides without friction the hollow sphere travels a maximum distance L 

 02

2 g sin 

up the incline before running out of speed. (C) Explain why, in terms

of physics concepts, the hollow sphere travels further up the incline when rolling without slipping than in the case where the incline is frictionless. (D) Which of the standard geometries (e.g. solid or hollow sphere or cylinder) would give the greatest value of L in part (A)? Explain your reasoning in terms of physics concepts. *45. Boulder Dash. Some monkeys are playing a friendly game of Boulder Dash, in which a monkey

2 5

 

tries to outrun a spherical boulder  mR2  rolling from rest down a 56-m long, 30º incline.

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30º (A) What is the speed of the boulder at the bottom of the incline? (B) How much time does the monkey have to safely reach the bottom? (Unfortunately, he isn’t given a head start.) *46. Glass Eyes. A monkey standing at the top of a 30º incline pulls the glass eyes out of their sockets.

 

One glass eye is a solid sphere  I s 

2  mR2  , while the other glass eye is a hollow sphere 5 

2  2  I h  mR  . Both glass eyes start from rest from the same position on the incline. They roll without 3   slipping a distance of 12 m down the incline. (A) Find the final speed of each sphere. Be sure to indicate which is which. (B) Use physics concepts to support your answer above for which glass eye wins the race. 47. One-Wheel Getaway. The 40-kg No-Mo Yo-Yo escapes on a 50-kg unicycle that has a 20-kg wheel (a thin hoop) with 70-cm diameter. No-Mo Yo-Yo coasts 100-m down a 30º incline. The wheel rolls without slipping. (A) What is the speed of the unicycle at the bottom of the incline? (B) What is the acceleration of the unicycle? (C) What is the force of friction acting on the wheel? (D) What would be the final speed if the wheel were rolled by itself (without the unicycle and monkey)? (E) What would be the unicycle’s speed if it were instead sliding without friction? 48. Yo-Yo of Doom. No-Mo Yo-Yo is an evil monkey with plans to destroy the world. Presently, he holds the free end of the 2-m cord of negligible mass wound around the Yo-Yo of Doom and releases it from rest. The Yo-Yo of Doom consists of two 30-g solid disks of 6-cm diameter joined by a 10-g hollow cylinder of 1-cm diameter. In order to save the world from destruction, the Powder Puff Pearls must stop the Yo-Yo of Doom before it completely unwinds. (A) What is the moment of inertia of the Yo-Yo of Doom about its axis of rotation? (B) Apply Newton’s second law to find the acceleration of the Yo-Yo of Doom. (C) Conserve energy to find the speed of the Yo-Yo of Doom at the bottom. (D) What is the tension in the cord? (E) How much time do the Powder Puff Pearls have to save the world? *49. Rolling Monkeyball. A 5-kg monkeyball (solid sphere) is initially at rest at Point A, on a horizontal surface. A monkey gives the monkeyball a 100 Ns horizontal impulse. After continuing horizontally for a time, the monkeyball rolls without slipping down a curved track, through a 30-m diameter loop, and then leaves the track horizontally at Point C. Point B is at the top of the loop, which is 80 m below Point A. Neglect the size of the monkeyball compared to the distances given. Point C is 20 m below Point B 109

Volume 1: Mechanics and 150 m above the floor. (A) What is the speed of the monkeyball at Point B? (B) What normal force is exerted on the monkeyball at Point B? (C) How far does the monkeyball travel horizontally while traveling through the air? *50. Monkey Car. A 50-kg monkey drives a 1150-kg car up a 20º incline, uniformly accelerating from rest to 40 m/s in 8 seconds. The wheels (solid disks) roll without slipping. Each wheel has a mass of 60kg and radius of 15 cm. (A) What is the acceleration of the car up the incline? (B) How far does the car travel up the incline? (C) What is the angular acceleration of the wheels? (D) How many revolutions do the wheels complete? (E) What torque does the motor deliver to the wheels? (F) How much work is done against gravity? *51. Monkey Shapes. A solid sphere traveling 5 m/s to the right collides head-on with a hollow sphere of equal mass and radius traveling 3 m/s to the left in a perfectly elastic collision. The spheres have uniform densities. The spheres roll without slipping. (A) Explain why the velocity equation is not valid even though this is a one-dimensional elastic collision between two objects. (B) Find the final velocity of each sphere. *52. Banana Juice. A monkey has three identical 100-g bottles: One bottle is empty, another bottle is filled with 300-g of liquid banana juice, and the third bottle has 300-g banana juice that is frozen. The three bottles roll from rest without slipping 3.6 m down a 30º incline. (A) Find the final speed of each bottle. Be sure to indicate which is which. (B) Explain the ordering of your answers in terms of physics concepts. *53. Monkey Lab. A monkey rolls a solid cylinder, hollow cylinder, solid sphere, and hollow sphere down a steady incline of 30º; these objects roll without slipping. He also slides a block of ice down the incline that is virtually frictionless. Derive symbolic equations for the acceleration of each of these objects in terms of g ; no other symbols may appear in your final answers. As always, simplify your final answers. When you are finished, use your answers to rank the objects from greatest to least acceleration. 54. Monkey-Go-Round II. A 60-kg monkey is placed at the center of a merry-go-round – a large disk with 200-kg mass and 10-m diameter. A turkey spins the merry-go-round at a rate of 0.1 rev/sec and lets go. As the merry-go-round spins, the monkey walks outward until he reaches the edge. It takes the monkey 3 seconds to walk from the center to the edge. (A) Conserve angular momentum for the system (merry-go-round plus monkey) to find the angular speed of the merry-go-round when the monkey reaches the edge. (B) What is the average angular acceleration of the merry-go-round while the monkey is walking? (C) How many revolutions does the merry-go-round make while the monkey is walking? *55. Musical Hamster. A hamster likes to run, but not in the traditional hamster wheel: She likes to run on an old-fashioned record player. A horizontal record is held in place by a small vertical axle. When the hamster climbs onto the record and runs in a clockwise circle, the record rotates counterclockwise. Neglect any friction between the record and the surface upon which it rests. The system is initially at rest. The 200-g hamster runs 50 cm/s in a circle with a radius of 6 cm, centered about the axle. The 400-g record has a diameter of 20 cm. What is the angular speed of the record? 110

Creative Physics Problems – Chris McMullen, Ph.D. 56. Lazy Turkey. A turkey throws a 20-kg monkey made out of silly putty at a 50-kg door with 60-cm width and 2-m height. The monkey travels 5 m/s horizontally when it strikes the door at a height of 1 m, 40 cm from the hinges. The monkey sticks to the door. The door is at rest prior to the collision. Take the system to include the monkey and the door. (A) Calculate the initial linear momentum of the system. (B) Calculate the initial energy of the system. (C) Calculate the initial angular momentum of the system relative to the hinges. (D) What is conserved for this process (from just before impact to after impact)? Justify your answer. (E) Calculate the angular speed of the door as it closes. (F) Calculate the percentage of energy lost or gained. 57. Turkular Motion. As illustrated below, a 20-kg monkey on a frictionless table is connected to a turkey on the floor below by a light, inextensible cord that passes through a hole in the table. The turkey is initially pulling the cord such that the monkey initially slides at a rate of 5 m/s in a circle of 80cm diameter centered about the hole. The turkey then increases the tension until the monkey slides in a circle 20-cm diameter. (A) Find the initial angular velocity of the monkey. (B) Find the initial moment of inertia of the monkey. (C) Find the initial acceleration of the monkey. (D) Find the initial tension in the cord. (E) Find the initial angular momentum of the monkey relative to the hole. (F) Find the initial net torque exerted on the monkey. (G) Which of the following are conserved as the tension is increased – linear momentum, energy, and/or angular momentum? Explain your answers. (H) Find the final velocity of the monkey. (I) Find the final angular velocity of the monkey. (J) Find the final moment of inertia of the monkey. (K) Find the final acceleration of the monkey. (L) Find the final tension in the cord. monkey

turkey

*58. Monkeyball Star. A monkey balances a spinning coconut on his fingertip. The coconut has a 2-kg shell and 1-kg filling. It has the shape of a sphere with 12-cm diameter. It is initial spinning with an angular speed of 24 rev/s. While it is spinning, another monkey gently places a thin ring on top of the coconut. The ring has a radius of 4 cm and a mass of 2 kg.

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(A) What is the angular speed of the system after the ring is added? (B) What percentage of kinetic energy does the system lose? (C) Is mechanical energy lost, conserved, or gained in this process? If it is not conserved, where does the missing energy go or come from? *59. Ice Sk-ape-ades. A monkey is performing her exercise in the ice skating competition of the Winter Monkolympics. In one part of her routine, she spins while standing tall, with her arms straight up in the air and her legs straight down, maintaining excellent balance by positioning her center of gravity above the point of contact between her skate and the ice. She completes 27 revolutions in 12 seconds. Then she quickly stretches her arms horizontally outward, extends one of her legs horizontally backward, and leans her torso and head horizontally forward, effectively increasing her moment of inertia by a factor of 3 – still maintaining her balance exquisitely. She spins like this for 8 seconds. How many revolutions does she complete in this extended position? *60. Uh Oh! A monkey6 is SO frustrated with his slow internet connection that he picks up his laptop, slams it against a brick wall, and then jumps high into the air and stomps on it. At that exact moment, the earth suddenly contracts until it has one-third of its initial radius. How long will a ‘day’ be now, and how many ‘days’ will there be in a year? *61. Monkey Math. Use the equations below to answer the following questions:

 F  3ˆi  4ˆj  5kˆ  s  2ˆi  6ˆj  kˆ  r  5ˆi  2ˆj

 v  5ˆi  8kˆ  p  3ˆi  9ˆj

U ( x)  3x 4  8x 2  16



where SI units have been suppressed for convenience. (A) Find the work done by F in displacing an









object along s . (B) Find the angle between F and s . (C) Find the power delivered by F in moving an







object with velocity v . (D) Find the torque exerted by F applied at r relative to the z-axis. (E) Find the     angular momentum of p relative to r . (F) Find the angle between p and r . (G) Find the force 6

The very same monkey is rumored to have slapped his television set just before a famous New York blackout.

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Creative Physics Problems – Chris McMullen, Ph.D. associated with the potential energy U (x) . (H) Find the stable minima of a particle with potential   energy U (x) . (I) Find the magnitude and direction of 2s  3r .

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8 Fluids (One Problem) *1. Antiquarian Physics. The year is DXXXIII (the number of years after the founding of Rome, ab urbe condita). You are a highly esteemed scientist known as Archimedes. King Hieron II has just received a new gold wreathe to be consecrated to the gods, and although the weight of the wreathe matches the weight of the gold given to the goldsmith, he suspects that it may have been adulterated with silver or lead. He has asked you to determine if wreathe is pure gold (without damaging the crown, of course). Sorry, Archimedes didn’t have a calculator. However, you may use your abacus.  The accepted densities of water and pure gold are: pure water: pure gold:

L rocks / cubic thumb CMLXV rocks / cubic thumb

Sorry, Archimedes didn’t use SI units.  Here is your data for the “weight” of the crown in air and wholly submerged underwater: true weight: apparent weight:

DCCC ± V rocks DCCXLV ± V rocks

Well, Archimedes didn’t use Roman numerals of course, but you’re probably more familiar with them than what Greeks used at this time… Actually, the Greeks may have used something more like our current system, so you won’t be penalized for converting all these numbers to decimals…   Should the goldsmith be executed? The goldsmith’s life rests in your hands. Is the “gold wreathe” made of pure gold, or was it adulterated with a less precious metal?

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Review 4: Rotation 1. Losing Time. The minute hand of the Official Exam Clock points to the 12 and is functioning normally until the exam begins, at which time the minute hand develops a uniform angular acceleration of 1.00 rev / hr2. The exam is over when the minute hand points to the 10. (A) What is the final angular speed of the minute hand? (B) How much exam time do the students lose? *2. Balancing Time. A 100-kg plank is 20 m long. A 50-kg stack of physics homework is placed on one edge of the plank, while a 75-kg stack of history papers is placed on the other edge. The plank, with the stack of physics homework on one end and stack of history papers on the other, is to be balanced on a fulcrum. Where must the fulcrum be placed to achieve static equilibrium? [The stacks are at the very edges of the plank.] *3. Wishing Well. Donate your time to the wishing well and make a wish. A 50-kg bucket full of watches is connected to a rope that is wound around a solid 10-kg cylinder with 30-cm diameter at the top of a well. The bucket of watches is then released from rest. Make reasonable assumptions concerning the unwinding rope. (A) Find the acceleration of the bucket of watches. (B) A splash is heard 3 seconds after the bucket of watches is released. How far did the bucket of watches fall? *4. Time Flies. Jack and Jill went up Time Hill to fetch a pail of hours. Jack fell down and broke his crown, and the pail came tumbling after. Jill chased the pail to no avail. The pail of hours has a mass of 20 kg and diameter of 25 cm. The pail began rolling from rest. The hill is 50 m high. The speed of the pail at the bottom of the hill is 25 m/s. The pail of hours does not have a simple geometric shape and therefore has an unknown formula for moment of inertia. What is the moment of inertia of the pail of hours? (Assume that the hours rotate with the bucket.) *5. Daylight Savings. There just aren’t enough hours in a day. Let’s change that! (A) Suppose that two extremely large arms are constructed such that the earth could stretch its arms out like a spinning ice skater. By what factor would the moment of inertia of the earth need to change in order for a ‘day’ to be 32 hours long? (B) An enormous train track is mounted over the earth’s equator and an enormous train heads east continually (never stopping). What effect, if any, would this have on the length of a day? Explain. *6. Time Conflict. A solid calculus hourglass rolls without slipping along a horizontal surface at a rate of 5 m/s. The solid calculus hourglass collides with a hollow physics hourglass initially at rest. The hollow physics hourglass has the same mass but twice the moment of inertia as the solid calculus hourglass. What is the final speed of each cylinder if the collision is perfectly elastic? 115

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*7. NgULaR ac ELER8I 0 N . Zeus, with a mass of 150 kg, rides in a chariot of mass 450 kg in a circle of 80-m diameter. After driving with a tangential acceleration of 10 m/s2 for 7 revolutions, his angular speed triples. (A) What is Zeus’s final angular speed? (B) What is Zeus’s initial tangential speed? (C) How much time elapses during this trip? (D) What is Zeus’s total final acceleration? 2

*8. R0 Li Ng 0 sLi INg . Zeus stands at the bottom of a 30º incline with a hollow globe of the earth. He bowls the globe up the incline with an initial speed of 20 m/s. Assume that the globe rolls without slipping for the entire trip. 2

2

?

20 m/s 30º (A) How far does the globe travel up the incline before running out of speed? (B) How far does the globe travel up the incline if instead it slides without friction? (C) Explain why, in terms of physics concepts, the globe travels further up the incline when rolling without slipping than in the case where the incline is frictionless. *9. 0 mENt Fin ErI I . A globe-on-a-stick consists of a uniform solid 40-kg globe with 30-cm radius and an 80-kg uniform rod with 120-cm length. The globe-on-a-stick is placed above a fulcrum as illustrated below.

120 cm

30 cm 40 cm (A) What is the moment of inertia of the globe-on-a-stick about the axis of rotation (through the fulcrum and perpendicular to the rod)? (B) What is the initial angular acceleration of the globe-on-a-stick? 116

Creative Physics Problems – Chris McMullen, Ph.D. *10. 0 mENt Fin ErI II . (A) A globe-on-a-stick consists of a solid globe of mass m and radius

R and a rod of mass 2m and length 4 R . The globe has uniform density, while the rod has nonuniform linear mass density  ( x)   x . The globe-on-a-stick is placed above a fulcrum as illustrated below.

4R R 2R

(i) What are the SI units of  ? (ii) Solve for  in terms of m and R . (iii) Express the moment of inertia of the globe-on-a-stick about the axis of rotation (through the fulcrum and perpendicular to the rod) in terms of m and R . (B) A solid disk has radius R and non-uniform density  (r )  r . (i) What are the SI units of  ? (ii) Solve for the mass m of the disk in terms of  and R . (iii) Express the moment of inertia of the disk about an axis through its diameter in terms of m and R . *11. NgULaR 0mENtU . A solid 30-kg globe with 60-cm radius spins on Zeus’s fingertip with a constant angular speed of 4 rev/sec. With his free hand, Zeus gently places a thin 10-kg ring with 40-cm radius on the globe such that the plane of the ring is horizontal. There is friction between the globe and ring, but not between Zeus’s finger and the globe.

(A) What is the final angular speed of the system (globe plus ring)? (B) Compute the percentage of kinetic energy that is lost or gained by adding the ring to the spinning globe.

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Volume 1: Mechanics *12. ½ aT  D' s mac hi NE . Zeus constructs one-half of an Atwood’s machine by suspending a 50-g globe from a cord wrapped around a 20-g solid disk, as illustrated below. 2

20 g R

50 g

(A) What is the acceleration of the 50-g mass? (B) What is the tension in the cord?

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Final Review: Motion

Every monkey in -ville celebrates physics on the slightest occasion, Discussing concepts in the elevator, solving problems on vacation. But there is one resident of -ville, with a brain sizes too small, Who does not share this love for physics, does not enjoy it at all. The chimp skips his lectures, complaining about physics stress, Homework takes too long, quizzes too hard, physics is pointless. “All these silly monkeys,” criticizes the chimp, “wasting their time, With projectiles, inertia, energy, and torque. It should be a crime!” It is at this time that the chimp’s lips slowly curl into a wry smile, Evil thoughts churning inside his head. Quick! 314, somebody dial! *The chimp parades around -ville along a semicircular arc, dragging Newton’s corpse behind him. Make a clear distinction between the conceptual meaning of the chimp’s average velocity and average speed as he completes one semicircle.

Explain why the chimp is accelerating when he travels along a semicircle with constant speed. What are the name and direction of this type of acceleration? 119

Volume 1: Mechanics The chimp throws physics textbooks out of the third-story window of the library. Why isn’t the acceleration of a physics textbook zero at the peak of its trajectory? Why isn’t the velocity of a physics textbook zero at the peak of its trajectory? The chimp fires a gun to pierce a physics chalkboard with bullets. Which of Newton’s laws explains why the gun recoils? Explain. The chimp flushes lecture notes down a toilet. Consider a plot of the velocity of the lecture notes as a function of time. How would you find the acceleration of the lecture notes at given point? How would you find the net displacement of the lecture notes? The chimp drives through -ville while honking his horn to disrupt physics conversations. The chimp travels 80 m at an angle of 120º. The chimp then travels 160 m at an angle of 240º. (A) Draw a sketch, clearly labeling the given vectors and the resultant. (B) Find the magnitude of the chimp’s net displacement. (C) Find the direction of the chimp’s net displacement.

When the chimp approaches the library, he uniformly decelerates for 8 seconds, traveling 96 m, until his speed is 4 m/s. Find (A) the initial velocity and (B) acceleration of the chimp. The chimp stands on the roof of the library, 80 m above the ground, and throws Sir Isaac Newton’s Principia at a rate of 60 m/s at an angle of 30º above the horizontal. (A) What is the speed of the Principia at the top of its trajectory? (B) What horizontal distance does the Principia travel before striking the ground?

60 m/s 30º

80 m

120

Creative Physics Problems – Chris McMullen, Ph.D. The chimp uses a 200-kg boulder to haul a 300-kg dumpster filled with physics exams up a 30º incline, as illustrated below. The coefficient of friction between the box of bananas and the incline is

1 . The 3

system is initially moving to the right.

pulley

30º (A) Find the acceleration of the system. (B) Find the tension in the connecting cord. The chimp collects all of -ville’s books on rocket science, and launches them into space. The satellite orbits the planet with a constant speed of 1000 m/s. (A) What is the altitude of the satellite? (B) What is the period of the satellite? The chimp flattens lab equipment with a bulldozer. Explain, conceptually, why a solid cylinder has a smaller moment of inertia than a hollow cylinder of the same mass and radius. The chimp even destroys the janitorial supplies so that the monkeys of -ville won’t be able to clean up the mess! The chimp saws a mop into two pieces at the mop’s center of mass. Do the two pieces have equal mass? Explain. If the masses are not equal, then what quantity is balanced? The chimp gives Cindy Lou  a spin on frictionless ice. What is the conceptual meaning of rotational inertia? If Cindy Lou  extends her arms outward, what will happen to her angular speed? Which of the main conservation laws applies to this process? With a giant torque, the chimp brings down Newton’s sacred apple tree. In the equation for torque, explain how r is defined. Also, explain how  is defined. The head of Frosty the Newton (a snowmonkey) is a 20-kg pumpkin. The chimp shoots a 5-kg arrow that strikes the head of the pumpkin. The speed of the arrow is 30 m/s just prior to impact. The arrow sticks inside the pumpkin as the pumpkin falls off of frosty. (A) What is the speed of the pumpkin as it falls off 121

Volume 1: Mechanics of Frosty the Newton? Neglect friction between the pumpkin and the large snowball on which it initially rests. (B) Find the percentage of kinetic energy lost during the collision.

30 m/s

The chimp finds all of -ville’s cool physics Frisbees and burns them in a bonfire. A cool physics Frisbee is no ordinary Frisbee: A cool physics Frisbee of radius R and uniform mass density features a circular

R  ,0  , as illustrated below. 2 

hole of radius R / 2 centered about 

Find the coordinates of the center of mass of the cool physics Frisbee. In the illustration below, the 50-kg chimp stands above the pivot, 40-kg Cindy Lou  hangs from one end of a 60-kg uniform rod that is 10 m long, and a 30-kg stack of physics textbooks lies 2 m from the opposite end. Also, the chimp pulls on a rope as shown with a force of 100 N. Stack of physics textbooks

Not drawn to scale

60º pivot

4m 3m 5m 122

Creative Physics Problems – Chris McMullen, Ph.D. (A) Find the torque exerted by Cindy Lou . (Relative signs are crucial.) (B) Find the torque exerted by the chimp’s weight. (C) Find the torque exerted by the tension in the rope (pulled by the chimp). (D) Find the torque exerted by the stack of physics textbooks. (E) Find the net torque exerted on the rod. The chimp climbs a curved hill until he is 50 meters above the ground, from which point he gives a large cylindrical boulder an initial speed of 10 m/s, headed toward -ville at the bottom of the hill. (A) Find the final speed of the cylindrical boulder. (B) How much faster would the cylindrical boulder reach the bottom if it were to slide down the hill in the absence of friction?

When the chimp sees Cindy Lou  at the bottom, he races down the hill, Surpassing the boulder, stopping it with all his might, and saving -ville! The chimp has a rotating brain. The chimp’s spherical brain has a mass of 500 g, radius of 3 cm, and angular speed of 8 rev/s. However, through the act of saving Cindy Lou  , the chimp’s brain grows 3 times larger and increases in mass by a factor of two. (A) What is the angular speed of the chimp’s rotating brain after increasing in size and mass? (B) What percentage of kinetic energy is lost or gained during this process?

Despite the massive destruction, every monkey in -ville is cheerful, Discussing concepts and solving problems, and more than an earful. The chimp realizes that it’s not about lectures, quizzes, or homework, No, physics is something much, much deeper than a lot of hard work.

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Up late studying, little time to sleep, Didn’t hear the alarm clock beep! Mad dash to school, late for class, Better hurry or you may not pass!

Spot a monkey, ask him for the time, He ignores you like a pantomime! Fall down a hole – isn’t life so grand? Welcome to Physics in Monkeyland! Monkey Directions

Although each question may not ask you to do so explicitly, Remember to explain all of your answers quite thoroughly. You may safely ignore any effects that air resistance may cause, Except for biological purposes (we don’t want your heart to pause). Monkeyland Constants Compute gravitational acceleration in Monkeyland on page two, 23 Knowing the planet’s mass, 6  10 kg , and radius, 4  10 m , too.

6

Although your weight is 490 N on earth, this value in Monkeyland You must yourself compute, even though the calculator is banned. *1.

A monkey throws a banana straight up, really high, At its peak the banana’s acceleration isn’t zero. Why? A monkey throws a banana at an angle from a treetop, At the top of its trajectory, its speed isn’t zero. Why not? While at rest, you are bumped into by a 200-kg fast-moving monkey. Compare the forces involved between you and the creature so furry. During the previous collision, who will change speed the most? Explain each answer clearly or be haunted by Newton’s ghost. A sailor monkey launches a cannonball straight up on a pirate ship, Traveling with constant velocity. Where ends the cannonball’s trip? A monkey with a bunch of bananas is stranded on frictionless ice. How will the poor monkey get home, and what will be the price?

124

Creative Physics Problems – Chris McMullen, Ph.D. Sit in a giant teacup, go for a spin, At a constant rate, again and again. What is the name of this kind of force? Is it the same kind that propels a horse? *2.

What’s the direction of this acceleration? Headed toward a holiday celebration?

A monkey sitting high up in a tree releases a pomegranate, Compute its acceleration from the mass and size of the planet. Knowing your weight on the earth, given on the title page, Find your mass and weight in Monkeyland, but not your age. A monkey, sitting directly above you, throws a banana up, vertically. Eight seconds later you catch it, 5 m below. Find the final velocity, Just before reaching your hand, of the banana in meters per second. Also, what is the height of the banana before it begins to descend?

*3.

From a clearing you walk into the trees, First 100 m at 120º and then 200 m at 240º. How far and which way should you head If you wish to return to where you started?

*4.

Atop a cliff toss the Principia at 60º above the horizontal at 8 3 m/s, How far from the base of the sixty-meter cliff does the book land? Suppose you could throw it the same speed in any direction: What angle provides the greatest final speed? Good question!

*5.

A crazy monkey named Mad Hopper grabs a 20-kg monkey by the tail, And spins it in a horizontal circle with a radius of

3 m. Hear it wail!

The period of the monkey’s uniform circular motion is 4 3 seconds long. What angle does the monkey’s tail make with the vertical? (What’s wrong?) If you find that  hurts your head, You might round it to ten instead. 2

*6.

In the monkey’s tail there is tension, Find it without a moment’s hesitation. A monkey pulls two boxes with a force of 100 2 N, as illustrated below. If the coefficient of friction is ¼, with what acceleration will they go? 125

Volume 1: Mechanics

100 2 N

20 kg

80 kg

45º

If the monkey pulls the boxes a total distance of 20 m horizontally, Find the work done by gravity, and the power delivered by the monkey. *7.

Choosing between a solid or hollow cylinder of the same mass and size, Which has a smaller moment of inertia? Correct explanation wins a prize! A monkey saws a rake into two pieces at its center of mass. Which end weighs more? Answer correctly in order to pass. A monkey gives you a spin on frictionless ice. “Stop me!” you plead. If you extend your arms outward, what happens to your angular speed? With a giant torque, a monkey brings down Newton’s sacred apple tree. Explain how r and  are defined in the equation for torque, eloquently. What is the conceptual meaning of rotational inertia? Does it tell you how much a monkey can hurt ya?

*8.

In a 200-kg car, your mass included, head 30 m/s west toward a feast, When you collide with a 400-kg car that was moving 20 m/s to the east. Find the final velocity of each car if (A) the cars stick together, And (B) if instead the collision is elastic. Oh, what a bother!

*9.

Climb onto the escalator that’s illustrated below. What is your acceleration when down you go?

counterweight

you

126

Creative Physics Problems – Chris McMullen, Ph.D. Counterweight’s mass = 30 kg Pulley’s mass = 10 kg Pulley’s diameter = 30 cm Pulley’s shape = hollow sphere Coefficient of friction = 0 Angle of incline = 30º *10.

Coast in a unicycle from rest down a hill from 600 meters high, Find the speed at the bottom. Here’s more info, don’t ask why: Unicycle’s mass, including the wheel, but not you = 50 kg Wheel’s mass = 40 kg Wheel’s shape = solid disk Wheel’s diameter = 80 cm

*11. Shown below on a frictionless table is a 20-kg monkey, Connected by a cord passing through a hole is a turkey. The monkey initially travels with a constant period of 2 seconds, In a circle with a diameter of one meter; but this fun soon ends: The turkey pulls down, shortening the cord, as the monkey cries, Until the monkey’s circle is just one-fourth of its original size. Which of the fundamental quantities is conserved as the tension is increased? Also, what is the speed of the monkey after the cord’s shortening has ceased? monkey

turkey

*12.

The pie-slice below, with density  (r ,  )  r and radius R , costs a dollar. Find the pie-slice’s moment of inertia about the x-axis in terms of m and R .

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Volume 1: Mechanics

y

 3

x

Apes from another Planet 13. Fire in the Sky. All sources of broadcast news are focused on a strange purple light seen orbiting the earth. The UFO is traveling with constant speed in a circular orbit with a period of 10  seconds. (A) What is the radius of the UFO’s orbit? (B) What is the speed of the UFO? 5

14. Alien Invasion. The UFO suddenly leaves its orbit and descends to the earth. During its final







approach to earth’s surface, it travels along the vector 3A  2B , where A has a magnitude of

 100 3 km and a direction of 210º and B has a magnitude of 300 km at a direction of 240º. At this

range, the UFO looks like a purple flying saucer. Find the magnitude and direction of the vector

  3A  2B .

15. E.T. The UFO is hovering steadily 6.0 m above the ground when a purple ape suddenly opens a door and jumps out with an initial velocity of 7.0 m/s at an angle of 30º above the horizontal. (A) How far does the purple ape travel horizontally? (B) What are the height, speed, and acceleration of the purple ape at the top of its trajectory? *16. Close Encounter. The purple ape meets an earthling and announces in a rather nasal voice, “Greetings, earthling. I am a purple-people eater from the planet Ban.” Fearing for her life, the 128

Creative Physics Problems – Chris McMullen, Ph.D. earthling runs for 4 seconds. She starts from rest and uniformly accelerates until her speed is 8 m/s. How far does she run? 17. Purple-People Eater. The purple ape chases the earthling, shouting, “Wait! You have nothing to worry about! I eat purple people, and you’re not purple!” The 90-kg purple ape running 12 m/s collides with the 50-kg earthling running 5 m/s in the same direction. The earthling bounces off the purple ape’s rubber-like belly in this elastic collision. (A) Find the final velocity of both the earthling and the purple ape. (B) If they are in contact for 500 ms during the collision, find the average force that each exerts on the other during the collision. 18. Attack of the Aliens. The earthling encounters a second purple ape. This purple ape pulls out a strange gun, fires a cone of purple light at the earthling, and drags the 50-kg earthling up the inclined ground with a constant speed of 4.0 m/s as illustrated below. The coefficient of friction between the ground and earthling is 1

3.  Fgun 30º

30º



What is the magnitude of the force Fgun that the ray gun exerts on the earthling? 19. Purple People-Eater. This second purple ape winds a rope all the way around the earthling such that she looks somewhat like a mummy. The earthling states, “I thought you only ate purple people,” to which the purple ape replies, “You have me confused with my cousin. I’m from planet Ana: I’m purple and eat people, whereas my cousin eats purple people (and coincidentally also happens to be purple.” “OMG!” exclaims the earthling, “You’re gonna eat me!” “Nah,” replies the purple ape, “I’m just playin’ with ya’. Boo!” The purple ape yanks on the rope and unwinds the earthling, from rest, for 12 seconds, until the final angular speed of the earthling is 36 rev/s. (A) How many revolutions does the earthling complete? (B) What is the earthling’s angular acceleration? 20. Cocoon. The purple apes retrieve a giant cocoon from their ship. The earthling asks, “Are you going to take over the earth by replacing people with aliens one by one?” “That’s silly,” reply the purple apes, “you’ve been watching too many movies.” The purple apes release the cocoon from rest, letting it roll 70-m along a 30º incline without slipping. They measure its final speed with a radar gun to be 20 m/s. They ask the earthling if she is intelligent enough to determine a formula for its moment of inertia from 129

Volume 1: Mechanics this information. Solve for the coefficient in the moment of inertia formula for the cocoon to derive a symbolic equation for its moment of inertia. 21. Body Snatchers. A 40-kg boy stands on the edge of a 200-kg merry-go-round in the shape of a uniform solid disk with a diameter of 10 m that is spinning with constant angular speed, completing 12 revolutions every minute. One of the purple apes points his ray gun at the merry-go-round, shoots, and suddenly zaps the boy, who is teleported off the merry-go-round and over to where the purple apes are standing. “Hey!” exclaims the girl, “I thought you said you weren’t going to snatch our bodies!” “Relax,” explains one of the purple apes, “We just want to see if all earthlings are intelligent.” (A) What is the period of the merry-go-round after the boy suddenly disappears from it? (B) How much kinetic energy does the system (merry-go-round plus boy) lose during this process? 22. Balance of Power. The purple apes place the 40-kg boy 1 m away from the right edge of a 12-m long uniform 110-kg plank and the 50-kg girl 4 m away from the left edge. The purple apes then lift the plank, with the boy and girl in place, and balance it on a fulcrum. The purple apes ask the boy to describe the physics concepts involved. (A) Find the location of the fulcrum. Draw and label a diagram to illustrate your answer. (B) If instead the fulcrum is placed at the midpoint, find the initial angular acceleration of the (initially horizontal) plank.

Monkhouse, Ph.D. Chimpton University’s physics department employs Dr. Monkhouse to evaluate students who fail physics. While he often behaves like a donkey, he is by far the best monkey in the business. *23. A patient has a pencil lodged up his nostril. He earned 59% overall. He aced the quizzes on graphing, center of mass, and moment of inertia, but bombed most of the other topics. Foremonk thinks that the student was so disappointed after messing up the easy problems that he shoved the pencil up his nostril, which resulted in a neurological problem. (A) Foremonk attempts to solve the problem by pulling the pencil out with a pair of pliers, exerting a



force F  6 y ˆj to displace the pencil from y  4 to y  9 , where the force is expressed in N when y is in cm. How much work does Foremonk do? Monkhouse calls Foremonk an idiot. Obviously, the student jammed the pencil up his nostril because he was frustrated that he just barely failed the course. Since the student aced all the quizzes that involve applications of calculus, mathematics is his strong point; having bombed everything else, the student has a clear case of conceptualitis. Monkhouse tells Chimperon to give the student some conceptual questions with which to practice. (B) Chimperon cuts a croquet mallet in two pieces at its center of mass and asks the patient which end weighs more. The patient says that they must be the same if it was the center of mass. Monkhouse

130

Creative Physics Problems – Chris McMullen, Ph.D. slaps himself in the forehead and mutters, “Silly monkey.” Reason which end weighs more in terms of physics concepts. Also, which physical quantity is balanced? (C) Chimperon shows the patient a video of an astronaut tightening screws on a spaceship in orbit around the earth with a wrench, when the astronaut carelessly loses her tie rope. Chimperon asks the patient how the astronaut might return to her spaceship. The patient suggests that she make swimming strokes with her arms. Monkhouse shakes his head and mumbles, “Hopeless.” Explain the physics concepts behind the correct solution. Monkhouse prescribes a physics therapist and IQ enhancement tablets. *24. Sick of physics, a student vomits. The vomit is launched with an initial velocity of

8 m/s at an 3

angle of 30º above the horizontal from a height of 1 m above the ground. Curiously, the vomit spatter spells out the word inertia. How far does the vomit travel horizontally before striking the floor? Monkhouse deems that this case is hopeless. Since the only way to learn physics is to solve problems until you turn blue in the face, there is no way to avoid physicsitis and still pass… *25. A student who only turned in one-third of his lab reports and one-fourth of his prelab exercises failed lab. Monkhouse grabs the student by the ear, hauls him into a rocket, and launches his lazy carcass into space. Now the student won’t have anything to do other than write his lab reports. The rocket is presently orbiting the earth, which has a mass of 6  10 kg , in a circular orbit with a period of 24

105  sec . (A) Find the speed of the satellite. (B) Find the radius of the satellite’s orbit. *26. A student has solved so many circular motion and rotation problems that she can no longer walk straight. Monkhouse, who is intrigued by this, asks her to walk for 30 minutes while he makes some measurements. She completes 90 laps and walks with a constant speed of 4  m/s . (A) How far does she travel? (B) What is her period? (C) What is her angular speed? (D) What is the radius? (E) What coefficient of friction between her ground and shoes is required to prevent her from slipping as she tries to complete her turn? Monkhouse treats her ailment by making her carry her backpack with her left shoulder instead of her right shoulder. *27. A student leaves all problems involving springs and rolling without slipping blank out of sheer terror. In order to help the student overcome his fear, Monkhouse forces the student to solve a problem involving both. One end of a 10 N/m horizontal spring is connected to a wall. The spring is compressed 21 cm from equilibrium and a 14-kg solid sphere with 6-cm diameter is placed at its free end. The system is released from rest and the sphere rolls without slipping as the spring stretches. Find the final angular speed of the sphere. *28. Monkhouse’s boss, Dr. Chimpie, wants him to diagnose a student who got all of his vector operations mixed up on the final. Monkhouse yawns, inquiring what’s so special about a case where the 131

Volume 1: Mechanics student accidentally adds instead of subtracts vectors and interchanges the cross and dot product. Dr. Chimpie holds up the student’s file and points out that the student actually added vectors for cross product problems and computed the scalar product for vector subtraction problems. Monkhouse can’t help but see for himself. (Unlike the student, however, you better solve these problems correctly.)

 A  3 ˆi  ˆj  2 3 kˆ  B  3 ˆi  3 ˆj  2kˆ





 





(A) Find the magnitude of 2A  B 3 . (B) Find A  B . (C) Find A  B . *29. Monkhouse walks into a student’s dorm room as part of his clinic duty. He picks up the clipboard, reads over the file, and says, “I hear you’re having trouble with Newton’s laws,” and before he finishes, the student starts reciting the most vile curses from Shakespeare. It turns out that the student does this anytime he hears Sir Isaac Newton’s name. Monkhouse cures this problem with hypnosis: Now the student sings beautiful arias when he hears Newton’s name, and instead curses à la Shakespeare when he hears his classmates disparage the subject of physics. In order to test the results of the hypnosis, Monkhouse asks the student to solve a problem involving one of Newton’s laws. (A) Derive an equation for the acceleration of the masses for Atwood’s machine in terms of m1 , m2 , mp and/or g , where the cord rotates with the pulley (a uniform solid disk) without slipping. No other variables may appear in your answer. (B) Show that your result simplifies correctly for the case where the cord slides around the pulley without friction. *30. One student has Monkhouse so flabbergasted that Monkhouse resorts to knocking some sense into the student. Monkhouse connects a 0.5-m long croquet mallet to the ceiling via a hinge (so that the mallet hangs at the bottom). The croquet mallet consists of a 0.5-m long 6-kg handle (a uniform rod) and a pointlike 30-kg head. Monkhouse tilts the croquet mallet such that it is horizontal (i.e. hinged 90º) and releases it from rest. At the bottom of its arc, it will knock some sense directly into the student’s forehead. (A) Find the moment of inertia of the croquet mallet about the rotation axis through the hinge. (B) Find the initial angular acceleration of the croquet mallet. Fortunately, for the student (or, unfortunately, perhaps), Monkhouse’s colleague, Wilmonk, intervenes, catching the mallet just before impact.

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Creative Physics Problems – Chris McMullen, Ph.D.

Answers to Selected Problems The 2nd Little Piggy (G) 28.6 s (H) 13.0 s, 51.5 s (I) 8.9 m/s (J) 1.4 m/s2 (K) 258 m (L) -1.7 m/s

Chapter 1: One-Dimensional Motion 3. Physics Makes You Run Around in Circles. (A) 1’16.9” (B) 24 laps (C) 180 m (D) (278 m,  105 m)

The 3rd Little Piggy (M) 0  t  8.0 s , 13 s  t  28 s (N) 20.0 s  t  39.5 s (O) 16 s  t  22 s (P) 6.0 m/s (Q) 8.0 s, 13.0 s, 28.0 s (R) -0.75 m/s2

4. Going Bananas! (A) 157 m (B) 100 m (C) 4.12 m/s (D) 6.43 m/s (E) 0 6. Monkey Climbing. Show work and/or explain your answers. (A) 4.2 m/s (B) 2.5 m/s2 (C) 3.4 s, 8.5 s (D) 3.4 s to 8.5 s (E) 2s to 4.8 s (F) 8.0 m/s (G) ~ (about) 45 m (H) ~ -1 to -3 m (I) 5.3 m/s (J) ~ -0.1 to -0.4 m/s

12. Monkey Skater I. (A) m/s3, s (B) 12 m/s (C) 12 m/s (explain this) (D) 2.0 m/s2 (E) -61.9 m (F) -7.33 m/s (G) 8.14 m/s

7. Wild Monkeys. Show work and/or explain your answers. (A) -6.0 m (B) 30 m (C) 0.67 m/s (D) -0.67 m/s (E) -0.7 m (F) 6 s to 12 s

13. Monkey Skater II. (A) m/s5/2 (B) 40 m/s (C) -8.0 m/s

8. Three Little Piggies. The 1st Little Piggy (A) -8.0 m (B) 3.6 m/s (C) 7.0 s (D) All of them (technically, in one case he is not moving), but exclude the junctions (E) 4.8 m/s (F) 12 m/s

16. Monkey Wish. (A) 30.7 m (B) 24.5 m/s 17. Monkey Jordan. 4.9 m 18. Banana’s Revenge. (Assume that the monkey lands on the ground below.) (A) 6.63 s 133

Volume 1: Mechanics (B) 40.1 m/s (C) 81.9 m (above the ground)

(B) 1.5 m/s2,

21 m/s 16

(C) 2.0 m/s2, 40 m (D) 22/3 m/s, 248/15 m (E) 0, 12/5 sec (F) -1/3 m/s

19. Ceiling. 75.9 m 24. Wait a Minute! 56.3 m 25. Towering Knight. (A) 19.8 m/s (B) 5.52 s

Chapter 2: Projectile Motion and Vectors 1. The Triangle. The following answers may not be in the proper order. (A) 2 / 3, 5 / 3, 5 / 2

26. Stairway to Heaven. 102-½ stairs

(B) 2 / 13,3 / 13,2 / 3

27. Court Jouster. (A) 107 m (B) 9.25 s (C) 23.1 m/s

(C) 1/ 4, 15 / 4, 15 (D) 7 / 10, 51 / 10,7 / 51 2. Trig Practice. The following answers may not be in the proper order. (A) 5 m,48.2º,41.8º (B) 12.9 mm,14.2mm,65º

33. Mighty Pen. (A) 28.5 m (B) (i) (4.62 m/s, upward) (ii) 10.3 m/s

(C) 3.28 km,2.29km,55º

34. Kidnapped! (A) 8.00 m/s2 (explain the sign) (B) -44.7 m/s (C) (i) 36.0 m (ii) 0 (iii) -8.00 m/s2 (and this sign)

(D) 4.90 cm,8.55cm,35º 3. The Banana Triangle. (A) 30º (B) (C) (D) 5 (E) 5/13 (F) 5/12 (G) (H) 45º (I) 45º

35. Dunk a Skunk. (A) 86.1 m (B) 28.5 m/s (C) -10.8 m/s (D) 21.1 m/s 36. Monkey Chase. (A) 5.46 s (B) 74.6 m (C) 27.3 m/s

4. Monkematics II. (A) 23.6º, 156º (B) 101º, -40.5º (C) 0.866 (D) 0.546 (E) 11.8º, 78.2º (F) 56.3º, 236º 5. Monkey Tails.

39. Stop, Thief! 0.225 m/s2 42. Monkeymatics I. (A) 0, 50 m/s2 134

Creative Physics Problems – Chris McMullen, Ph.D. (A) (246 m, 172 m) (B) (204 m, 78.7º) (C) (469 m, 52.5º)

31. Sick of Physics. (A) 64.4 m (B) 81.2 m (C) 8.45 m/s (D) -9.81 m/s2 (E) 81.5 m (F) -52.2º

6. Monkey Madness. (A) 233º, 307º (B) 0, 120º, 240º 7. Monkey Vectors I. (365 m, -84.6º)

32. Knights in Smoldering Armor. (A) (69.4 m/s, -66.9º) (B) (69.4 m/s, -66.9º) (C) (69.4 m/s, -64.4º)

8. Royal Vectors I. (223 crowns, 173º) 9. Gold Strike! (102 N, 194º)

33. Scaredy-Cat. (A) 52.4º (B) 11.7 m (C) 11.7 m (D) yes

10. Family Jewels. (A) (548 m, 195º) (B) (–530 m, – 139 m) 11. Gravitational Pull. (102 kN, 194º)

34. Serenade. 17.3 m/s

19. Monkeyin’ Around. (A) 30.5 m (B) (24.7 m, 76.0º) (C) (12.4 m/s, 76.0º) (D) (4.0 m/s2, 90º)

35. Bomb’s Revenge. 4.43 km 36. Football’s Revenge. (A) 95.6 m (B) 25 m/s (C) 221 m (D) 8.83 s

20. Royal Vectors II.

    = –1.84 ˆi – 1.54 ˆj 2  (B) P(2 ) = (249, 5.79º, 90.0º)   (C) P  2Q = (x3 + 12 x) ˆi + (4 x) ˆj – [21 cos (2 x) + 3] kˆ / 2     (D) 3Q   2P  = (34.0, -31.5º, 45.1º) 2 2 (A) R

42. Golf Ball’s Revenge. 68.4 m Review 1: Motion 1. Tug of War. (A) 96.9 N (B) 184º

21. Silly Physics. 4.0, -30º. Warning: In the past, some students have obtained these answers by sheer accident with a fundamentally incorrect strategy.

2. Drinking Well. (A) 114 m (B) 53.5 m/s (C) 14.2 m/s downward (D) 22 m/s

135

Volume 1: Mechanics 3. Helicopter Rescue I. (A) -1.4 m/s2 (B) 6.5 s (C) 28 m

18. Trig Torment. 238 N 19. To Steal the Treasure. (A) 161 N (B) 0.805 m/s2 (C) 6.95 m/s

5. Banana Split. (A) 11.9 s (B) 141 m

25. To Bring Home Glory. (A) 12.1 m/s2 (B) 196 N

6. Elevator. (A) 8.89 s (B) 152 m (C) 17.1 m/s (D) 113 m (E) (43.7 m/s, -67.0º)

28. Defying Gravity.

P  (M  m) g tan

38. Green Cheese. (A) 1.62 m/s2 (use a formula – don’t divide by 6) (B) 1.99 × 1020 N

22. Bumblebee Hunting. m/s, rad/s, -2  m/s2, 16 /  m

39. Monkey Earth. (B) 2.64 × 106 m

25. Slothward. 5/4 sec Chapter 3: Newton’s Laws of Motion

40. Law of Attraction. (A) 1.60 nN (B) 1.22 Gg

2. Lunar Dummy. 81.6 kg, 801 N, 81.6 kg, 133 N.

42. Planet Mnqy I. (A) 5.45 m/s2 (B) 18.3 kg, 82.6 N

3. Astronautic Dummy. 123 kg, 1.54 kN. 6. 00. (A) 5.10 m/s2

(B) 894 N, 2.11 kN

8. Special Delivery. (A) 51.1 cm/s2

(B) 255 N, 766 N

44. Magic Bananas. 90% of the earth-moon distance, closer to the moon (but a percentage will not suffice) 45. Giant Leap for Monkeykind.

9. Treasure Chest.

7 M 8  41GMm ˆ i (B) F   450 R02 (A)

(A) a  1.64 m/s (B) T  164 N

2

13. Minimal Effort. (A) 21.8º

Chapter 4: Uniform Circular Motion

16. Play Time. a1  g (sin   cos )

1. Father Time. (A) 0.00175 rad/sec

17. Neanderthal Escalator. ≥ 290 kg 136

Creative Physics Problems – Chris McMullen, Ph.D. (B) 3.67 rad

(C) 0.669 N (D) 5.28  1020 N (E) 2.95  10-7 rad/sec (F) 1.94  103 m/s (G) 5.72  10-4 m/s2 (H) 5.28  1020 N (I) 162 s

2. Round-N-Round. (A) 0.199 rad/s (B) 29.9 km/s (C) 5.95 mm/s2 (D) 3.56  1022 N 4. Banana Dance. (A) 5.24 rad/sec (B) 7.85 m/s (C) 13.4º (D) 1.54 m (1.50 m is incorrect) (E) 41.1 m/s2 (F) centripetal (G) inward (horizontally toward the center of the circle – not along the tail) (H) 8.46 N (8.22 N is incorrect) 9. Centrifuge. T  2

Review 2: Newton’s Laws 9. Bloody Physics. (A) g 8 (B)  7g 8 13. The End. (A) 2.5 m/s2 (B) 100 N Chapter 5: Conservation of Energy

s R g

2. Fun, Fun, Fun! (A) 2.41 kJ (work is negative) (B) 0 (C) -3.22 kJ (work is positive) (D) 4.74 hp

10. Swinging Around. 15.9 m/s (r is not 10 m!) 12. Racecar Physics. (A) 18.9 m/s (this must be the Granny Gran Prix…) (B) 29.6 m/s

6. Monkey Product. (A) -96.0 J (B) (6.40 N, 38.7º) (C) (35.8 m, 153º) (D) 115º (E) -96.0 J. (This answer is exact.)

13. To Spin a Tail. (C) 392 N (D) 17.0 m/s2 (E) centripetal (not centrifugal) (F) inward (horizontally to the right for the solid position shown) (G) 5.42 m/s

8. More Fun! (A) -7.00 J (work is positive) (B) 72.0 J (work is negative) (C) 24.0 W, 6.00 W

17. Geosynchronous Satellite. (A) 3.08 km/s (B) 3.59  104 km

13. Banana Peels. (A) -124 J (B) 40.0 J (C) 164 J (D) 40.5 m/s (E) 124 J

19. Banana Planet. (A) 2.93  103 m/s2 (B) 0.119 kg 137

Volume 1: Mechanics (F) 0 (G) 25.1 W

32. Monkey Coaster I. (A) 30 m/s

5800 N (B) 7

23. Banana Path. You should not round until you obtain your final answer. If you compute an intermediate quantity, keep every digit on your calculator. Additional digits have been included in brackets so that you can check your answers to better precision (since you can get a nearly correct answer with an incorrect solution). However, if you like points, you better follow the rules for significant figures in your final answers. (A/B) 0.541[756185] m, 0.534[155744] m @ 33.6[9006753]º (C) 98.[5970735] %

(C) 500 N/m 33. Free Fall I. (A) 40 m/s (B) -320 kJ (C) 2.0 34. Free Fall II. (A) 40 m/s (B) 15 N/m2 35. Monkey Slide I. (A) 15 m/s (B) 5.0 m

24. Monkey Rushmore. (A) 10.7 m/s (B) 69.0º

36. Monkey Slide II. (A) 553 N (B) 58.9 kJ (C) 11.1 kW (D) 1200 kg m/s (E) 12.0 kJ (F) 58.9 kJ (G) -9.88 kJ (H) 45.1 m/s (I) 33.8 m/s (J) 650 N (K) 58.3 m

25. Banana Sled I. (A) -240 J (B) 29.0 m/s 26. Banana Sled II. 38.5 m/s 27. Pendulemur I. (B) 8.88 m/s 28. Pendulemur II. (A) h0  L(1  cos ) , h  0





(B) v 0  0 , v  θˆ 2 gL(1  cos )

39. Eskimonkey. (A) mgR (B) 0 (C) mgR cos

  (C) a 0   g sin  θˆ , a  2 g (1  cos )rˆ   (D) T0  mg cos rˆ , T  mg3  2 cos rˆ 2 3   4   21.1º .  

(E)   cos 1 

2 gR(1  cos ) (E) mg(3 cos  2) (D)

Half the

gR cos

height does not correspond to half the angle.

(F)

29. Simple Harmonic Banana. (not necessarily in order) 6.25 m/s2, 0, 1.25 m/s2, 0, 39.5 cm/s, 38.7 cm/s

(G) cos 1  

2 3

138

Creative Physics Problems – Chris McMullen, Ph.D. 22. Walking the Plank. (B)  0.667 m/s (C) 133 J gained

Chapter 6: Systems of Objects 2. Center of Monkeys II. (A) 15 m away from the 30-kg monkey, 5 m away from the 90-kg monkey. (B) (9.17 m, 5.83 m) (C) (1.53L, 0.767 L)

23. Monkey Mayhem I. (A)  5.33 m/s (B) 3 m/s, 36.9º

 R  ,0   6 

(D)  

24. Monkey Mayhem II.

 22 m/s, 118 m/s

(E) 118 N, 275 N . (F) 75 kg

29. Look Here.  11. Buried Treasure.

32 12  , 0, 3 5

11 4 v0 , v0 3 3

34. Monkey Pile. (A) (1.10 m/s, 18.0º) (B) 822 J lost (C) -156 kg m/s

13. Balancing Act I. (A) kg/m3

3m 7 L3 45L ˆ (C) i 28 (B)

35. Rescue Aperation.

 5  4 cos  0   9  

(A)   cos 1  (B) 56.3º (C) 45.7º

14. Balancing Act II.

3m 7 R03 45R0 ˆ (B) i 14 (A)

Review 3: Momentum

Conservation of Energy and

6. Banana Jump. 0.40 m 15. Banana Creations. (A)

 2 2R       ,4  

(These

are

in

7. Stop the Box. (A) 15 m/s (B) 80 m/s

polar

coordinates.) (B)

8. Monkey Golf. (A) 10.0 m/s (B) 5.0 m/s

3R ˆ k 8

21. Bouncing Coconut. (A)  0.700 kg m/s

9. Monkey Power. 2.0 hp

(B) 0.626 kg m/s (C) 44.2 N (D) 0.981 J

10. Coconut Cream Pie.

139

Volume 1: Mechanics

 L 3    6    L 3 . (B)  0,   4  

18. Vector Product.

(A)  0,

Intermediate answer: m 



(A) τ  208ˆi  390ˆj  104kˆ (B) 82.0º 23. Monkey of Inertia I.

L3 8

(A)

.

15 2 mL 8 2

(B) 5mL

2

(C) 3mL

Chapter 7: Rotation

2

(D) 124mL 1. Tired. (A) 132 m (B) 50.3 rad/sec

2

(E) 24mL

2

(F) 92mL (G)

2. Physics Makes You Dizzy I. (A)

3 rad/s 2 25

13 mR 2 24

29. No-Mo Baby Toys.

3 mR 2 4 13 (B) mR 2 24 2 (C) mR 2 3 m 2 (D) (L  W 2 ) 12  m 3m2  2 (E)  1  L 5   2

(B) 9.22 m/s2 [You can get a close answer with an incorrect solution…+

(A)

3 rev 4 (D) 48 m (C)

 6  2  m/s 5   3 (F) m/s 2 125 2

(E) 

(G) 100 s

87 rad/s 250

13. Rotational Equilibrium. (A) 12,000 Nm, 0, -12,000 Nm, 0 (B) 304 N

39. Monkey Dunking. (A) 5.31 rev (B) 196 N (D) 22.5 kg m2 (F) 4.00 m/s2 (G) 2.67 rad/s2 (H) 20.0 m/s (I) 13.3 rad/s (J) 396 N, 436 N

15. Uniform Boom. (A) 5.30 kN (B) (4.08 kN, 5.50º)

43. Monkey Bowling I. (A) 37.4 m/s, 6.86 m/s (B) 34.7 m/s

(H)

12. Happy Monksgiving! -1200 Nm, 0, -500 Nm, 1600 Nm, 3000 Nm, 2900 Nm

140

Creative Physics Problems – Chris McMullen, Ph.D. (C) 0.226

Review 4: Rotation

MR 2

1. Losing Time. (A) 2.85 mrad/s (What does the ‘m’ mean?) (B) 12.0 min

47. One-Wheel Getaway. (A)

2 gh

m = 28.3 m/s m  mw

(B) 4.01 m/s2 (C) 80.3 N (sum the torques) (D) gh = 22.1 m/s (E)

Final Review: Motion 13. Fire in the Sky. (A) 100,000,000 m (B) 2000 m/s

2 gh = 31.3 m/s

48. Yo-Yo of Doom. (A) 273 g cm2 (B) 0.592 m/s2 (C) 1.54 m/s (D) 0.645 N (E) 2.60 s

14. Alien Invasion. (A) 300 km (B) 120º 15. E.T. (A)

54. Monkey-Go-Round II. (A) 0.0625 rev/sec (B) -0.0125 rev/sec2 (C) 0.244 rev

21 3 m 4

(B) 49/80 m 17. Purple-People Eater. (A) 14 m/s (B) 900 N

56. Lazy Turkey. (A) 100 kg m/s (B) 250 J (C) 40 kgm2/s (D) Angular momentum (E) 4.35 rad/s (F) 65% lost

18. Attack of the Aliens. (A) 125 N (B) 250 3 N 19. Purple People-Eater. (A) 216 rev (B) 3.0 rev/s2

57. Turkular Motion. (A) 12.5 rad/s (B) 3.20 kg m2 (C) 62.5 m/s2 (D) 1.25 kN (E) 40.0 kg m2/s (F) 0 (H) 20.0 m/s (I) 200 rad/s (J) 0.200 kg m2 (K) 4.00 km/s2 (L) 80.0 kN

20. Cocoon. ¾ 21. Body Snatchers. (A) 25/7 s (B) 28 J 22. Balance of Power. (A) 6.5 m (B) 25/63 rad/s2

141