PHYSICS WORKBOOK-I FOR 11TH GRADE IBDP STUDENTS

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TED ANKARA COLLEGE FOUNDATION HIGH SCHOOL PHYSICS DEPARTMENT

PHYSICS WORKBOOK-I FOR 11TH GRADE IBDP STUDENTS |TED Ankara College Foundation High School Physics Department

2020-2021 ACADEMIC YEAR

CONTENT

1. Measurement and uncertainty

2

Uncertainty exercise

3

Error propagation exercise

4

Graph straightening exercise

5

Using graph exercise

7

Graphing rules

13

Data analysis exercise

18

Measurement and uncertainty exercise

20

Analysis of an experiment

39

Guidelines for Physics individual investigation

41

Individual investigation criteria

46

2. Vectors/Force-Moment and equilibrium/Center of gravity

48

3. Uniform linear motion

78

4. Relative motion

98

5. Newton’s laws of motion

104

6. Motion near earth surface

124

7. Work, energy and power

144

8. Impulse and momentum

168

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MEASUREMENT AND UNCERTAINTY

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UNCERTAINTY EXERCISE A student is asked to determine the density of a piece of metal in the form of cylinder.

d

measurement

magnitude

uncertainty

d

3.15 cm

±0.01 cm

h

12.6 cm

±0.1 cm

mass

266 g

±0.002 cm

h

(a)

Determine the percentage uncertainty in h.

(b)

Determine the percentage uncertainty in mass.

(c)

Determine the percentage uncertainty in d.

(b)

Calculate the density of the cylinder and its absolute uncertainty.

Density: __________________ ± ___________ _________ Uncertainty unit

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ERROR PROPOGATION EXERCISE 1. Five people measure the mass of an object. The results are 0.56g, 0.58g, 0.58g, 0.55g, 0.59g. How would you report the measured value for the object’s mass?

2. Juan measured 8 floor tiles to be 2.67±0.03m long. What is the length of one floor tile?

3. The first part of a trip took 25 ± 3s, and the second part of the trip took 17 ± 2s. a. How long did the whole trip take?

b. How much longer was the first part of the trip than the second part?

4. The sides of a rectangle are measured to be 4.4±0.2cm and 8.5±0.3cm. Find the area of the rectangle.

5. A car traveled 600 ± 12m in 32 ± 3 s. What was the speed of the car?

6. The radius of a circle is measured to be 2.4±0.1cm. What is the area of the circle?

7. The time (t) it takes an object to fall freely from rest a distance (d) is given by the formula: 2𝑑

𝑡 = √ 𝑔 , where g is the acceleration due to gravity. A ball fell 12.5 m ± 0.3 m. How long did this take?

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GRAPH STRAIGHTENING EXERCISE Use the information provided to determine to determine what to plot to “straighten the line” for each of the following equations and then to give meaning of the slope of the straight line. 1. d= 𝑣 × 𝑡

d

d t

Slope= ______________

d t

Slope= ______________

F t

Slope= ______________

A t

Slope= ______________

t

𝑣2

2. d = 2𝑎

d

v

3. F = 4𝜋 2 𝑚𝑟𝑓 2

F

f

4. A = 6𝐼 2

A

I

V

5. V =

V t

Slope= ______________

4𝜋𝑅3 3

R

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𝑄2

6. F= 𝑘 𝑑2

F

F t

Slope= ______________

Q

𝑄

7. E = 𝑘 𝑑2

E

E t

Slope= ______________

F t

Slope= ______________

a t

Slope= ______________

d

8. 𝐹 = 𝐺

𝑀.𝑚 𝑅2

F

R

9. 𝑎 =

𝑣2 2𝑑

a

d

1

10. 𝐸 = 2 𝑚𝑣 2

E

E t

Slope= ______________

v

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USING GRAPH EXERCISE Activity 1: Determine the slope of the line in the following graphs and indicate the units on the slope.

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Activity 2: In each question, a data set, the graph of data set and an explanation for the relation among the data set are presented below. One of these representations is not like the others. Choose the one that doesn't belong in each question.

Representation 1

Representation 2

Representation 3

The relationship between x and y is linear

1

2

The relationship between x and y is linear

3

As the value of x is doubled, the value of y is halved

4

As the value of x is doubled, the value of y is doubled

5

As the value of x is doubled, the value of y stays the same

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Which one doesn’t belong?

Representation 1

Representation 2

Representation 3

As the value of x is doubled, the value of y stays the same

6

7

As the value of x is doubled, the value of y is doubled

8

The relationship between x and y is linear

9

As the value of x is doubled, the value of y is doubled

10

As the value of x is doubled, the value of y is quadrupled

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Which one doesn’t belong?

Representation 1

10

Representation 3

The quantities of x and y are inversely proportional.

11

As the value of x is doubled, the value of y is quadrupled

12

As the value of x is doubled, the value of y is quadrupled

13

Representation 2

As the value of x is doubled, the value of y is qhalved

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Which one doesn’t belong?

Activity 3: One of these representations is not like the others. Choose the one that doesn't belong and sign it.

Representation 1

Representation 2

Representation 3

1

2

3

4

5

6

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Representation 4

Representation 1

Representation 2

Representation 3

7

8

9

10

11

12

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Representation 4

GRAPHING RULES Rules for Graphing

1.

Use a RULER and GRAPH PAPER.

2.

Select a TITLE.

3.

Position the ORIGIN for the graph to FILL THE PAPER.

4.

Select a UNIFORM SCALE for the x-axis and the y-axis.

5.

LABEL the x-axis and the y-axis. What are the variable quantities?

6.

Put the INDEPENDENT VARIABLE on the X-AXIS.

7.

Put the DEPENDENT VARIABLE on the Y-AXIS

8.

Put the UNITS on the appropriate scales. While putting name and unit on the axes, a variety of notations can be used; (T/0C ; T(0C) ; T in 0C ; etc.)

9.

Complete the graph with a suitable SMOOTH CURVE.

10.IDENTIFY THE RELATIONSHIP (direct, power, inverse, etc.) 11.IDENTIFY THE MATHEMATICAL RELATIONSHIP. (State an equation if possible) 12.Put only one graph on a page. 13.Use only one side of the paper for the graph.

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Scaling for the graph The scale direction must be conventional (i.e. increasing from left to right)

5

4.2

4.2

4.0

4.0

3.8

3.8

3.6

3.6

3.4

3.4

3.2 0

3.2

4 3 2 1 1 2 – unconventional Not acceptable

0

1

2

3

4

5

Acceptable – conventional scale direction

scale direction Scales should be labelled reasonably frequently.

10

10

8

8

6

6

4

4

2

2 0

0

0

0

20

5

10

15 20

25

Acceptable – scales have regular labels.

Not acceptable – too many large squares with no label.

There should be no holes in scale.

10

10

8

8

6

6

4

4

2

2

0

0

0 5

10

15 25

Not acceptable – nonlinear scale on the x-axis

30

0

5

10

15 20

Acceptable – scales have regular labels.

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25

The graph is expected to be plotted in max scale.

4.2

4.2

4.0

4.0

3.8

3.8

3.6

3.6

3.4

3.4

3.2

0

1

2

3

4

5

3.2

0

1

2

3

4

5

Acceptable – points fill more than half the graph grid in both the x and y directions

Not acceptable – scale in the y-direction is compressed

The line must be thin and linear. Thick/ hairy/ point-to-point/kinked lines are not credit.

4.2

4.2

4.0

4.0

3.8

3.8

3.6

3.6

3.4

3.4

3.2

0

1

2

3

4

5

Not acceptable – joining point to point

3.2

0

1

2

3

Not acceptable – thick line

10 8 6 4 2 0

0

5

10

15 20

4

25

Acceptable – ‘hairy’ curve 15 |TED Ankara College Foundation High School Physics Department

5

Line (or curve) best fit There must be a reasonable balance of points about the line. It is often felt that candidates would do better if they were able to use a clear plastic rule so that points can be seen which are on both sides of the line as it is being drawn.

4.2

4.2

4.0

4.0

3.8

3.8

3.6

3.6

3.4

3.4

3.2

0

1

3.2 2

3

4

5

Not acceptable – too many points above the line

0

1

2

4.0 3.8 3.6 3.4

0

1

2

3

4

4

Acceptable balance of points about the line

4.2

3.2

3

5

Not acceptable – forced line through the origin (not appropriate in this instance)

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5

Determining gradients All the working must be shown. A ‘bald’ value for the gradient may not be credited. It is helpful to both candidates and examiners if the triangle used to find the gradient were to be drawn on the graph grid and the co-ordinates of the vertices clearly labelled.

4.2

4.2

4.0

4.0

3.8

3.8

3.6

3.6

3.4

3.4

3.2

0

1

2

3

4

5

Not acceptable – the triangle used is too small

3.2

0

1

2

3

4

Acceptable – a large triangle is used

4.2 The hypotenuse of the triangle should be greater than HALF the length of the graph line.

4.0 3.8

If plots are used which have been taken from the data table of results, then they must lie on the line of best fit.

3.6 3.4

3.2

0

1

2

3

4

5

Not acceptable – the data points which do not lie on the line of best fit are used

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5

DATA ANALYSIS EXERCISE The table below gives values of the falling time of a stone for different values of initial height (h) where the stone is dropped. (Uncertainties in measurement are also stated.)

(a)

Initial Height / m (± 1.10-3)

2

4

6

8

10

12

Falling time / sec (± 10 %)

0.64

0.90

1.10

1.27

1.43

1.56

On the grid below,draw the axes and plot the data points to show the variation with initial height H of the falling time t. Add uncertainties in falling time and draw a curve that best fits the points you have plotted on the grid

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(b) The relation between the falling time (t) and the initial height (H) is given by t = Draw the straight line that best fits on the graph.

(c) Use your graph to determine constant acceleration of free fall (g). (d) Calculate the falling time for this stone, when its initial height is 15m.

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2H g

MEASUREMENT AND UNCERTAINTIES 1. Specimen 2016 P3 The speed of sound in air, v, was measured at temperatures near 0C °. The graph shows the data and the line of best-fit. The error bars for temperature are too small to be shown.

A student suggests that the speed of sound is related to the temperature in degrees Celsius by the equation v=a+bθ where a and b are constants. (a) (i) Determine the value of the constant a, correct to two significant figures. [1]

(ii) Estimate the absolute uncertainty in b. [3]

(iii) A student calculates that b=0.593ms-1 0C-1. State, using your answer to (a)(ii), the value of b to the correct number of significant figures.[1]

(b) (i) Estimate the temperature at which the speed of sound is zero. [1]

(ii) Explain, with reference to your answer in (b)(i), why the student’s suggestion is not valid. [2]

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2. Specimen 2016 P3 A student uses an electronic timer in an attempt to estimate the acceleration of free-fall g. She measures the time t taken for a small metal ball to fall through a height (h) of 0.50 m. The percentage uncertainty in the measurement of time is 0.3 % and the percentage uncertainty height is 0.6 %. (a) Using h =

1 2 gt , calculate the expected percentage uncertainty in the value of g. [1] 2

(b) State and explain how the student could obtain a more reliable value for g. [3]

3. Specimen 2016 P3 In an experiment to measure the specific heat capacity of a metal, a piece of metal is placed inside a container of boiling water at 100 C°. The metal is then transferred into a calorimeter containing water at a temperature of 10 C°. The final equilibrium temperature of the water was measured. One source of error in this experiment is that a small mass of boiling water will be transferred to the calorimeter along with the metal. (a) Suggest the effect of the error on the measured value of the specific heat capacity of the metal. [2]

(b) State one other source of error for this experiment. [1]

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4. IB Exam May 2016 P3 A student investigates the oscillation of a horizontal rod hanging at the end of a vertical string. The diagram shows the view from above.

The student starts the rod oscillating and measures the largest displacement for each cycle of the oscillation on the scale and the time at which it occurs. The student begins to take measurements a few seconds after releasing the rod.The graph shows the variation of displacement x with time t since the release of the rod. The uncertainty for t is negligible. (a) Draw the line of best-fit for the data. [1]

(b) Calculate the percentage uncertainty for the displacement when t = 40 s. [2]

(c) The student hypothesizes that the relationship between x and t is x = test the hypothesis x is plotted against

a , where a is a constant. To t

1 as shown in the graph. t

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(i) The data point corresponding to t = 15s has not been plotted. Plot this point on the graph above. [1] (ii) Suggest the range of values of t for which the hypothesis may be assumed to be correct. [2]

5. IB Exam May 2016 P3 A student measures the refractive index of the glass of a microscope slide. He uses a travelling microscope to determine the position x1 of a mark on a sheet of paper. He then places the slide over the mark and finds the position x2 of the image of the mark when viewed through the slide. Finally, he uses the microscope to determine the position x3 of the top of the slide.

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The table shows the average results of a large number of repeated measurements.

(a) The refractive index of the glass from which the slide is made is given by

x 3 − x1 .Determine x3 − x2

(i) the refractive index of the glass to the correct number of significant figures, ignoring any uncertainty. [1]

(ii) the uncertainty of the value calculated in (a)(i). [3]

(b) After the experiment, the student finds that the travelling microscope is badly adjusted so that the measurement of each position is too large by 0.05 mm. (i) State the name of this type of error. [1]

(ii) Outline the effect that the error in (b)(i) will have on the calculated value of the refractive index of the glass. [2]

(c) After correcting the adjustment of the travelling microscope, the student repeats the experiment using a glass block 10 times thicker than the original microscope slide. Explain the change, if any, to the calculated result for the refractive index and its uncertainty. [2]

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6. IB Exam November 2015 P2 Data analysis question. An experiment is undertaken to investigate the relationship between the temperature of a ball and the height of its first bounce. A ball is placed in a beaker of water until the ball and the water are at the same temperature. The ball is released from a height of 1.00 m above a bench. The maximum vertical height h from the bottom of the ball above the bench is measured for the first bounce. This procedure is repeated twice and an average hmean is calculated from the three measurements. The procedure is repeated for a range of temperatures. The graph shows the variation of hmean with temperature T.

(a) Draw the line of best-fit for the data. [1] (b) State why the line of best-fit suggests that h mean is not proportional to T. [1]

(c) (i) State the uncertainty in each value of T. [1] (ii) The temperature is measured using a liquid in glass thermometer. State what physical characteristic of the thermometer suggests that the change in the liquid’s length is proportional to the change in temperature. [1] (d) Another hypothesis is that hmean= KT3where K is a constant. Using the graph above, calculate the absolute uncertainty in K corresponding to T = 50°C. [4]

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7. Data analysis question. The table below gives values of the stopping distance of a car for different values of its initial speed. The car applies break with a constant acceleration in each trial. (Uncertainties in measurement are also stated.) Initial Speed/kmh-1 (± 8 %)

20

40

60

80

100

120

Stopping Distance/km (± 1.10-6)

0.006

0.016

0.030

0.048

0.070

0.096

(a)

On the grid above, draw the axes and plot the data points to show the variation with initial speed V of the stopping distance x. [3]

(b)

Draw a curve that best fits the points you have plotted on the grid. [2]

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The relation between the stopping distance (x) and acceleration (a) is given by x =

The data points of v2 versus x graph are plotted below. Draw the straight line that best fits on the graph. [1]

X (km)

(c)

V2 2a

V2 (kmh-1)

(d)

By calculating the relevant uncertainty in V2, add error bar to the data point (10000, 0,070). [2]

(e)

Use your graph to determine constant acceleration of the car. [3]

(f)

Calculate the stopping distance for this car, when its initial speed is 150kmh-1. [2]

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8.

Which of the following is equivalent to the joule? B. N m–2

A. N m2

9.

11.

12.

D. kg m2 s–2

The magnitude of the mass of the universe is of the order of A. 1020 kg.

10.

C. kg m s–2

B. 1030 kg.

C. 1040 kg.

D. 1050 kg.

Which one of the following contains three fundamental units? A.

Metre

Kilogram

Coulomb

B.

Second

Ampere

Newton

C.

Kilogram

Ampere

Kelvin

D.

Kelvin

Coulomb

Second

Which of the following contains one fundamental and one derived unit? A.

ampere

kilogram

B.

ampere

coulomb

C.

joule

newton

D.

joule

coulomb

The resistive force F acting on a sphere of radius r moving at speed v through a liquid is given by F = cvr where c is a constant. Which of the following is a correct unit for c? B. N s–1

A. N

13.

C. N m2 s–1

D. N m–2 s

An object falls for a time of 0.25 s. The acceleration of free fall is 9.81 m s–2. The displacement is calculated. Which of the following gives the correct number of significant digits for the calculated value of the displacement of the object? A.

1

B.

2

C.

3

D.

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4

14.

The diagram below shows the position of the meniscus of the mercury in a mercury-inglass thermometer.

2

T / °C

6

4

8

10

Which of the following best expresses the indicated temperature with its uncertainty?

15.

A.

(6.0 ± 0.5)°C

B.

(6.1 ± 0.1)°C

C.

(6.2 ± 0.2)°C

D.

(6.2 ± 0.5)°C

The length of a rod is measured using part of a metre rule that is graduated in millimetres, as shown below.

cm

2

3

4

5

6

7

8

9

10

Which one of the following is the measurement, with its uncertainty, of the length of the rod?

16.

A.

5  0.1 cm

B.

5  0.2 cm

C.

5.0  0.1 cm

D.

5.0  0.2 cm

Values of current I in an electrical component and of the corresponding potential difference V across the component are plotted on a graph. Error bars for each point have been included. Which one of the following shows the best-fit line for the plotted points? A.

I

B.

I

V

C.

V

D.

I

I

V

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V

17. The grid below shows one data point and its associated error bar on a graph. The x-axis is not shown. 5.0 4.0 3.0 2.0 1.0

Which of the following is the correct statement of the y-value of the data point, with its uncertainty? A. 3  0.2

18.

19.

20.

B. 3.0  0.2

C. 3.0  0.20

D. 3.00  0.20

Which of the following will reduce random errors in an experiment? A.

Using an instrument having a greater precision

B.

Checking the calibration of the instrument used

C.

Checking for zero error on the instrument used

D.

Repeating readings

An ammeter has a zero offset error. This fault will affect A.

neither the precision nor the accuracy of the readings.

B.

only the precision of the readings.

C.

only the accuracy of the readings.

D.

both the precision and the accuracy of the readings.

The reading of a constant potential difference is made four times by a student. The readings are 1.176 V, 1.178 V, 1.177 V and 1.176 V. The student averages these readings but does not take into account the zero error on the voltmeter. The average measurement of the potential difference is A.

precise and accurate.

B.

precise but not accurate.

C.

accurate but not precise.

D.

not accurate and not precise.

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21.

22.

The time period T of oscillation of a mass m suspended from a vertical spring is given by 𝑚 the expression 𝑇 = 2𝜋 𝑘 , where k is a constant. Which one of the following plots will give rise to a straight-line graph? A.

T2 against m

B.

T against

C.

T against m

D.

T against m

m

A particle is moving in a circular path of radius r. The time taken for one complete revolution is T. The acceleration a of the particle is given by the expression a=

4 2 r . T2

Which of the following graphs would produce a straight-line?

23.

A.

a against T

C.

a against

1 T

B.

a against T2

D.

a against

1 T2

The frequency f of waves of wavelength λ travelling on the surface of deep water is given by f=

g 2

where g is the acceleration of free fall. Which of the following will yield a straight-line graph? y-axis

x-axis

A.

f2

1 

B.

f2



C.

f



D.

f

1 

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24.

The current in a resistor is measured as 2.00 A ± 0.02 A. Which of the following correctly identifies the absolute uncertainty and the percentage uncertainty in the current? Absolute uncertainty

Percentage uncertainty

A.

± 0.02 A

±1 %

B.

± 0.01 A

± 0.5 %

C.

± 0.02 A

± 0.01 %

D.

± 0.01 A

± 0.005 %

25.

The current, I, through a resistor is measured with a digital ammeter to be 0.10 A. The uncertainty in the calculated value of I2 will be A. 1 %.

26.

B. 2 %.

B. 12 %.

B. ±8 mm3

30.

D. 2 %.

C. ±400 mm3

D. ±600 mm3

Natalie measures the mass and speed of a glider. The percentage uncertainty in her measurement of the mass is 3% and in the measurement of the speed is 10%. Her calculated value of the kinetic energy of the glider will have an uncertainty of A. 30%.

29.

C. 8 %.

The length of each side of a sugar cube is measured as 10 mm with an uncertainty of ±2 mm. Which of the following is the absolute uncertainty in the volume of the sugar cube? A. ±6 mm3

8.

D. 20 %.

A body accelerates from rest with a uniform acceleration a for a time t. The uncertainty in a is 8 % and the uncertainty in t is 4 %. The uncertainty in the speed is A. 32 %.

27.

C. 5 %.

B. 23%.

C. 13%.

D. 10%.

A student measures a distance several times. The readings lie between 49.8 cm and 50.2 cm. This measurement is best recorded as A.

49.8  0.2 cm.

B.

49.8  0.4 cm.

C.

50.0  0.2 cm.

D.

50.0  0.4 cm.

The period T of oscillation of a mass m attached to the end of a spring is given by

T = 2 m , where k is an accurately known constant. The mass is 0.500  0.045 kg. k What is the percentage uncertainty in the calculated value of the period? A. 3.0

B. 4.5

C. 9.0

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D. 18

31.

The volume V of a cylinder of height h and radius r is given by the expression V = r2h. In a particular experiment, r is to be determined from measurements of V and h. The uncertainties in V and in h are as shown below. V

7

h

3

The approximate uncertainty in r is A.

32.

10.

B.

5.

C.

4.

D.

2.

Two lengths, a and b, are measured to be 51±1 cm and 49±1 cm respectively. In which of the following quantities is the percentage uncertainty the largest? A. a + b

B. a – b

C. a × b

D.

a b

33. The masses and weights of different objects are independently measured. The graph is a plot of weight versus mass that includes error bars.

These experimental results suggest that the A.

measurements show a significant systematic error but small random error.

B.

measurements show a significant random error but small systematic error.

C.

measurements are precise but not accurate.

D.

weight of an object is proportional to its mass.

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34.

35.

Which of the following is a valid statement? A.

A measurement that is not precise can be accurate.

B.

A measurement that is precise is always accurate.

C.

A measurement that is not precise will always be inaccurate.

D.

Repeated measurements will always increase accuracy and precision.

In an experiment to measure the acceleration of free fall at the surface of the Earth the following results were obtained. Acceleration of free fall / m s–2 7.69 7.70 7.69 7.68 7.70 The results are A.

accurate and precise.

B.

inaccurate but precise.

C.

accurate but imprecise.

D.

inaccurate and imprecise.

Textbook (TSOKOS) Homework:

Please solve the questions 23-31 on page 20, the question 7 on page 32 the questions 8-11 on page 33.

34 |TED Ankara College Foundation High School Physics Department

KEY

35 |TED Ankara College Foundation High School Physics Department

4

4

4

36 |TED Ankara College Foundation High School Physics Department

5

6.

7. (a) (b)

Stopping distance/km

0.100 0.096

20

40

60

80

100

120

Inıtial speed /kmh-1

(c) A suitable straight line must be plotted 37 |TED Ankara College Foundation High School Physics Department

(d) 𝑉 2 = 10000 ± 16% and so ∆𝑉 2 = 1600𝑚𝑘ℎ−2 ;

[1]

Length of each horizontal bar should be qual to 1.6 units.

[1]

1

1

(e) 𝑉 2 = 2𝑎𝑥 ; 𝑥 = 2𝑎 𝑉 2 ; 𝑠𝑙𝑜𝑝𝑒 = 2𝑎 = 6.7 × 10−6 ℎ 𝑎=

1 2×𝑠𝑙𝑜𝑝𝑒

=

1 2×6.7×10−6

(f) 𝑥 = 𝑠𝑙𝑜𝑝𝑒 × 𝑉 2

; [1]

; 𝑎 = 7.5 × 104 𝑘𝑚ℎ−2

and so

𝑥 = 6.7 × 10−6 × 1502

[2]

;

𝑥 = 0.15𝑘𝑚 = 150𝑚 [2]

_________________________________________________________________________ 8.D

9. D

10. C

11. B

12. D

13. B

14. C

15 . C

16 . B

17 . B

18. D

19. C

20. B

21. A

22. D

23. A

24. A

25. D

26. B

27. D

28 . B

29. C

30. B

31. B

32. B

33. A

34. A

35. B

______________________________________________________________________

38 |TED Ankara College Foundation High School Physics Department

ANALYSIS OF AN EXPERIMENT The presentation and analysis of experimental results is an essential part of Physics. In the table below are the results of experiment. You are asked to present and analyze these results in a form which will enable you to predict the outcome of similar experiments.

d/cm (±0,1)

h/cm (±0,5%) 30.0

10.0

4.0

1.0

1.5

73.0

43.5

26.7

13.5

2.0

41.2

23.7

15.0

7.2

3.0

18.4

15.0

6.8

3.7

5.0

6.8

3.9

2.2

1.5

Table of times (in seconds) for water to poor out of a can for varying heights of water in the can and varying diameters of hole at bottom of the can. The experiment was an investigation of the time it takes water to pour out of a can through a hole in the bottom. As you would expect, this time depends on the size of the hole and the amount of water in the can.

h

To find the dependence on the size of the hole, four large cylindrical containers of water of the same size were emptied trough relatively small circular openings of different diameters. To find the dependence on the amount of water, the same containers were filled to different heights.

d

Each measurement was repeated several times, and the averages of the times (in seconds) that each container took to empty have been entered in the table. A stop watch operated by a human hand cannot be trusted to measure less than a tenth of a second. The last digit in each time entry in the table may be error by one unit either way. Therefore, the relative (or fractional) error is larger for shorter times than for longer times. You are asked to plot necessary graphs by using Vernier Logger Pro data-collection and analysis program and obtain an equation for the relation between variables of experiment. The following graphs are expected to be plotted and analyzed: 1. First, plot the time versus the diameter of opening for a constant height you choose. You will get a smoot curve. From your graph you will see that t decreases rather rapidly with increasing d. This suggests an inverse relationship. 1

2. Plot time (t) versus reciprocal of d ( 𝑑 ). Examine this graph. What do you conclude about the relation of t and d? If the graph is not linear then you cannot say that time is inversely proportional to d. 39 |TED Ankara College Foundation High School Physics Department

1

3. This time try a plot of time (t) versus reciprocal of d2 ( 𝑑2 ). Examine the graph line and decide whether t is inversely proportional to d2 or not. 4. Now, to investigate the dependence of time on height of water (h), plot t versus h graph when the diameter of hole is fixed at a value you prefer. You will get a smoot parabola. By examining this curve try to guess right mathematical expression between t and h. If you succeed, check by further graphing (follow 5) to see if your prediction is correct or not. 5. In order to verify your guess on the relationship between t and h, make necessary conversions on graph in (4) to strengthen the graph line. As a result, you should have a straight-line graph of t versus suitable power of h. 6. In order to find the general expression for the time of flow as a function of both h and d, 𝑡=a

ℎ𝑚 𝑑𝑛

a linear graph of three variables should be constructed. At first determine the value of m and n by ℎ𝑚

using proper graphs you have plotted. Plot t versus 𝑑𝑛 to obtain a linear graph. The gradient of this graph is equal to a. Determine the gradient of line. 7. At the end, determine the function of time for water to empty through a hole at the bottom of a can depending on the height of water in can and the diameter of hole -t(h,d).

40 |TED Ankara College Foundation High School Physics Department

GUIDELINES FOR PHYSICS INDIVIDUAL INVESTIGATIONS Introduction to the new assessment model for IA The new assessment model uses five criteria to assess the final report of the individual investigation with the following raw marks and weightings assigned: Personal Engagement 2 (8%)

Exploration 6 (25%)

Analysis 6 (25%)

Evaluation 6 (25%)

Communication 4 (17%)

Total 24 (100%)

Levels of performance are described using multiple indicators per level. In many cases the indicators occur together, but not always. Also, not all indicators are always present. This means that a candidate can demonstrate performances that fit into different levels. To accommodate this, the IB assessment models use mark bands and advise examiners and teachers to use a best-fit approach in deciding the appropriate mark for a particular criterion. Please read the guidance on using mark bands shown above before starting to mark, it is also essential to be fully acquainted with the marking of the exemplars in the Teacher Support Material. The precise meaning of the command terms used in the criteria can be found in the glossary of the subject guides. Personal engagement This criterion assesses the extent to which the student engages with the exploration and makes it their own. Personal engagement may be recognized in different attributes and skills. These could include addressing personal interests or showing evidence of independent thinking, creativity or initiative in the designing, implementation or presentation of the investigation. 0

The student’s report does not reach a standard described by the descriptors below.

1

The evidence of personal engagement with the exploration is limited with little independent thinking, initiative or insight. ▪ The justification given for choosing the research question and/or the topic under investigation does not demonstrate personal significance, interest or curiosity. ▪ There is little evidence of personal input and initiative in the designing, implementation or presentation of the investigation.

2

The evidence of personal engagement with the exploration is clear with significant independent thinking, initiative or insight. ▪ The justification given for choosing the research question and/or the topic under investigation demonstrates personal significance, interest or curiosity. ▪ There is evidence of personal input and initiative in the designing, implementation or presentation of the investigation.

41 |TED Ankara College Foundation High School Physics Department

Exploration This criterion assesses the extent to which the student establishes the scientific context for the work, states a clear and focused research question and uses concepts and techniques appropriate to Diploma level. 0 1-2

The student’s report does not reach a standard described by the descriptors below. ▪ ▪

▪

▪

3-4

▪ ▪

▪

▪

5-6

▪ ▪

▪

▪

The topic of the investigation is identified and a research question of some relevance is stated but it is not focused. The background information provided for the investigation is superficial or of limited relevance and does not aid the understanding of the context of the investigation. The methodology of the investigation is only appropriate to address the research question to a very limited extent since it takes into consideration few of the significant factors that may influence the relevance, reliability and sufficiency of the collected data. The report shows evidence of limited awareness of the significant safety, ethical or environmental issues that are relevant to the methodology of the investigation * The topic of the investigation is identified, and a relevant but not fully focused research question is described. The background information provided for the investigation is mainly appropriate and relevant and aids the understanding of the context of the investigation. The methodology of the investigation is mainly appropriate to address the research question but has limitations since it takes into consideration only some of the significant factors that may influence the relevance, reliability and sufficiency of the collected data. The report shows evidence of some awareness of the significant safety, ethical or environmental issues that are relevant to the methodology of the investigation*. The topic of the investigation is identified, and a relevant and fully focused research question is clearly described. The background information provided for the investigation is entirely appropriate and relevant and enhances the understanding of the context of the investigation. The methodology of the investigation is highly appropriate to address the research question because it takes into consideration all, or nearly all, of the significant factors that may influence the relevance, reliability and sufficiency of the collected data. The report shows evidence of full awareness of the significant safety, ethical or environmental issues that are relevant to the methodology of the investigation.

* This indicator should only be applied when appropriate to the investigation. See exemplars in TSM.

42 |TED Ankara College Foundation High School Physics Department

Analysis This criterion assesses the extent to which the student’s report provides evidence that the student has selected, recorded, processed and interpreted the data in ways that are relevant to the research question and can support a conclusion.

0

1-2

The student’s report does not reach a standard described by the descriptors below.

▪ ▪ ▪ ▪

3-4

▪

▪

▪ ▪

5-6

▪ ▪

▪ ▪

The report includes insufficient relevant raw data to support a valid conclusion to the research question. Some basic data processing is carried out but is either too inaccurate or too insufficient to lead to a valid conclusion. The report shows evidence of little consideration of the impact of measurement uncertainty on the analysis. The processed data is incorrectly or insufficiently interpreted so that the conclusion is invalid or very incomplete.

The report includes relevant but incomplete quantitative and qualitative raw data that could support a simple or partially valid conclusion to the research question. Appropriate and sufficient data processing is carried out that could lead to a broadly valid conclusion but there are significant inaccuracies and inconsistencies in the processing. The report shows evidence of some consideration of the impact of measurement uncertainty on the analysis The processed data is interpreted so that a broadly valid but incomplete or limited conclusion to the research question can be deduced.

The report includes sufficient relevant quantitative and qualitative raw data that could support a detailed and valid conclusion to the research question. Appropriate and sufficient data processing is carried out with sufficient accuracy so as to enable a conclusion to the research question to be drawn that is fully consistent with the experimental data. The report shows evidence of full and appropriate consideration of the impact of measurement uncertainty on the analysis The processed data is correctly interpreted so that a completely valid and detailed conclusion to the research question can be deduced.

43 |TED Ankara College Foundation High School Physics Department

Evaluation This criterion assesses the extent to which the student’s report provides evidence of evaluation of the investigation and the results regarding the research question and the accepted scientific context.

0

1-2

The student’s report does not reach a standard described by the descriptors below.

▪ ▪ ▪

▪

3-4

▪ ▪ ▪

▪

5-6

▪ ▪ ▪

▪

A conclusion is outlined which may not be relevant to the research question or may not be supported by the data presented. The conclusion is erroneous or superficial compared to the accepted scientific context. Strengths and weaknesses of the investigation, such as limitations of the data and sources of error, are outlined but are restricted to an account of the practical or procedural issues faced. The student has outlined very few realistic and relevant suggestions for the improvement and extension of the investigation.

A conclusion is described which is relevant to the research question and supported by the data presented. A conclusion is described which makes some relevant comparison to the accepted scientific context. Strengths and weaknesses of the investigation, such as limitations of the data and sources of error, are described and provide evidence of some awareness of the methodological issues* involved in establishing the conclusion. The student has described some realistic and relevant suggestions for the improvement and extension of the investigation.

A conclusion is described and justified which is relevant to the research question and supported by the data presented. A conclusion is correctly described and justified through relevant comparison to the accepted scientific context. Strengths and weaknesses of the investigation, such as limitations of the data and sources of error, are discussed and provide evidence of a clear understanding of the methodological issues* involved in establishing the conclusion. The student has discussed realistic and relevant suggestions for the improvement and extension of the investigation.

44 |TED Ankara College Foundation High School Physics Department

Communication This criterion assesses whether the investigation is presented and reported in a way that supports effective communication of the focus, process and outcomes. 0

The student’s report does not reach a standard described by the descriptors below.

1-2

The presentation of the investigation is unclear, making it difficult to understand the focus, process and outcomes. ▪ The report is not well structured and is unclear: The necessary information on focus, process and outcomes might be missing or is presented in an incoherent or disorganized way. ▪ The understanding of the focus, process and outcomes of the investigation is obscured by the presence of inappropriate or irrelevant information. ▪ There are many errors in the use of subject specific terminology and conventions*.

3-4

The presentation of the investigation is clear. Any errors do not hamper understanding of the focus, process and outcomes. ▪ The report is well structured and clear: the necessary information on focus, process and outcomes is present and presented in a coherent way. ▪ The report is relevant and concise thereby facilitating a ready understanding of the focus, process and outcomes of the investigation. ▪ The use of subject specific terminology and conventions is appropriate and correct. Any errors do not hamper understanding.

*e.g. incorrect/missing labelling of graphs, tables, images; use of units, decimal places. For issues of referencing and citations refer to the academic honesty section.

45 |TED Ankara College Foundation High School Physics Department

INDIVIDUAL INVESTIGATION CRITERIA Descriptor

0

1

2

Evidence of personal engagement with exploration.

Standard not reached.

Limited with little independent thinking, initiative or insight.

Clear with significant independent thinking, initiative or creativity.

The justification given for choosing the research question and/or the topic under investigation.

Standard not reached.

Does not demonstrate personal significance, interest or curiosity.

Demonstrates personal significance, interest or curiosity.

Evidence of personal input and initiative in the designing, implementation or presentation of the investigation.

Standard not reached.

Little

A lot

Exploration Descriptor The topic of the investigation is identified and research question.

0 Standard not reached

1-2 Some relevance is stated but not focused.

3-4 Relevant but not fully focused.

5-6 Relevant and fully focused.

The background information provided for the investigation is.

Standard not reached

Superficial or of limited relevance and does not aid the understanding of the context of the. investigation

Mainly appropriate and relevant and aids the understanding of the context of the investigation.

Entirely appropriate and relevant and enhances the understanding of the context of the investigation.

Appropriateness of the methodology of the investigation.

Standard not reached

Limited

Mainly

Highly

Consideration of factors that may influence the relevance reliability and sufficiency of collected data.

Standard not reached

Few factors considered.

Some factors considered.

Nearly all factors considered.

Evidence of awareness of the significant safety, ethical or environmental issues that are relevant to the methodology of the investigation*.

Standard not reached

Limited

Some

Full

Analysis Descriptor Raw data is

0 Standard not reached.

1-2 Insufficient to support a valid conclusion.

3-4 Relevant but incomplete. Could support a simple or partially valid conclusion.

5-6 Sufficient. Could support a detailed and valid conclusion.

Data processing

Standard not reached.

Basic, inaccurate or too insufficient to lead to a valid conclusion

Appropriate and sufficient. Could lead to a broadly valid conclusion but significant inaccuracies and inconsistencies in the processing.

Appropriate and sufficient accuracy enables a conclusion to the RQ to be drawn that is fully consistent with the experimental data.

Impact of uncertainties

Standard not reached.

Little consideration.

Some consideration.

Full and appropriate consideration.

Interpretation of processed data

Standard not reached.

Incorrect or insufficient invalid or very incomplete

Broadly valid limited conclusion.

Correct valid and detailed conclusion.

46 |TED Ankara College Foundation High School Physics Department

Evaluation Descriptor Conclusion

0 Standard not reached

1-2 Outlined but may not be relevant to the research question or may not be supported by the data.

3-4 Described, relevant to the research question and supported by the data.

5-6 Described in detail and justified, entirely relevant to the RQ fully supported by the data.

Conclusion

Standard not reached.

Erroneous or superficially compared to the accepted scientific context.

Some relevant comparison to accepted scientific context.

Justified through relevant comparison to the accepted scientific context.

Strengths and weaknesses of the investigation, such as limitations of the data and sources of error, are

Standard not reached.

Outlined but are restricted to an account of the practical or procedural issues faced.

Described and provide evidence of some awareness of the methodological issues* involved in establishing. the conclusion

Discussed and provide evidence of a clear understanding of the methodological issues* involved in establishing. the conclusion

Realistic and relevant suggestions for the improvement and extension of the investigation.

Standard not reached.

Very few outlined.

Some described.

Are discussed.

Communication Descriptor Presentation of the investigation

0 Standard not reached.

1-2 Unclear, making it difficult to understand the focus, process and outcomes.

3-4 Clear. Any errors do not hamper understanding of the focus, process and outcomes.

Structure

Standard not reached.

Not well structured and is unclear: the necessary information on focus, process and outcomes is missing or is presented in an incoherent or disorganized way.

Well-structured and clear: the necessary information on focus, process and outcomes is present and presented in a coherent way.

Relevance

Standard not reached.

The understanding of the focus, process and outcomes of the investigation is obscured by the presence of inappropriate or irrelevant information.

Relevant and concise thereby facilitating a ready understanding of the focus, process and outcomes of the investigation.

Terminology

Standard not reached.

There are many errors in the use of subject specific terminology and conventions*.

The use of subject specific terminology and conventions is appropriate and correct. Any errors do not hamper understanding.

47 |TED Ankara College Foundation High School Physics Department

VECTORS FORCE AND TRANSLATIONAL EQUILIBRIUM TORQUE AND ROTATIONAL EQUILIBRIUM CENTER OF GRAVITY

48 |TED Ankara College Foundation High School Physics Department

VECTORS 1. Can two vectors of different magnitude be combined to give a zero resultant?

2. Can the difference of two vectors have a greater magnitude than the sum of the same vectors?

3. Can three vectors of different magnitude be combined to give a zero resultant?

4. A and B are two points on the plane. Their coordinates are A (14, 8) and B (2, 3). If an object travels from A to B, what is the displacement of this object?

5. A room has the dimensions 3m, 4m, and 12m. A fly starting at one corner ends up at a diametrically opposite corner. What is the magnitude of its displacement?

6.

Three vectors, which are on the same plane, are given in Figure. Find, a) A + B + C B = 12 u

C=8u

b) A + B -2C

A=5u

7. Four vectors are on the same plane as in Figure. Find, C

a) A + B + C + D B

D A

b) A + C – D c) D – B – C

49 |TED Ankara College Foundation High School Physics Department

8. A, B and C are three vectors on the same plane. The vector A, the sum of vectors A and B (A+B), and B and C (B+C) are given in Figure. Find the magnitudes and the directions of B and C. A

A+B

B+C

9. F1, F2, F3 are three vectors, they are placed on the same plane. Find the magnitude and direction of F1 vector. F1 +F2 1 unit

F2 +F3 F3

10. Two vectors A and B are on the same plane as seen in Figure. What are the magnitude and direction of the resultant vector? B=10 N

370 A = 16 N

11. Four vectors are on the same plane. Find the magnitude and direction of A+B+C+D. C=1N

D=2N

450

B=2N

450 450

A=4N

12. A, B, C, D, and E are five vectors. They are on the same plane. What is the magnitude of A+B+C+D+E? C=3m B=1m

D=1m 0

E=3m

600

30

600

A=2m

50 |TED Ankara College Foundation High School Physics Department

FORCE AND EQUILIBRIUM 13. The system shown below is in equilibrium. Find the frictional force acting on B. 370

B

A=120N

14. A sphere is hung by a rope on a wall as in Figure. The weight of the sphere is 24 N. a) What is the tension in the rope?

37

b) What is the reaction force exerted by the wall on the sphere?

15. A small object of weight 10 N rests in equilibrium on a rough inclined plane as in Figure. Calculate the magnitude of the frictional force.

37 

16. System in Figure is in equilibrium. Weight of uniform sphere is 80 N.

37 

a)

Find the tension in the rope.

b)

Find the reaction force exerted by the inclined plane.

51 |TED Ankara College Foundation High School Physics Department

17. This question is about the equilibrium of forces. (a) (i) State the difference between a scaler and a vector quantity.

(ii) State two examples of a scaler quantity and a vector quantity.

(b) Figure below shows a ship fitted with a sail attached to a cable. The force of the wind on the sail assists the driving force of the ship’s propellers.

The mass of ship is 1ton and the cable exerts a steady force of 2.8kN on the ship at angle of 350 above a horizontal line. (i) Calculate the horizontal and vertical components of this force

(ii) While the ship is floating on the sea, what is the buoyant force applied by the sea on the ship?

(iii) Is the ship in equilibrium in this position? Explain your reasoning.

52 |TED Ankara College Foundation High School Physics Department

18. The helicopter shown in Figure A is moving horizontally through still air. The lift force from the helicopter’s blades is labelled A.

Figure B

Figure A (a) Name the two forces B and C that also act on the helicopter.

B

...........................................................................................................

C

...........................................................................................................

(b) State the condition for translational equilibrium

(c) The force vectors are also shown arranged as a triangle in Figure B. State and explain how Figure B shows that the helicopter is moving at a constant velocity.

(d) The lift force, A, is 9.5 kN and acts at an angle of 74° to the horizontal. Calculate the weight of the helicopter. Give your answer to 2 significant figures.

53 |TED Ankara College Foundation High School Physics Department

TORQUE AND EQUILIBRIUM 19. Calculate the total moment on the stick about point O in terms of “Fa”. The stick is equally divided and its weight is neglected. 2F

4F a

300 O

1500 F

3F

20. If the system given in Figure is in equilibrium and the bar is weightless, determine the location of pivot point. 30 N B

A 2m

2m

3m 10 N

20 N

50 N

21. A uniform rod of length 2m and weight 20N rests horizontally on supports A and B and the system is in equilibrium. A load of 10 N is attached to the rod at a distance 0.4 m from A. Find the forces exerted on the rod by the supports. 0.4 m A

B W =10 N

22. The rod AB is weightless. To keep the rod in equilibrium what must be value of the minimum and maximum forces of X? A

B

X=? 20N

54 |TED Ankara College Foundation High School Physics Department

23. ABC is an equilateral triangle of side 2m. Find the total moment about point A created by the exerted forces. C

2N

4N 1N A 3N

B

24. Weight of each uniform square plate is 3N. System is in equilibrium. Find tension T.

T T1

370

T2

25. System shown in the figure is in equilibrium. Weight of the equally spaced uniform rod is 10 N. Find the tension in the rope. F=60N

530

26. The rigid stick OK is weightless and can rotate about point O. Find P in terms of F. K

F

P 530 O

55 |TED Ankara College Foundation High School Physics Department

27. A weightless and equally divided rod is in equilibrium, what is tension T?

T=? P=20N 37

28. Figure below shows a pole vaulter holding a uniform pole horizontally. He keeps the pole in equilibrium by exerting an upward force, U, with his leading hand, and a downward force, D, with his trailing hand.

Calculate for the situation in figure above (a) the force, U

(b) the force, D

56 |TED Ankara College Foundation High School Physics Department

29. A waiter holds a tray horizontally in one hand between fingers and thumb as shown in diagram. 0.01m

0.25m P

W

P, Q and W are the three forces acting on the tray. (a) State two relationship between the forces that must be satisfied if the tray is to remain horizontal and in equilibrium.

(b) If the mass of tray is 0.12kg, calculate the magnitude of forces P and Q.

(c) The waiter places a glass on the tray. State and explain where the glass should be positioned on the tray if the force P, is to have the same value as in part (a).

30. The diagram shows the rear door of a station wagon supported horizontally by a strut. The mass of the door is 18 kg and the compression force in the strut is 450 N. At what distance, x, from the hinge is the center of gravity of the door located?

57 |TED Ankara College Foundation High School Physics Department

31. Figure below shows a supermarket trolley

10cm

40cm

50cm

The weight of the trolley and its contents is 150N. (a) State the condition in order that the trolley can remain stable.

(b) P and Q are the resultant forces that the ground exerts on the rear wheels and front wheels respectively. Calculate the magnitude of forces P and Q.

(c) Calculate the minimum force that needs to be applied vertically at A to lift the front wheels off the ground.

(d) State and explain, without any calculation, how the minimum force that needs to be applied vertically at A to lift the rear wheels off the ground compares to the force you calculated in (c).

58 |TED Ankara College Foundation High School Physics Department

32. The arm pulls against the strap as shown so that the scale reads a force of 100 N.

F =100 N Strap

28 cm 32 cm biceps

4 cm

humerus

elbow joint

(a) On the diagram above, draw in force vectors to represent all the forces acting on the forearm. State what object exerts each force. (b) Calculate the force exerted by the biceps muscle.

(c) Explain why, in this particular situation, the weight of the forearm does not play a role in determining the force exerted by the biceps muscle.

(d) Figure below shows the force, F, on a bicycle pedal. Force, F, is constant in magnitude and direction while the pedal is moving downwards. State and explain how moment of F changes as the pedal moves through 800, from the position shown.

59 |TED Ankara College Foundation High School Physics Department

33. A uniform plank of wood of mass 32 kg and length 4.0 m is used by a boy to help him cross a ditch. In the ditch there is a rock, which is used to support the plank horizontally 0.80 m from one end, as shown in figure below. The other end of the plank is supported by the bank. The boy has a mass of 48kg.

plan

4.0m 0.8m bank ditch

rock

(a) Define the moment of a force about a point.

(b) State the condition for rotational equilibrium.

(c) Calculate the vertical supporting force from the rock when the plank is placed in position as shown in figure. (The boy is not on the plank yet.)

(d) The child comes up on the plank and walks 1m from the left end. Calculate the vertical supporting forces from both rock and bank on the plank.

(e) While the child is walking through the right end of the plank, (i) how do the supporting forces from rock and from bank change?

(ii) Is the left end of the plank rised and disconnected from the ground?

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34. Two identical rods are soldered at an angle of 143 degrees and then they are attached to the wall by the means of a hinge. The system can rotate freely around the hinge. Take sin (37) = cos (53) = 0.6 and sin (53) = cos (37) = 0.8 for this question.

l l

F

horizontal

1430

A

The system is kept in equilibrium as shown in figure above by the help of a horizontal force F. Each of the rods has a weight of 12 Newtons. (a) State the condition for rotational equilibrium.

(b) Calculate the magnitude of horizontal force F.

(c) Use the concept of equilibrium to explain why a force must act on the rod at point A.

(d) Calculate (i)

the horizontal component of force exerted on the rod at point A.

(ii)

the vertical component of force exerted on the rod at point A.

61 |TED Ankara College Foundation High School Physics Department

35. The crate of width (a) and height (h) and weight (W) is held equilibrium by a force F as shown in Figure.

F α

h

A

a

Given are: Weight (W) = 2000 N Angle (α) = 37o Width (a) = 4 m Height (h) = 2 m

(a) Find the magnitude of the force F necessary to keep the crate in the position shown.

(b) Find the support force at point A.

62 |TED Ankara College Foundation High School Physics Department

CENTER OF GRAVITY 36. This question is about center of gravity and rotational equilibrium.

(a) Describe what is meant by the center of gravity of an object.

(b) A sheet of cardboard is pivoted at point P and is held in the position shown in the diagram below. P

The centre of gravity of the sheet is at point C. (i) On the diagram above, draw an arrow labelled W to represent the weight of the cardboard.

C

(ii) The cardboard sheet is now released. Explain why, when the sheet comes to rest, point C will be below point P.

(c) Why does the Leaning Tower of Pisa not topple over?

37. A bar 5m long has its center of gravity 1.5m from its heavy end. It is placed on a pointed support 1.5m from its light end and a weight of 400N is loaded at the light end. The bar is balanced in this way. What is the weight of the bar?

38. A 200 cm homogenous bar is folded 50 cm on itself from one end. How far is the center of newly formed body shifted from that of the original center of gravity?

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39. A rectangle is formed by a uniform wire as shown in figure I and is changed as shown in figure II: What is the displacement of the center of gravity? 4 cm

4 cm

2 cm

Fig II

Fig I

2 cm

40. Uniform square lamina of side 24 cm is attached to a uniform triangle lamina. Referring the figure, find the distance between the center of gravity of the newly formed body and center of the square.

41. Circular uniform wires made of the same material are joined as shown in Figure. What is the distance between O1 and center of the newly formed system? O1 3r

O2

O3

r

2r

42. Two uniform circular plates made of same material are combined as shown in figure. How many cm away is the center of gravity of combination from C1?

C1

C2

R1 = 4 cm

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43. ABCD is a uniform rectangular plate. The shaded part of this plate is cut off and taken away from the plate. How many cm will the center of gravity shift? (π=3) 20 cm

10 cm

44. K is the center of gravity of a uniform rectangular plate. The shaded part of this plate is cutoff. What is the shifting of center of gravity of this plate after cutting off? 36 cm K

18 cm

18 cm

18 cm

45. The circular plates given in Figure are made up of different materials but have equal thicknesses. When they are joined rigidly as in Figure, determine the position of center of gravity of newly produced body with respect to O1. R1 = 2r, d1 = d

O1

R2 = r, d2= 2d

O2

46. If the shaded part of the uniform circular plate with radius of 6cm is cut off and stuck again as in the figure, find how far is the center of gravity shifted?

r=6cm 60

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47. This question is about the stability of a loaded truck.

A loaded truck is parked on a sloping ground as seen in figure above. The center of gravity of loaded truck depends on how the truck is packed. It is 4.0m high, 2.4 m wide and its center of gravity is 2.2m above the ground. (a) The truck in figure is in stable equilibrium. What condition should be satisfied in order that truck can stay without tipping over?

(b) Calculate the steepest angle (θ) that the truck can be parked on without tipping over?

48. A system is formed by attaching a sphere whose radius is 4r and many smaller spheres whose radii are r. All of the spheres are made up of the same material. In order that the center of gravity of the system becomes at point K, how many small spheres should be used?

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49. A square paper with side length 24cm is folded up from its corners as shown in the figure. What are the direction and the magnitude of the displacement of the center of the mass of the plate relative the point O?

50. A cylinder of length 4r and radius r is removed from a cylinder of length 4r and the radius 2r and then attached to the same cylinder as shown in figure. What is the change in the position of the center of mass of the system? 4r

r r

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MULTIPLE CHOICE QUESTIONS ON CENTER OF GRAVITY AND EQUILIBRIUM CONDITIONS 1. Two forces of magnitudes 7 N and 5 N act at a point. Which one of the following is not a possible value for the magnitude of the resultant force? A. 1 N

B. 3 N

C. 5 N

D. 7 N

2. Three coplanar forces, each of magnitude 10 N, act through the same point of a body in the directions shown. 10N

10N 300 300

What is the magnitude of the resultant force? A. 0N

B. 1.3N

C. 7.3N

D. 10N

10N 3. A 220 N bag of potatoes is suspended from a rope as shown in the diagram. A person pulls horizontally on the bag with a force of 80 N. What is the tension in the rope? A. 1.4×102 N

B. 2.2×102 N

C. 2.3×102 N

D. 3.0×102 N

4. Two forces act on an object as shown. Find the magnitude of the third force required to achieve translational equilibrium A. 15 N

B. 33 N

C. 47 N

D. 65 N

5. A mass of 5.0 kg is suspended from a cord as shown in the diagram below. What horizontal force F is necessary to hold the mass in the position shown?

A. 28 N

B. 35 N

C. 40 N

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D. 70 N

6. A mass suspended by a string is held 24°from vertical by a force of 13.8 N as shown. Find the mass.

A. 0.6 kg

B. 1.5 kg

C. 3.4 kg

D. 4.5 kg

7. A 25 kg block is pulled by a horizontal force. The supporting rope can withstand a maximum tension force of 620N. To what maximum angle, θ, can the block be pulled without the rope breaking?

A. 220

B. 230

C. 670

D. 880

8. In which direction should a force act at point P to hold the boom in equilibrium so that the force will be a minimum?

A. 1

B. 2

C. 3

D. 4

9. An 85.0 kg mountaineer remains in equilibrium while climbing a vertical cliff. The tension force in the supporting rope is 745 N. Find the magnitude of the reaction force, F, which the cliff exerts on the mountaineer's feet. A. 88.0N

B. 373 N

C. 479 N

D. 554 N

10. What is the unit of torque? A. N×m

B. N/m

C. N×s

D. N/s

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11. An artist must push with a minimum force of 75 N at an angle of 45°to a picture to hold it in equilibrium. The coefficient of friction between the wall and the picture frame is 0.30. What is the mass of the picture?

A. 1.6 kg

B. 2.3 kg

C. 3.8 kg

D. 6.9 kg

12. A body is in rotational equilibrium when A. Στ=0

B. ΣF=0

C. ΣP=0

D. ΣEk =0

13. A body is in static equilibrium when A. Στ = 0 only.

B. ΣF= 0 only.

C. ΣF= 0 and Στ = 0.

D. ΣF= Στ

14. Which of the four problems shown requires the application of torque?

15. A force F is applied to a uniform horizontal beam as shown in the diagram below.

Which of the following is a correct expression for the torque on the beam about pivot point P due to this force? A. Fsinθ × d

B. Fsinθ × d/l

C. Fcosθ × d

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D. Fcosθ × d/l

16. The diagram shows four forces applied to a circular object. Which of the following describes the resultant force and resultant torque on the object?

30N

resultant force

resultant torque

A

zero

zero

B

zero

non-zero

C

non-zero

zero

D

non-zero

non-zero

20 N

20 N 30 N

17. A beam is to be kept horizontal by a cord. In which of the four situations shown below will the tension in the cord be least?

18. The diagram shows the forces acting on a massless ladder resting on the floor and a frictionless inclined surface. As a person walks up the stationary ladder, what happens to the magnitude of the forces FN1 and FN2? Magnitude of FN1

Magnitude of FN2

A.

Decreases

Decreases

B.

Decreases

Increases

C.

Increases

Decreases

D.

Increases

Increases

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19. A trailer carrying a boat is supported by a scale which initially reads 48 kg. The boat (and therefore its center of gravity) is moved 0.15 m further back on the trailer. The scale now reads 37 kg. Find the mass of the boat.

A. 440 kg

B. 1600 kg

C. 1700 kg

D. 3400 kg

20. An L-shaped rigid lever arm is pivoted at point P. Three forces act on the lever arm, as shown in the diagram.

P 5N 2m 2m 10N 3m

1m 20N

What is the magnitude of the resultant moment of these forces about point P? A. 30 Nm

B. 35 Nm

C. 50 Nm

D. 90 Nm

21. A uniform 2.5 kg beam, pivoted at its right end, is held in a horizontal position by a cable as shown in the diagram.

If the cable is attached 0.10 m to the left of the beam's center of gravity, how long is the beam? A. 0.34 m

B. 0.60 m

C. 1.2 m

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D. 9.6 m

22. A 35 kg uniform plank is balanced at one end by a 55 kg student as shown.

What is the overall length of this plank? A. 2.6m

B. 3.3m

C. 5.4m

D. 6.7m

23. A uniform 1. 5 kg beam hinged at one end supports a 0.50 kg block. The beam is held level by a vertical 0.80kg rod resting on a Newton scale at the other end.

What is the reading on the scale? A. 8.6N

B. 9.1N

C. 16N

D. 27N

24. A boom hinged at P is held stationary, as shown in the diagram below.

If the tension in the supporting cord, attached three-quarters of the way along the boom from P, is 720 N, what is the weight of the boom? A. 720 N B. 1080 N C. 1440 N D. 2160 N

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25. A uniform beam of weight 100N is pivoted at P as shown. Weights of 10N and 20N are attached to its ends . The length of the beam is marked off at 0.1m intervals. At which point should a further weight of 20N be attached to achieve equilibrium?

0.1 m

A.

B. 0.6 m

C. P

D. 0. 4m

10 N

20 N

26. The motorcycle shown has a mass of 200 kg and a wheel base of 1.8 m.

If the rear wheel exerts a 1200 N force on the ground, find how far the motorcycle's center of gravity is located from the front wheel. A. 0.7 m

B. 0.9 m

C. 1.1 m

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D. 1.2 m

KEY FOR PROBLEMS VECTORS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

No Yes Yes 13 units 13m (a) 5 units (b) 21 units (a) 2D (b) -B (c) A |B| = √5 units and |C| = 1unit 1 unit to right 10N, 370 to the horizontal in north-west direction 5N, 370 to the horizontal in north-east direction √13 N

FORCE AND EQUILIBRIUM 13. 14. 15. 16.

160N (a) 30N 6N (a) 60N

(b) 18N (b) 100 N

17.

(a) (i) vector has direction and a scalar does not (ii) scalar examples; any two e.g. speed, mass, energy, time, power vector examples; any two e.g. displacement, velocity, acceleration, force or weight (only gravity is not awarded) (iii) (b) (i) horizontal comp. = 2.8 cos 35 = 2.3 kN (2293.6N) vertical comp. = 2.8 sin 35 = 1.6 kN (1606.0N) (ii) Fb =10-1.6 = 8.4kN (8394N)

18.

(a) B: drag force / air resistance C: weight / mg / gravitational force (b) sum of all forces acting on the body should be zero (c) closed triangle (of vectors) so forces are in equilibrium / resultant force is zero / forces balance (so, moving at constant velocity) (d) W = 9500 × sin 74 = 9100 N (9132N is not awarded) 2 sf

TORQUE AND EQUILIBRIUM 19. 20. 21. 22. 23. 24. 25.

5fa in ccw direction 4.2m away from point A 18N and 12N 4N and 10N √3 Nm in ccw direction 21N 50N

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26. 27. 28.

2F 30N (a) 200 × 1.85 = U × 1.1 U = 336.36N (b) D = 336.36 – 200 = 136.6N

29.

(a) The sum of three forces should be equal to zero and the total torque of three forces should be equal to zero (b) P × 0.1 = 1.2 × 0.25 P = 1.8N Q = 1.8+1.2 = 3N (c) It should be placed just at the position of fingers (where force Q is also exerted)

30.

450 × sin(32) × 0.36 = 180 × x

x = 0.48m

31.

(a) The extension of weight vector should pass through between the wheels in order that the total moment on the trolley is able to be zero. (b) P × 0.90 = 150 × 0.50 P = 83 N (83.3 N) Q × 0.90 = 150 × 0.540 Q = 67 N (66.7 N) or Q = (160 − 83) = 67 N (c) (minimum) force × 0.10 = 150 × 0.40 force = 600 N (d) force is less because distance to pivot is larger ; smaller force gives large enough moment

32.

(a) FSTRAP (directed towards right hand side), Weight (towards downward), FBICEPS (towards left) (b) 100 × 32 = F × 4 F = 800N (c) Since the weight passes through the elbow joint, its moment around joint is zero and also weight has no horizontal component (d) It increases in the first half of motion and then decrease again

33.

(a) (b) (c) (d)

34.

(a) (b) (c) (d)

35.

(a) (F × sin(37) × 4) + (F × cos(37) × 2) = 4000 F=N (b) Rx = 4000 - 1000 × sin (37) = 1400N Ry = 1000 × cos (37) = 800N R = 1612N

The product of force by the distance of force to the pivot point. The net / resultant / total force exerted on the body should be zero 320 × 2.0 = (4-0.8) × N N= 200N 320 × 2.0 + 480 × 1.0 = R × 3.2 R = 350N B = 480+320-350 B = 450N (e) (i) Supporting force from rock increases, supporting force from bank decreases. (ii) When the boy is at exactly the right end of the plank, bank does not apply any force on the bank. The net / resultant / total torque/moment on the body should be zero [12 × cos(37) × d/2] + [12 × (cos(37) × d + d/2)] = F. sin(37) × d F = 34N The total force exerted on the body must be zero. Hor. comp. = F = 34N Ver. Comp. = 12 + 12 = 24N

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CENTER OF GRAVITY 36.

(a) The point where all gravitational attraction on a body is assumed to be concentrated is called the center of gravity of that object. (b) (i) A downward arrow at point C (ii) The moment of weight around point P becomes zero in equilibrium position. Therefore, the extension of weight vector should pass through the pivot point (point P) (c) The weight vector is in the direction of passing through the bottom. Therefore, the momet of weight is in the opposite direction to the moment of reaction force by the ground.

37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47.

300 N 12.5 cm a/3 3.2cm 3r 0.4 cm 3 cm 2 cm r 2 cm (a) The total moment on truck should be equal to zero. Therefore, the weight vector should pass through tires of truck. (b) tan 𝜃 = 1.2 θ=28.60 2.2

48.

64m × 4r = xm × xr

x = 16

49.

Center of gravity is shifted 1cm left

50.

3x = (4r – x)

𝑥=𝑟

KEY for multiple choice questions 1. A

2. C

3. C

4. C

5. B

6. C

7. C

8. C

14. D

15. A

16. D

17. D

18. B

19. A 20. A 21. B

9. D

10. A

11. D

12. A

13. C

22. D

23. C

24. D

25. B

26. C

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UNIFORM LINEAR MOTION

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GRAPH EXERCISES 1. Draw velocity-time graphs for the following position-time graphs

2. Draw velocity-time and acceleration–time graphs for the following position-time graphs.

(A) x(m)

(A) x(m)

V(m/s)

32

16

(V0 = 0 )

16 t(s)

0

t(s)

t(s) 0

4

(B) x(m)

8 a(m/s2)

V(m/s)

(B)

V(m/s)

4

28

t(s)

14

t(s)

t(s) 0

(C)

t(s)

7

x(m)

(C) x(m)

V(m/s)

24

t(s)

+8

(V0 = 0 ) 4 t(s) 2

t(s) 0

6

-8

(D)

a(m/s2) V(m/s)

x(m)

t(s)

V(m/s)

t(s)

24 t(s) 0

4 10

t(s)

18

(D) x(m)

- 16

(E)

t(s)

0

x (m)

16 0 4

a(m/s2)

V(m/s) 6

10 t (s)

t(s)

V(m/s) t(s)

-8 t(s)

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PROBLEMS 1. Velocity-time graph of an object is given in figure. The object is at position Xo = 10 m at time to=0.

(a) Draw position versus time (x-t) graph of the object.

(b) Draw acceleration versus time (a-t) graph of the object.

(c) Calculate the displacement of object in 9 seconds.

(d) Calculate the average speed of the object in 9 seconds.

(e) Indicate the direction of the object in the following thime intervals: 0-3 sec : __________ 3-6 sec : __________ 6-9 sec : __________

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2. Acceleration versus time graph of an object is given. At t=0, V = 4 m/s. a(m/s2) 1 0

16 t(s)

8

-2

(a) Draw velocity versus time (v-t) graph of the object.

(b) Draw position versus time (x-t) graph of the object. The object is at the position of x=0 initialy.

3. V-t graph of a body is given in figure below. V(m/s) 10 12 16 0

8

t(s)

-10

a) What is its average velocity during its whole motion?

b) What is its average speed during its whole motion?

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4.

Velocity-time graph of two moving objects are given. At t=0 (L) is 200 m IN FRONT OF (K). At what time are they next to each other? V(m/s) K

40 30

L t(s)

20 0

10

5.

Two cars K and L are at rest and 250 m apart. They start to approach each other with constant accelerations of 2 m/s2 and 3 m/s2 respectively. Find the speed of each at the instant they meet.

6.

A pedestrian is running at his maximum speed of 6 m/sec to catch a bus stopped by a traffic light. When he is 25 meters away from the bus, the light turns green and the bus accelerates uniformly at 1 m/sec2. Can the pedestrian catch the bus? (a) If yes, how far would he have to run to catch the bus? (b) If no, what would be his frustration distance (closest approach) to the bus?

7.

A car moving at 24 m/sec slows down uniformly and stops in 3 sec. Find;

(a) The acceleration

(b) The stopping distance

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8.

A car moving at a speed of 20 m/s slows down uniformly and stops in a distance of 40 m.

(a) How long does it take to stop for this car?

(b) What is its acceleration of the car?

(c) What is the speed of the car 2 seconds before it stops?

(d) How far does it travel during the last second of motion?

9.

A body accelerates uniformly for 4 seconds. It covers 21 meters in the first 2 seconds, and it moves 39 meters in the last 2 seconds of its motion. What is the acceleration of the body?

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10.

(a)

(i) Define acceleration.

(ii) State why acceleration is a vector quantity. (b) State what feature of a velocity-time graph may be used to calculate (i) acceleration: (ii) displacement:

(c) The graph in Figure 1 shows how the displacement of a runner from a fixed point, along a straight track, varies with time.

Without calculating, sketch on the grid in Figure 2 a graph to show how the velocity of the same runner varies over the same period. The time scales are the same on the both axes.

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11. The distance-time graphs for two runners, A and B, in a 100m run are shown.

distance/m

time/s

(a) Explain how the graph shows that athlete B accelerates throughout the race.

(b) Estimate the max distance between the athletes.

(c) Calculate the speed of athlete A during the race.

(d) The acceleration of athlete B is uniform for the duration of the race. (i) State what is meant by uniform acceleration

(ii)

Calculate the acceleration of athlete B.

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12. Two cars, A and B, are travelling along the outside lane of a motorway at a speed of 30.0 m s–1. They are a distance d apart. 6. 30.0 m s–1

30.0 m s–1

B

A

d

The driver of car A sees a slower vehicle move out in front of him, and brakes hard until his speed has fallen to 22.0 m s–1 . The driver of car B sees car A brake and, after a reaction time of 0.900 s, brakes with the same constant deceleration as A. The diagram below shows velocity-time graphs for car A (solid line) and car B (broken line). Velocity/m s–1

30

25 A

B

20

15

10

5

0 0

1

2

3

4

5

6 Time/s

(a) Find the deceleration of the cars whilst they are braking.

PhysicsAndMathsTutor.com

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6

(b) What does the area under a velocity-time graph represent?

(c) Determine the shaded area.

(d) State the minimum value of the initial separation d if the cars are not to collide. Explain how you arrived at your answer.

(e) Suppose that, instead of only slowing down to 22.0 m s–1 , the cars had to stop. Add lines to the grid above to show the velocity–time graphs in this case. (Assume that the cars come to rest with the same constant deceleration as before.) Explain why a collision is now more likely.

13. A cheetah accelerating uniformly from rest reaches a speed of 29ms-1 in 2.0 s and then maintains this speed for 15 seconds. (a) Calculate (i) its acceleration

(ii) the distance it travels while accelerating

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(iii) the distance it travels while it is moving at constant speed

(b) The cheetah and an antelope are both at rest and 100m apart. The cheetah starts to chase the antelope. The antelope takes 0.5s to react. It then accelerates uniformly for 2 seconds to a speed of 25ms-1 and then maintains this speed. The graph shows the speed-time graph for the cheetah.

(i)

Using the same axis, plot the speed-time graph for the antelope during the chase.

(ii)

Calculate the distance covered by the antelope in the 17 seconds after the cheetah started to run.

(iii)

How far apart are the cheetah and the antelope after 17 seconds?

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14. The diagram shows a velocity-time graph for a ball bouncing vertically on a hard surface. The ball was dropped at t = 0 s.

v/ms-1

5

t/s 0.5

1.0

1.5

2.0

2.5

-5

(a) At what time does the graph show the ball in contact with the ground for the third time? (b) The downward sloping lines on the graph are straight and parallel with each other. Why?

(c) Show that the height from which the ball was dropped is about 1.2 m.

(d) Sketch a displacement-time curve on the axes below for the first second of the motion.

Displacement /m 0.5

1.0

t/s

(e) What is the displacement of the ball when it finally comes to rest?

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15. A lorry is travelling at 25 m s–1 down a mountain road when the driver discovers that the brakes have failed. She notices that an escape lane covered with sand is ahead and stops her lorry by steering it on to the sand.

sand

Escape lane

(a) The lorry is brought to a halt in 40 m. Calculate the average deceleration of the lorry.

(b) Suggest how the depth of the sand affects the stopping distance. Justify your answer.

(c) Suggest how the inclination of the escape lane affects the stopping distance. Justify your answer.

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16. Joseph runs along a straight track. The variation of his speed v with time t is shown below. Which of the followings is correct for 25 seconds?

17. A car of mass 1000kg accelerates on a straight, flat, horizontal road with an acceleration a=0.3ms-2. The driving force F on the car is opposed by a resistive force 500N.

The net resultant force on the car is A. 200N

B. 300N

C. 500N

D. 800N

18. The graph shows the variation with time t of the acceleration a of an object. Which of the following is the change in velocity of the object in the time interval 0 to 4s?

A. -8ms-1

B. -4ms-1

C. +8ms-1

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D. +4ms-1

19. A car accelerates from rest. The acceleration increases with time. Which graph shows the variation of the speed v of the car with time t?

20. The graph shows how the displacement d of an object varies with time t. The tangent to the curve at time t1 is also shown.

Which of the following gives the speed of the object at point P? A. the gradient at P

B. the shaded area

C. 1/ gradient at P

D. d1/t1

21. A net force of magnitude 4.0N acts on a body of mass 3.0kg for 6.0s. The body is initially at rest. Which of the following is the speed of the body after 6.0s interval?

A. 0.50ms-1

B. 2.0ms-1

C. 4.5ms-1

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D. 8.0ms-1

22. The graph is speed versus time graph for an object that is moving in a straight line. The distance traveled by the object during the first 4.0s is A. 80m

B. 40m

C. 20m

D. 5m

23. A ball, initially at rest, is dropped in the air from a great height. Air resistance is not negligible. Which one of the following graphs best shows the variation with time t of the acceleration a of the ball?

24. If a moving object is subject to a constant force, which of the following can be correctly deduced from Newton’s first law? A.

The object continues to move a changing velocity.

B.

The object continues to move a constant velocity.

C.

The object continues to move a changing direction.

D.

The object continues to move in the same direction.

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25. The graph shows how the velocity of a particle varies with time. Which of the following graphs correctly shows how the acceleration of the particle varies with?

26. The graph below shows the variation with time t of the velocity v of an object moving along a straight line. The displacement of the object between t=0 and t= 6.0s is A. 2m

B. 12m

C. 20m

D. 24m

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KEY FOR PROBLEMS 1.

(a) An appropriate graph. (b) An appropriate graph. (c) 27 m (d) 3ms-1 (e) + / + / 2. An appropriate graphs for both (a) and (b). 3.

(a) 5 ms-1 (b) 7. 5 ms-1

4.

30 seconds

5.

20 ms-1 and 30 ms-1

6.

No. Closest approach is 7 m.

7.

(a) a = -8 ms-2 (b) X = 36 m

8.

(a) 4 sec. (b) a = 5 ms-2 (c) 10 ms-1 (d) X = 2.5 m

9. 10.

4.5 ms-2 (a) (i) rate of change of velocity (ii) acceleration has magnitude and direction. (b) (i) acceleration is the gradient (or slope) of the graph (ii) displacement is the area under the graph. (c) (linear) increase to t = 2.0 ± 0.2 s uniform velocity between 2.0 s and 6.0 s zero velocity after t = 8.0 s

11.

(a) The slope of curve (rate of displacement covered per unit time) increases during motion. (b) 6.5x4=26m (c) 72/8=9ms-1 (d) (i) constant rate of change in velocity (ii) 100 = (a.112 )/2 a=1.6ms-2

12.

(a) Acceleration = gradient / suitable eqn. of motion Correct substitutions [ 0.9 for t is wrong] 6.1 – 6.3 m s–2 [no ecf]

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(b) Distance/displacement (c) Shaded area: 6.9 – 7.5 m

[Allow 1 mark for 5.5 – 6.1 cm2.]

(d) Same as above [ecf] Area is the extra distance car B travels/how much closer they get (e) Graph: Both sloping lines continued down to time axis [by eye] Explanation: graphs is larger/B travels faster for longer/B still moving when A stops. Extra distance B goes is larger/ > ‘7.2’ Initial separation must be greater. 13.

(a) (i) a=29/2=14.5ms-2 (ii) x=14.5x4/2=29m (iii) 29x15=435m (b) (i) at t=0.5s, v=0 ; at t=2.5 s, v= 25ms-1. Linear graph between t=0.5s and t=2.5s. Horizontal line at v=25ms-1 after t=2.5s. (ii) x= (25x2/2) + (14.5x25) = 387.5m (iii) Xcheetah=(29x2/2) + (15x29) = 464m Δd = 100 – (464 - 387.5) = 23.5m

14.

(a) t = 2.1s (b) Represents acceleration of the ball. Force on ball or gravitational field strength or acceleration is constant or uniform (c) Relevant equation or correct area. Displacement /m

0

0.

5

1.

0

t/s

(d) Substution correct displacement scale as shown above First half of curve correct Second half correct with reduced height (e) –1.25 m (correct magnitude and direction) [Look at candidate ’s displacement origin] 15.

(a) Average deceleration Select υ 2 = u2 + 2ax, ½ m υ 2 = Fx and F = ma OR equations of motion. Correct substitutions of 40 m and 25 m s–1. a = 7.8 m s–2 [If a = –7.8 m s–2 → 2/3]

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(b) Depth of sand and stopping distance More sand ⇒ shorter stopping distance/stops more quickly/slows down faster Because lorry sinks further/ bigger resisting (c) Inclination and stopping distance Steeper inclination ⇒ longer stopping distance/stops more slowly/slows down slower Because parallel component of weight to the road increases /smaller normal (or reaction force) and smaller friction force (or resisting force)/net force on lorry decreases

KEY for multiple choice questions: 16. C 17. B 18. C 19. B 20. A 21. D 22. B 23. C 24. A 25. C 26. B

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RELATIVE MOTION

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PROBLEMS 1. Cars K and L start moving with constant velocities 3 m/s and 4 m/s relative to ground in the given directions. (a) What is the velocity of K with respect to L? N

3 m/s

K

W

E L

(b) What is the velocity of L with respect to K?

4 m/s S

2. While car K is moving in the direction of north-east with a velocity of V 2 with respect to ground, car L is moving due north with a velocity of V relative to ground. (a) What is the velocity of L with respect to an

N

observer looking from car K?

𝑉√2

W

K

450

E

V

(b) What is the velocity of car K with respect to an

L

observer looking from car L?

S

3. Velocities of car K and L are given relative to ground. What is the velocity of car K relative to car L? N VK = 10 m/s VL = 2 m/s

370 E

W

S

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4. In the given figure X measures the velocity of Y as V1 in the 1st part of the road and as V2 in the 2nd part of the road. If velocities of X and Y given in the figure are relative to ground what is the ratio V1/V2?

II

I X Y VX = 3 m/s VY = 4 m/s

5. A train travels due east at 30 m/s (relative to ground) in a rain that is blown toward the south by the wind. The path of each raindrop makes an angle of 370 with the vertical, as measured by an observer stationary on the earth. An observer on the train, however, sees the drops fall perfectly vertically. Determine the speed of the raindrops relative Earth.

6. The driver of the car A moving towards east at 30 m/s thinks that car B is moving towards north at 40 m/s. What is the velocity and direction of car B with respect to the ground?

7. Velocity of a swimmer with respect to the river is 3 m/s. The velocity of the river is 1 m/s to the right. Swimmer starts from point (K) goes to (L) and comes back to point (K). How long is his swimming time?

vriver=1 m/s K

L 60 m

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8. A river is flowing at 4 m/minute as shown. A boat moves from point K to L with a constant velocity of (v) relative to river in 10 minutes. When it reaches (L) it turns round and moves for another 5 minutes at this same constant speed. Where is it now with respect to point K? Vriver=4 m/min L

K 80 m

9. A river is flowing at 4 m/s. A boat is moving at 5 m/s, always moving at 90 0 with the current. How far away from point (A) will he land in meters? A vriver=4m/s 80 m

vboat=5m/s

10. A man wants to cross a 400 m wide river, which is flowing at a speed of 2 m/s. He starts at point A aligning himself an angle of 530 with the shore (horizontal). If velocity of the man is 10 m/s relative to river, find the followings. VR = 2 m/s VM = 10 m/s

d = 400 m

530 A

(a) How much time does it take the man to travel shore to shore?

(b) What distance is the man drifted when he reaches the opposite shore?

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11. If the man in question 10 were aligned as in the below figure and speed of the river were to be 4 m/s, what would be the answers of (a) and (b)? VR = 4 m/s VM = 10 m/s

d = 400 m

530 A

12. In the given figure an object starts to swim in the river with the given velocity relative to river. K Vstream = 1 m/min 30 m

5 m/min 370

(a) How much time does it take the object to travel shore to shore?

(b) What distance is the object drifted when he reaches the opposite shore?

13. Two boats K and L cross the river with velocities of v and v/2 relative to water. O

d

VK=v

O

Vriver=v/2

K Fig.I

d

VL=v/2

Vriver=3v

L Fig.II

The boat K arrives at the opposite side at a distance of X from point O. How far does the boat L arrive at the opposite side from O in terms of X?

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KEY FOR PROBLEMS 1.

(a) 5 m/s, due NW

(b) 5 m/s, due SE

2.

(a) V, due west

(b) V, due east

3.

6√2

4.

1/5

5.

50 m/s

6.

50 m/s, 53 east of north

7.

45 s

8.

at K

9.

64m right side of point A

m , due NE s

10. (a) 50 s (b) 400 m lefthand side 11. (a) 50 s (b) 100 m righthand side 12. (a) 10 min.

(b) 30 m

13. 12x

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NEWTON’S LAWS OF MOTION

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PROBLEMS 1. An object of 2 kg is at rest at point A. 6 N force, which is parallel to the plane, acts on the body between A and B. What is the velocity of the object at point B if the plane is smooth? F= 6N B

A 24 m

2. A block with the mass of 3 kg slides on a horizontal plane. The force of friction between the plane and the block is ff = 3N. At what distance from point A does the block stop? V0 = 14 m/s Ff = 3N A

3. A constant pull of 40N, applied on an angle of 370 to a 10kg mass, gives the block a uniform acceleration of 3m/s2. Find the magnitude of force of friction. F = 40 N 370

nN

4. What must be the magnitude of F if the block of mass 4 kg moves with constant velocity on the horizontal plane? F 370  = 0.5

5. A car of mass 900 kg slows down uniformly and displaces a distance of 4 m during the last second of its motion. What is the magnitude of frictional force between the car and the surface? (Assume that the car is moving on horizontal plane).

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6. 3 kg object is placed on a smooth inclined plane as in Figure. (a) Show all the forces acting on the object including the normal force. (b) Find the acceleration and the direction of motion of the object. 370

7. A 4 kg block is sliding down with constant velocity on an inclined plane. What force of friction is required to maintain this motion?

300

8. A 2 kg block is sliding down with constant velocity on an inclined plane. How large a parallel upward force is required to pull it up with an acceleration of 2m/s2?

530

9. 1kg object is held on the wall under the effect of 10 N force as in Figure. There is no friction between the surface of the wall and the object. wall

(a) Show all the forces acting on the object including the normal force. 1 kg

10 N

(b) Find the acceleration of the object.

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10. Two blocks are pulled by a force of 50 N as in Figure. Find the tension in the cord. F = 50 N 1 kg

T =?

4 kg

370

 = 0.5

11. What is the tension in the cord, when the system is released? 6 kg  = 0.5

4 kg

12. What are the tensions in the cord, T1, T2 and T3, when the system is released?

T1

T2

2 kg

6 kg

T3 2 kg

13. After the wooden block is removed; 2

2 kg

(a) In what direction will the system move and which object will hit the ground? (b) What will be the acceleration of the system?

1

3 kg

Wooden block

9m

(c)

How long will it take for the object to hit the ground?

ground

(d) With what velocity will the object hit the ground?

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14. Two blocks of masses 2kg and 1kg are in contact on a smooth plane. If a force of 6N is applied on m1 find the reaction force between the blocks. m1=2kg m2=1kg

F=6N

15. Identify at least five of action-reaction force pairs in the following diagram.

16. Find the tension in the rope connecting the body to the ceiling of the elevator, if the elevator decelerates at the rate of 4m/s2 in the downward direction.

1kg

17. What is the apparent weight of a man of 60 kg in an elevator, if the elevator decelerates at the rate of 3 m/s2 in the downward direction?

18. An elevator and its load weight a total of 3200 N. Find the tension T in the supporting cable when the elevator originally moving downward at 20 m/s is brought the rest with constant acceleration in a distance of 50 m. T

W

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19. In order to investigate Newton’s second law, David arranged for a heavy trolley to be accelerated by small weights, as shown below. The acceleration of the trolley was recorded electronically. David recorded the acceleration for different weights up to a maximum of 3.0 N. He plotted a graph of his results. acceleration

heavy trolley

pulley

weight

(a)

Describe the graph that would be expected if two quantities are proportional to one another. [2]

(b)

David’s data are shown below, with uncertainty limits included for the value of the weights. Draw the best-fit line for these data. [2]

1.40 acceleration / ms–2 1.20 1.00 0.80 0.60 0.40 0.20 0.00 0.00

0.50

1.00

1.50

2.00

2.50 weight / N

(c)

Use the graph to (i)

estimate the value of the frictional force that is acting on the trolley. [1]

(ii)

estimate the mass of the trolley. [2]

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20. A trolley moves down a slope, as shown in figure below.

The slope makes an angle of 250 with the horizontal. A constant resistive force FR acts up the slope on the trolley. At time t=0, trolley has velocity v=0.50ms-1 down the slope, At t=4.0s, v=12ms-1 down the slope. (a)

(i) Show that the acceleration of trolley down the slope is approximately 3ms-2.

(ii) Calculate the distance x moved by the trolley down the slope from the time t=0 to t=4s.

(iii) Sketch the variation with time t of distance x moved by trolly.

(b) The mass of the trolley is 2.0kg. (i) Show that the component of the weight of the trolley down the slope is 8.3N

(ii) Calculate the resistive force FR.

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21. A helicopter has a cable hanging from it towards the sea below as shown in figure below. A man of mass 80 kg rescues a child of mass 50.5 kg. The two are attached to the cable and are lifted from the sea to the helicopter. The lifting process consists of an initial uniform acceleration follwed by a period of constant velocity and then completed by a final uniform deceleration.

(a) Calculate the combining weight of the man and child. [1]

(b) Calculate the tension in the cable during (i) the initial acceleration of 0.570 ms-2. [2]

(ii) the period of constant velocity of 2.00 ms-1. [2]

(c) During the final deceleration the tension in the cable is 1240 N. Calculate this deceleration. [2]

(d)

(i) Calculate the time over which the man and child are (1) moving with uniform acceleration. [1]

(2) moving with uniform deceleration. [1]

(ii) The time over which the man and child are moving with constant speed is 20s. On figure below, sketch a graph to show the variation with time of the velocity of the man and child for the complete lifting process.

2.0 Velocity (ms-1) 1.0

time (s)

0 0

5

10

15

20

25

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30

35

22. The diagram below represents a balloon just before take-off. The balloon’s basket is attached to the ground by two fixing ropes.

balloon

basket

fixing rope

50

fixing rope

50 ground

There is a force F vertically upwards of 2.15x103 N on the balloon. The total mass of the balloon and its basket is 1.95x102 kg. (a)

State the magnitude of the resultant force on the balloon when it is attached to the ground. [1]

(b)

Calculate the tension in either of the fixing ropes. [3]

(c)

The fixing ropes are released and the balloon accelerates upwards. Calculate the magnitude of this initial acceleration. [2]

(d)

The balloon reaches a terminal speed 10 seconds after take-off. The upward force F remains constant. Describe how the magnitude of air friction on the balloon varies during the first 10 seconds of its flight. [2]

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23. This question is about force and energies. (a)

(b)

A system consists of a bicycle and cyclist travelling at a constant velocity along a horizontal road.

(i)

State the value of the net force acting on the cyclist. [1]

(ii)

On the diagram draw labelled arrows to represent the vertical forces acting on the bicycle. [2]

(iii)

With reference to the horizontal forces acting on the system, explain why the system is travelling at constant velocity. [2]

The total mass of the system is 70 kg. The total resistive force acting on the system is 40 N and its speed is 8.0 m s–1. The cyclist stops pedalling and the system comes to rest. (i)

Calculate the magnitude of the initial acceleration of the system. [2]

(ii)

Estimate the distance taken by the system to come to rest from the time the cyclist stops pedalling. [2]

(iii)

State and explain one reason why your answer to (b)(ii) is only an estimate. [2]

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MULTIPLE CHOICE QUESTIONS ON NEWTON’S LAWS OF MOTION 1.

Stephen pushes two boxes P and Q, that stay in contact, along a rough table, with a force F of 30 N. Box P has a mass of 2.0 kg and box Q has a mass of 4.0 kg. Both boxes move with constant speed.

The resultant force on box Q is A. 0 N.

2.

B. 5.0 N.

C. 15 N.

D. 30 N.

The diagram shows a girl attempting (but failing) to lift a heavy suitcase of weight W. The magnitude of the vertical upwards pull of the girl on the suitcase is P and the magnitude of the vertical reaction of the floor on the suitcase is R. Which equation correctly relates W, P and R?

3.

4.

A. W = P + R

B. W > P + R

C. W < P + R

D. W = P = R

Objects A and B collide together. They end up joined together and stationary. During the collision, a force +F is exerted on object A by object B. According to Newton’s third law, there will also be a force of A.

–F acting on object B.

B.

–F acting on object A.

C.

+F acting on object B.

D.

+F acting on object A.

A lamp of weight W is suspended by a wire fixed to the ceiling. With reference to Newton’s third law of motion, the force that is equal and opposite to W is the A.

tension in the wire.

B.

force applied by the ceiling.

C.

force exerted by the lamp on the Earth.

D.

force exerted by the Earth on the lamp.

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5.

6.

7.

If a moving object is subject to a constant force, which of the following can be correctly deduced from Newton’s first law? A.

The object continues to move with a changing velocity.

B.

The object continues to move with a constant velocity.

C.

The object continues to move with a changing direction.

D.

The object continues to move in the same direction.

The net force acting on a body is zero. Which of the following quantities must also have zero magnitude for this body? A.

Momentum

B.

Velocity

C.

Speed

D.

Acceleration

Two unequal masses M and m are joined by a light inextensible string. The string passes over a light frictionless pulley as shown. pulley

m

M

The masses accelerate when released. Which diagram is the correct free-body diagram for the two masses? A.

M

B.

m

M

C.

m

M

D.

m

M

m

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8.

A light inextensible string has a mass attached to each end and passes over a frictionless pulley as shown. pulley

string

mass M

The masses are of magnitudes M and m, where m < M. The acceleration of free fall is g. The downward acceleration of the mass M is A.

(M − m)g . (M + m)

B.

C.

(M + m)g . (M − m)

D.

(M − m)g M

.

Mg . (M + m)

mass m

9.

A block on a frictionless horizontal table is attached by a light, inextensible string to an object P of mass m that hangs vertically as shown below. pulley

The pulley has zero friction and the acceleration of free fall is g. The acceleration of the block and object P is

M

A. g.

B.

m g. M

D.

m+M g. m

P mass m

C.

10.

m g. m+M

A block of mass m is pulled along a horizontal, frictionless surface by a force of magnitude F. The force makes an angle with the vertical. F block

The magnitude of the acceleration of the block in the horizontal direction produced by the force F is A.

F . m

B.

F sin θ . m

C.

F cos θ . m

D.

F tan θ . m

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11.

The diagram below shows five wooden blocks joined by inelastic strings. A constant force accelerates the blocks to the right on a frictionless horizontal table.

W

X

Y

accelerating

Z

force table

In which string is the tension the greatest? A. W

12.

B. X

C. Y

D. Z

Mandy stands on a weighing scale inside a lift (elevator) that accelerates vertically upwards as shown in the diagram below. The forces on Mandy are her weight W and the reaction force from the scale R. The reading of the scale is

lift acceleration

A.

R + W.

B.

W.

C.

R.

D.

R – W.

scale

13.

An elevator (lift) is used to either raise or lower sacks of potatoes. In the diagram, a sack of potatoes of mass 10 kg is resting on a scale that is resting on the floor of an accelerating elevator. The scale reads 12 kg. The best estimate for the acceleration of the elevator is A. 2.0 m s–2 downwards. elevator

10 kg

B. 2.0 m s–2 upwards. scale

C. 1.2 m s–2 downwards. D.

1.2 m s–2 upwards.

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14.

A ball of weight W is dropped on to the pan of a top pan weighing balance and rebounds off the pan. At the instant that the ball has zero velocity when in contact with the pan, the scale will read A. zero. pan

B. a value less than W but greater than zero. C. W.

00.00

15.

D. a value greater than W.

A block of mass M is held at rest on a horizontal table. A heavy chain is attached to the block with part of the chain hanging over the table. The block and the chain can slide without friction. block table A.

v

A.

B.

v

v

B.

v

chain

0

0

0

0

t

0

0

0

t

0

t

t

0

t

t

The block is released. Which one of the following graphs best represents the variation with time t of the speed v of the block as it moves on the table? A.

v

A.

0 0

C.

B.

v

v

B.

0

0 0

t

0

t

v C.

v

0

0 0

C.

0

D.

v

D.

0

0 t

v

0

t

t

t

0

v

0

D. von a horizontal C. Av box of mass 80 kg rests D. v 16. rough surface. A string attached to the box passes over a smooth pulley and supports a 2.0 kg mass at its other end.

v

0 0

0

0 0

t

t

0

0 0

t

t

When the box is released, a frictional force of 6.0 N acts on it. What is the acceleration of the box? A. 1.4ms-2

B. 1.6ms-2

C. 2.4ms-2

D. 2.8ms-2

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17. Newton’s third law of motion is often summarized as ‘Every action (force) has an equal and opposite reaction.’ A book rests on a table. If the weight of the book is the ‘action’ force, what is the ‘reaction’ force? A. the pull of the book on the Earth B. the pull of the Earth on the book C. the push of the book on the table D. the pull of the table on the book

18. What is the condition for an object to be in equilibrium? A. The object’s velocity and the resultant torque on it must be both zero. B. The object’s velocity must be zero. C. The resultant force and the resultant torque on the object must be both zero. D. The resultant force on the object must be zero.

19. A shot-put champion accelerates a 7.0 kg ball in a straight line. The ball moves from rest to a speed of 12 ms-1 in a distance pf 1.2 m. What is the average force on the metal ball? A. 70 N

B. 210 N

C. 420 N

D. 840 N

20. A car of mass 750 kg has a horizontal driving force of 2.0 kN acting on it. It has a forward horizontal acceleration of 2.0 ms-2.

What is the resistive force actimg horizontally? A. 0.50 kN

B. 1.5 kN

C. 2.0 kN

D. 3.5 kN

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KEY FOR PROBLEMS 1.

12 m/s

2.

98 m

3.

2N

4.

40 N

5.

7200 N

6.

6 m/s2, downward

7.

20 N

8.

36 N

9.

10m/s2

10.

11 N

11.

36 N

12

32 N, 32 N, 8 N

13.

(I) ; 2m/s2 ; 3 s ; 6 m/s

14.

2N

15.

Five appropriate pairs should be stated

16.

14 N

17.

780 N

18.

4480 N

19. (a) a straight line (linear graph); through the origin (0;0)

2

(b) any straight line; that fits within ALL the error bars; 1.40 acceleration / ms–2 1.20 1.00 0.80 0.60 0.40 0.20 0.00 0.00

0.50

1.00

1.50

2.00

2.50 Weight / N

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(c)

(i)

0.3 N; Accept 0.25 N - 0.35 N. NB

(ii)

realization that mass = (gradient)–1;

1

Award this mark for full Newton II equation to give mass = 1.4 kg (Accept 1.2 kg - 1.6 kg.);

2 max

Use of F = ma for 1 data point receives [0] Watch for ecf from candidate’s own line. [7] 20.

(a) (i) 𝑎 =

12−05 4

= 2.88𝑚𝑠 −2 1 2

(ii) 𝑠 = 0.5 × 4 + × 3 × 42 = 26𝑚 (iii)

(b) (i) Wx =2 x 9.8 x sin(25) = 8.3N (ii) F=8.3 - FR = 2 x 3

21.

F=2.3N

(a) W = (50.5 + 80) x 9.8 = 1300N (b) (i) F= 130.5 x 0.570 = 74.4 0 T – 1300

T = 1374.4N

(ii) T = mg = 1300N (c) mg – T = ma (d) (i) (1) 0.570 = 2/t (2) 0.46 = 2/t

1300 – 1240 = 130.5 x a

a= 0.46ms-2

t = 3.5s t = 4.35s

(ii)

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22.

(a)

zero;

1

(b)

resultant vertical force from ropes = = 237N; equating their result to 2T sin50; ie 2T sin 50 = 237 calculation to give T = 154.7N (150N);

3

Accept any value of tension from 130 N to 160 N. Award [2] for missing factor of 2 but otherwise correct ie 309 N. (c)

correct substitution into F = ma; to give a =

237 = 1.21 ms − 2 ; 1.95 10 2

2

Watch for ecf. NB Depending on value of g answer will vary from 1.0(3) ms-2 to 1.2(3) ms-2 all of which are acceptable. (d)

23.

(a)

statement that air friction increases with increased speed seen / implied; in 10 seconds, friction goes from 0 N to 237 N / force increases from zero until it equals the net upward accelerating force;

(i)

zero;

2 [8]

1

(ii)

correct position and labelling of weight/gravity force/mg; two reactions drawn as shown; force downwards on pedals; Ignore any other vertical forces and all horizontal forces. The total upward vector lengths should approximately equal the downward vector lengths.

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2 max

(iii)

(b)

drag force = thrust/forward force/driving force; net force=zero therefore acceleration is zero

(i)

 40  acceleration =   = 0.57 m s–2  70 

(ii)

use of F∆s =

2 2

1 mv2; 2

56m;

2

Or v2 = u2 + 2as equivalent seen and substituted correctly; 56m; (iii)

sensible physical reason e.g. air resistance / bearing friction/ brakes’ effectiveness varies with speed

2 [11]

KEY for multiple choice questions 1. A

2. A

3. A

4. C

5. A

6. D

7. B

8. A

9. C

10. B

11. D

12. C

13. B

14. D

15. C

16. A

17. A

18. C

19. C

20. A

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MOTION NEAR EARTH SURFACE

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PROBLEMS 1. A ball was dropped from an 80 m tall building. (a) What is the acceleration of the ball when t = 1 s?

(b) What is the velocity of the ball when t = 1 s?

(c) What is the displacement of the ball when t = 1 s?

(d) How long will it take for the ball to hit the ground?

2. Ball 1 is released from rest and ball 2 is thrown upward with a velocity of 20 m/s at the same time. If both of them hit the ground at the same time, what would be the initial height of Ball 1 (h)? Ball 1 V0 = 20 m/s

h=?

Ball 2

3. At the instant object at point A is released object at point B is thrown downward. If both objects hit the ground at the same time, what is the initial speed of the object at point B? (g = 10m/s2)

B V0 = ? A 300 m 180 m

ground

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4. Two balls are thrown towards each other as in Figure at the same instant. When and where do they collide?

30 m/s 30 m 30 m/s

5. The object that is 45 m above the ground is thrown with a velocity of 40 m/s, as shown in the figure.

(a)

In how many seconds does it hit the ground?

(b)

How many meters it can rise from the ground at most?

V0 = 40 m/s

h = 45 m

(c) Find the velocity when it hits the ground.

6. A stone is thrown horizontally from the top of a tower 320m high and hits the ground 480m away from the bottom of the tower. (a) Find the time required for the stone to reach the ground.

(b)

What is the initial speed of the stone?

(c)

Find the speed of the stone when it strikes the ground.

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7. A fight plane traveling horizontally at a speed of 140 m/s and at an altitude of 2000 m above the ground aims to bombard a spy car which is traveling at 40 m/s. V = 140 m/s

horizontal

h = 2000 m V = 40 m/s

X=?

If pilot wants to achieve his aim, what should be the horizontal distance between the fight plane and the car at the instant bomb is released?

8. The object is thrown from the point K on the inclined plane with a velocity of 40 m/s. It hits the plane at point L. Find the time that is needed for the object to move from point K to L. K

40 m/s

L 370 9. A ball is projected above horizontal as in Figure. Velocity of the ball is 20 m/s at its maximum height.

Vh = 20 m/s V

80 m

Find; (a) Time of flight of its motion.

(b)

Maximum height of the ball that it can reach.

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(c) Velocity, V, of the ball at the moment it is thrown.

10. The balloon is rising with the constant speed of 30 m/s. While it is 80 m high, the boy in it throws a stone horizontally with a speed of 20 m/s with respect to balloon.

(a) VS = 20 m/s

VB = 30 m/s

How many seconds later does the stone hit

the ground?

80 m (b)

How far does the stone move in horizontal?

(c) What is the vertical component of its velocity when it hits the ground?

11. A balloon and the stone attached to it with a rope are moving upward with constant speed of 20 m/s. When the distance between the stone and the ground level is h rope is broken, at the same instant an object is thrown upward with a speed of 60 m/s. If meeting time of the object and the stone is 3 s, what is the height of the balloon (h) when the rope is broken? Vb = 5 m/s

stone

h V0 = 60 m/s

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12. Linear motion (a)

Define the term acceleration. [2]

(b)

An object has an initial speed u and an acceleration a. After time t, its speed is v and it has moved through a distance s. The motion of the object may be summarized by the equations v = u + at, s=

(c)

1 2

(v + u )t.

(i)

State the assumption made in these equations about the acceleration a. [1]

(ii)

Derive, using these equations, an expression for v in terms of u, s and a. [2]

The shutter speed of a camera is the time that the film is exposed to light. In order to determine the shutter speed of a camera, a metal ball is held at rest at the zero mark of a vertical scale, as shown below. The ball is released. The shutter of a camera is opened as the ball falls. 0 cm

scale

camera 196 cm 208 cm

The photograph of the ball shows that the shutter opened as the ball reached the 196 cm mark on the scale and closed as it reached the 208 cm mark. Air resistance is negligible and the acceleration of free fall is 9.81 m s–2.

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(i)

Calculate the time for the ball to fall from rest to the 196 cm mark. [2]

(ii)

Determine the time for which the shutter was open. That is, the time for the ball to fall from the 196 cm mark to the 208 cm mark. [2]

13. Antonia stands at the edge of a vertical cliff and throws a stone vertically upwards. v = 8.0ms –1

The stone leaves Antonia’s hand with a speed v = 8.0ms–1. The acceleration of free fall g is 10 m s–2 and all distance measurements are taken from the point where the stone leaves Antonia’s hand. (a) Ignoring air resistance calculate (i) the maximum height reached by the stone. [2] Sea

(ii) the time taken by the stone to reach its maximum height. [1]

(b) The time between the stone leaving Antonia’s hand and hitting the sea is 3.0 s. Determine the height of the cliff. [3]

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14. An experiment is conducted on the surface of the planet Mars. A sphere of mass 0.78kg is projected almost vertically upwards from the surface of the planet. The variation with time t of the vertical velocity v in the upward direction is shown in Fig.2.1.

(a) State the time t at which the sphere reaches its maximum height above the planet’s surface. [1]

(b) Determine the vertical height above the point of projection at which the sphere finally comes to rest on the hill. [3]

(c) Determine the acceleration of free fall on the surface of Mars.[2]

(d) Calculate the force acting on the sphere for the first 3.5 seconds of its motion. [2]

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15. A stone is thrown horizontally from the top of a vertical cliff of height 33 m as shown below.

18 m s –1

The initial horizontal velocity of the stone is 18 m s–1 and air resistance may be assumed to be negligible.

(a) State values for the horizontal and for the vertical acceleration of the stone. [2]

33 m

Horizontal acceleration: ____________

Vertical acceleration

: ____________

sea level

(b) Determine the time taken for the stone to reach sea level. [2]

(c) Calculate the distance of the stone from the base of the cliff when it reaches sea level. [1]

(d) Ignoring air resistance, calculate the speed of stone when it strikes the sea. [2]

(e)

Use your answer in (d) to calculate the angle that the velocity of stone makes with the surface of the sea. [2]

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16. A small steel ball is projected horizontally from the edge of a bench. Flash photographs of the ball are taken at 0.10 s intervals. The resulting images are shown against a scale as in the diagram below. 0

20

distance / cm 40 60

80

100

0

20

40

60 distance / cm 80

100

120

140

(a)

Use the diagram to determine (i)

the constant horizontal speed of the ball. [2]

(ii)

the acceleration of free fall. [2]

(b)

Mark on the diagram the position of the ball 0.50 s after projection. You should carry out any calculations so that you can accurately position the ball. [3]

(c)

A second ball is projected from the bench at the same speed as the original ball. The ball has small mass so that air resistance cannot be neglected. Draw on the diagram the approximate shape of the path you would expect the ball to take. [3]

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17. This question is about projectile motion. A marble is projected horizontally from the edge of a wall 1.8 m high with an initial speed V. V

A series of flash photographs are taken of the marble. The photographs are combined into a single photograph as shown below. The images of the marble are superimposed on a grid that shows the horizontal distance x and vertical distance y travelled by the marble.

1.8 m

The time interval between each image of the marble is 0.10 s. ground

0

0.50

x/m 1.0

1.5

2.0

0

–0.50

y/m

–1.0

–1.5

–2.0

(a)

(b)

(c)

On the images of the marble at x = 0.50 m and x = 1.0 m, draw arrows to represent the horizontal velocity VH and vertical velocity VV.

(2)

On the photograph, draw a suitable line to determine the horizontal distance d from the base of the wall to the point where the marble hits the ground. Explain your reasoning.

(3)

Use data from the photograph to calculate a value of the acceleration of free fall. (3)

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18. This question is about projectile motion. A ball is kicked at an angle to the horizontal. The diagram below shows the position of the ball every 0.50 s. 30 25 20 vertical displacement / m

15 10 5 0

0

30 10 20 horizontal displacement / m

40

The acceleration of free fall is g = 10 m s–2. Air resistance may be neglected. (a)

(b)

Using the diagram determine, for the ball (i)

the horizontal component of the initial velocity. [1]

(ii)

the vertical component of the initial velocity. [2]

(iii)

the magnitude of the displacement after 3.0 s. [2]

On the diagram above draw a line to indicate a possible path for the ball if air resistance were not negligible. [2]

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19. This question is about trajectory motion. Antonia stands at the edge of a vertical cliff and throws a stone upwards at an angle of 60° to the horizontal. v = 8.0ms –1 60°

The stone leaves Antonia’s hand with a speed v = 8.0 m s–1. The time between the stone leaving Antonia’s hand and hitting the sea is 3.0 s. The acceleration of free fall g is 10 m s–2 and all distance measurements are taken from the point where the stone leaves Antonia’s hand. Sea

Ignoring air resistance calculate (a)

the maximum height reached by the stone. [3]

(b)

the horizontal distance travelled by the stone. [2]

20. A ball is projected from ground level with a speed of 28 m s–1 at an angle of 300 to the horizontal as shown below. 28 ms-1 h 300 1.6m There is a wall of height h at a distance of 16 m from the point of projection of the ball. Air resistance is negligible. (a)

Calculate the initial magnitudes of (i)

the horizontal velocity of the ball; [1]

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(ii)

(b)

the vertical velocity of the ball. [1]

The ball just passes over the wall. Determine the maximum height of the wall. [3]

21. Motion of a ball A ball of mass 0.25 kg is projected vertically upwards from the ground with an initial velocity of 30 m s–1. The acceleration of free fall is 10 m s–2, but air resistance cannot be neglected. The graph below shows the variation with time t of the velocity v of this ball for the upward part of the motion. v / ms–1

30.0

25.0

20.0

15.0

10.0

5.0

0.0 0.0

0.5

1.0

1.5

2.0

(a)

State what the area under the graph represents. [1]

(b)

Estimate the maximum height reached by the ball. [1]

2.5

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3.0 t/s

(c)

Determine, for the ball at t = 1.0 s, (i)

the acceleration; [3]

(ii)

the magnitude of the force of air resistance. [2]

(d)

Use the graph to explain, without any further calculations, that the force of air resistance is decreasing in magnitude as the ball moves upward. [2]

(e)

The diagram below is a sketch graph of the upward motion of the ball. Draw a line to indicate the downward motion of the ball. The line should indicate the motion from the maximum height of the ball until just before it hits the ground. (2) v / ms–1 30 20 10 0.0 0.0

2.0

4.0

t/s

–10 –20 –30

(f)

State and explain, by reference to energy transformations, whether the speed with which the ball hits the ground is equal to 30 m s–1. [2]

(g)

Use your answer in (f) to state and explain whether the ball takes 2.0 s to move from its maximum height to the ground. [2]

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KEY FOR PROBLEMS

1.

(a) 10 m/s2 (b) 10 m/s (c) 5 m (d) 4 s

2.

80 m

3.

20 m/s

4.

They collide 13.75 m above the ground and 0.5 s later

5.

(a) 9 s (b) 125 m (c) 50 m/s

6.

(a) 8 s (b) 60 m/s (c) 100 m/s

7.

2000 m

8.

6s

9.

(a) 4 s (b) 20 m

(c) 202 m/s

10. (a) 8 s (b) 160 m

(c) 50 m/s

11. 165 m 12. Linear motion (a)

(b)

(c)

change in velocity / rate of change of velocity; per unit time / with time; (ratio idea essential to award this mark)

2

(i)

acceleration is constant / uniform;

1

(ii)

t=

(i)

1.96 = x 9.81 x t2; t = 0.632s;

2

(ii)

time to fall (1.96 + 0.12) m is 0.651s; shutter open for 0.019s; If the candidate gives a one significant digit answer treat it as an SD-1. Award [0] if the candidate uses s = 1 at2 and s = 12cm.

2

(v − u ) ; 2s and t = clear working to obtain v2 = u2 + 2as; (u + v ) a

2

2

[9] 13.

(a)

(b)

v2 ; to give h = 3.2 m; 2g

(i)

h=

(ii)

0.80 s;

2 1

time to go from top of cliff to the sea = 3.0 – 1.6 = 1.4 s; recognize to use s = ut + 1 at2 with correct substitution, 2

s = 8.0 × 1.4 + 5.0 × (1.4)2; to give s = 21 m;

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3 [6]

14.

(a)

2.4s

1

(b)

h=

(c)

a= 2.4 = 3.75 m𝑠 −2

(d)

F = m × a = 0.78 × 2.4 = 2.925 N

9×2.4 6×1.6 − 2 2

=6m

3

9

2 9

2 [8]

15.

(a)

horizontally: zero; vertically: 9.8(10) ms-2 (downwards);

2

N. B. Part (b) and part (c) to be marked independently of part (a).

16.

33 = 12  9.8 t 2 ;

(b)

s = ut + 12 at 2

(c)

s = ut = 18x 2.6 = 47 (46.8) m;

(a)

(i)

t = 2.6s;

distance 72 or ; time 0.40 = 180 cm s–1;

2 1 [5]

speed =

2

Award [1 max] if time incorrect.

(ii)

s=

1 2

gt2 or 80 =

1 2

× g × 0.42; g = 10 m s–2;

2

Award answer with no working [0]. If it is clear that same mistake as in (i) has been made for the timing, then award full marks in (ii).

(b)

horizontal distance moved = 90 cm; (allow ecf from (a) (i)) vertical distance moved = 125 cm; (allow ecf from (a) (ii)) correct plot from candidate’s working;

3

Award full marks if the plot is correct but there is no working shown.

(c)

sketch: overall reasonable shape (smooth curve “below” given path); horizontal distance moved always decreasing when compared to given path; angle to vertical always greater than given path;

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3 [10]

17.

(a) 0

0.50

x/m 1.0

1.5

2.0

0

–0.50

y/m

–1.0

–1.5

–2.0

Mark both together. VH : horizontal arrows equal in length; VV : two vertical arrows, the one at 1.0 m noticeably longer than the one at 0.5 m;

2

If arrows correct but wrong point(s) award [1].

(b)

(c)

curve that goes through all data points; stops at y = 1.8m as this is the height of the wall; from graph d = 1.5(± 0.1)m;

3

travels vertically 1.8m in 0.6s / 1.25m in 0.5s; 2s g= 2 ; t to give g = 10 (±1) m s–2;

3

Award [2 max] for any time shorter than 0.5 s. [8]

18.

(a)

40 = 8.0 m s–1 5.0 Accept use of other values leading to the same answer. vx =

1

(ii)

vy = 0 = u0y – 10 × 2.5; u0y = 25 m s–1; Accept use of other values leading to the same answer.

2

(iii)

the x and y components of displacement at 3.0 s are 24 m, 30 m;

(i)

so the magnitude is (b)

24 2 + 30 2 = 38 m

maximum height reached is less; asymmetric with shorter range;

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2

2 [7]

19.

(a)

vV = 8.0 sin 60 = 6.9 m s–1; v2 h = 2 g ; to give h = 2.4 m ;

3

Award [1] if v = 8.0 m s–1 to get h = 3.2 m is used. (b)

vH = 8.0 cos 60; range = vHt = 8.0 cos 60 × 3 = 12 m;

2

Award [1] if v = 8.0 m s–1 to get R = 2.4 m is used. [5]

20.

(a)

(b)

21.

(i)

horizontal: 24 ms-1;

1

(ii)

vertical: 14 ms-1;

1

appropriate use of kinematic equation; correct substitution; h = 7.1m;

3 [5]

Motion of a ball (a)

the maximum height reached by the ball / the displacement in the first 2s / the distance travelled;

(b)

30 m; 1 Accept answers in the range 25m to 30m

(c)

(i)

1

drawing tangent at t = 1.0s; using a sufficiently large triangle (at least 6 cm hypotenuse);

(ii)

(d)

accept answers in the range a = 13.3-15ms-2;

3

R + mg = ma; R = 3.75 - 2.50 = 1.2N (Watch ecf from (i))

2

slope of the graph is decreasing; the force of air resistance must decrease as well;

2

(e) smooth curve at t = 2.0 s; terminating between 4.25 s and 4.50 s; (Award second marking point only if first is correct)

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2

(f)

(g)

it will be less; because mechanical energy / kinetic energy is being transformed into thermal energy (in the particle and air); Award [0] for an answer without justification. the areas under the graph for the upward and downward motion must be the same; from the way the curve slopes it follows that the time must be longer than 2.0s;

2

2

or the average speed on the way down is less; and so the time taken is longer; [15]

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WORK, ENERGY AND POWER

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PROBLEMS 1. An object at rest having a mass of 2 kg on a frictionless horizontal plane is pulled by a constant force 5 N from position A to B as shown in figure. 2kg

F = 5N

A

B

5m

(a)

What is the kinetic energy of the object at point B?

(b)

What is the velocity of the object at point B?

2. An object of mass 2 kg is moving on a frictionless horizontal plane. Its velocity is increased from 4 m/s to 12 m/s in 8 seconds by the help of a horizontal force acting on it. Vf=12 m/s at t = 8s

Vi=4 m/s at t = 0 2 kg

F=? 64m

(a) What is the work done by the applied force on the object during the 8 seconds?

(b) What is the magnitude of the resultant force acting on the object in 8 seconds?

3. A man pulls a block as in the figure. Under this force the block displaces 3m on the horizontal. If the coefficient of friction is 0.5; 40N

m=4kg

(a) How much work is done by the man?

370 (b) Find the work done by frictional force. 3m

(c) Find the final speed of the block.

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4. A one dimensional variable force acts on an object moving along a straight line. If the initial and the final speeds are 2m/s and 3m/s respectively what is the mass of the object? F(N) 10 X(m) 4

8

10

12

-10

5. The graph shows the kinetic energy versus position of a 2 kg object which is at rest on the horizontal plane at t=0. Find the time needed for the object to cover 40 m. Kinetic energy (Joule)

400

0

40

20

Position (m)

6. The object which is thrown with an initial velocity of 8 m/s can only reach to a height of 2.5 m on the inclined plane. With what speed will the object return to horizontal plane, if there is friction only on the surface of inclined plane.

8 m/s

2.5 m

7. A block of 1kg is released from top of the inclined plane and reaches the ground with a speed of 2m/s. Find the force of friction. m=1kg 2m 2m/s 300

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8. A body of mass 2 kg starts from rest and it is accelerated to a velocity of 10 m/s while going up along an inclined plane as shown. If frictional coefficient of the surface is 0,5, find the work done by F. Vf =10 m/s

8m

 = 0,5

F 530

9. Graph shows the kinetic energy-potential energy change of a 2 kg object which is thrown obliquely. Energy (j) If the time of flight of the object is 6 seconds, what will be the horizontal displacement of the object at the end of 6 seconds?

Kinetic energy

Potential energy

9 3

6

Time(s)

10. In a truck-loading station at a post office, a 2 kg package is released from rest at point A on a track that is one quadrant of a circle of radius 1 m. It slides down the truck and slides on a level surface a distance of 4 m to point C, where it comes to rest. (a) How much work is done by friction as the A body slides down the circular are from A to B? 1m (Only BC part is frictional)

B

4m

C

(b) What is the coefficient sliding friction on the horizontal surface?

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11. The frictionless system is released from rest. Find with which speed the object m 1 will hit the ground? m2=3kg

m1=1kg h=2m

12. The system is released from rest. Find with which speed the object m1=3kg will hit the ground? m2=2kg =0.5 m1=3kg h=2m

13. The system is released from rest with the 12 kg block 3 m above the floor. Find the velocity with which the block strikes the floor. (Neglect the friction of the pulley.)

12kg

4kg

3m

14. The frictionless system is released from rest. Find with which speed the object m1=3kg will hit to the ground? What is the maximum height object m2 can reach after m1 hit the ground?

m2=1kg

m1=3kg h=2m

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15. The frictionless system is released from rest. Find with which kinetic energy the object m1=3kg will hit the ground?

m2=2kg

m1=3kg h=2.5m

53

16. The m=2kg object is thrown with initial velocity of 8m/s as it is shown. If there is no friction, find the maximum compression in the spring? k=200N/m m=2kg

V=8m/s

h=3m

17. 1 kg object is released on the inclined plane as in Figure and it compresses the spring with the given spring constant by 0.05 m. What is the force of friction acting on the object along the inclined plane? VO = 0 1 kg

0.45 m k = 1000 N/m 0.05 m 37 

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18. An athlete trains by dragging a heavy load across a rough horizontal surface.

F 250

The athlete exerts a force of magnitude F on the load at an angle of 25° to the horizontal. (a)

Once the load is moving at a steady speed, the average horizontal frictional force acting on the load is 470 N. Calculate the average value of F that will enable the load to move at constant speed. [2]

(b)

The load is moved a horizontal distance of 2.5 km in 1.2 hours. Calculate

(c)

(i)

the work done on the load by the force F. [2]

(ii)

the minimum average power required to move the load. [2]

The athlete pulls the load uphill at the same speed as in part (a). Explain, in terms of energy changes, why the minimum average power required is greater than in (b)(ii). [2]

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19. A granite block is suspended at rest just below the surface of water tank (Figure i). The block is now released and falls 0.80m to the bottom (Figure ii).

0.80m

Figure (i)

Figure (ii)

(a) The volume of block is 3.0x10-3m3, and the density of granite is 2700 kgm-3. Calculate the gravitational potential energy lost by the block as it falls. [3]

(b) Although the water level has not changed, the water has gained gravitational potential energy. Explain why. [1]

(c) The gravitational potential energy gained by water is less than the gravitational potential energy lost by block. Explain this. [2]

20. This question is about the law of conservation of energy. (a) State the law of conservation of energy. [2]

(b) Figure below shows a block on a horizontal table top initially held against a spring, so that the spring is compressed. The other end of the spring is fixed to a wall. When released the block is pushed away by the spring. When the spring reaches its naturel length the block leaves the spring and then slides along the table top. A constant frictional force acting between the moving block and the table top eventually brings the block to rest.

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(i) When the block leaves the spring, the block has a kinetic energy 2.2J. The mass of the block is 0.40kg. Calculate the maximum velocity of the block. [1]

(ii) The block travels 1.2m after leaving the spring before coming to rest. Show that the frictional force between the block and table top is about 1.8N. [1]

(iii) The spring was initially compressed through 0.20m. The constant frictional force acts on the block whenever it is moving. Calculate the elastic potential energy in the spring when in its initial compressed position. Assume the spring has negligible mass. State an appropriate unit for your answer. [2]

(iv) The force exerted on the block by the spring is proportional to the compression of the spring. Calculate the maximum force exerted on the block by the spring. [1]

(v) Find the spring constant. [2]

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21. The diagram below represents an escalator. People step on to it at point A and step off at point B. B

30 m

A

(a)

40°

The escalator is 30 m long and makes an angle of 40° with the horizontal. At full capacity, 48 people step on at point A and step off at point B every minute. (i)

Calculate the potential energy gained by a person of weight 700 N in moving from A to B. [2]

(ii)

Estimate the energy supplied by the escalator motor to the people every minute when the escalator is working at full capacity. [1]

(iii)

State one assumption that you have made to obtain your answer to (ii). [1]

The escalator is driven by an electric motor that has an efficiency of 70%. (b)

(c)

(i)

Using your answer to (a)(ii), calculate the minimum input power required by the motor to drive the escalator. [3]

(ii)

Explain why it is not necessary to take into account the weight of the escalator when calculating the input power. [1]

Explain why in practice, the power of the motor will need to be greater than that calculated in (b)(i) [1]

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22. Mechanical power (a)

Define power. [1]

(b)

A car is travelling with constant speed v along a horizontal straight road. There is a total resistive force F acting on the car. Deduce that the power P to overcome the force F is P = Fv. [2]

(c)

A car drives up a straight incline that is 4.8 km long. The total height of the incline is 0.30 km.

4.8km 0.3km

The car moves up the incline at a steady speed of 16 m s–1. During the climb, the average friction force acting on the car is 5.0 x 102 N. The total weight of the car and the driver is 1.2 x 104 N. (i)

Determine the time it takes the car to travel from the bottom to the top of the incline. [2]

(ii)

Determine the work done against the gravitational force in travelling from the bottom to the top of the incline. [1]

(iii)

Using your answers to (c)(i) and (c)(ii), calculate a value for the minimum power output of the car engine needed to move the car from the bottom to the top of the incline. [4]

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(d)

From the top of the incline, the road continues downwards in a straight line. At the point where the road starts to go downwards, the driver of the car in (c), stops the car to look at the view. In continuing his journey, the driver decides to save fuel. He switches off the engine and allows the car to move freely down the hill. The car descends a height of 0.30 km in a distance of 6.4 km before levelling out.

6.4km 0.3km

The average resistive force acting on the car is 5.0 x 102 N. Estimate

(e)

(i)

the acceleration of the car down the incline. [5]

(ii)

the speed of the car at the bottom of the incline. [2]

In fact, for the last few hundred metres of its journey down the hill, the car travels at constant speed. State the value of the frictional force acting on the car whilst it is moving at constant speed. [2]

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23. The graph represents the motion of a car of mass 1.4x103 kg, traveling in a straight line.

(a) Describe, without calculation, how the resultant force acting on the car varies over this 10 second interval. [2]

(b) Calculate the maximum kinetic energy of car. [2]

(c) 4 seconds later, when the car is traveling at its heighset speed, the useful power developed by the engine is 24 kW. Calculate the driving force required to maintain this speed. [2]

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MULTIPLE CHOICE QUESTIONS 1.

The graph below shows the variation with displacement d of the force F applied by a spring on a cart. 5

The work done by the force in moving the cart through a distance of 2 cm is

4 3 F/N

A. 10 × 10–2J.

B. 7 × 10–2J.

C. 5 × 10–2J.

D. 2.5 × 10–2J.

2 1 0 0

2.

1 2 d / 10–2 m

3

The diagram below shows the variation with displacement x of the force F acting on an object in the direction of the displacement. Which area represents the work done by the force when the displacement changes from x1 to x2? F

A. QRS

B. WPRT

R Q

S

C. WPQV

P

0 W 0

3.

V x1

T x2

D. VQRT

x

The variation with time of the vertical speed of a ball falling in air is shown below. Speed

0 0

T

time

During the time from 0 to T, the ball gains kinetic energy and loses gravitational potential energy ΔEp. Which of the following statements is true? A. ΔEp is equal to the gain in kinetic energy. B.

ΔEp is greater than the gain in kinetic energy.

C.

ΔEp is equal to the work done against air resistance.

D.

ΔEp is less than the work done against air resistance.

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4.

The point of action of a constant force F is displaced a distance d. The angle between the force and the direction of the displacement is θ, as shown below.

F

d Which one of the following is the correct expression for the work done by the force?

A. Fd

5.

6.

B. Fd sin θ

C. Fd cos θ

D. Fd tan θ

Which of the following is a correct definition of work? A.

Product of force and distance

B.

Product of force and distance moved in the direction of the force

C.

Product of power and time

D.

Product of force and displacement

The output power of an electric motor is determined using the arrangement shown below. motor

wheel belt

W1 W2

The belt has weights W1 and W2 attached to its ends. The wheel has circumference S. When the wheel is rotating at R revolutions per second, the belt is stationary. Which one of the following is a correct expression for the output power of the motor?

A. W1 x SR

B. W2 x SR

C. (W2 − W1) x SR

D. (W2 + W1) x SR

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7.

An object of mass m falls from rest in a vacuum. As the object falls it loses an amount E of gravitational potential energy. The speed of the object is then

2E . m

A. 8.

B.

m . 2E

C.

2E . m

D.

m . 2E

Forces of magnitude F1 and F2 act tangentially on the edge of a wheel of circumference S. The wheel is made to complete one revolution about its centre, in the direction shown below. Which one of the following is a correct expression for the work done on the wheel?

F1

9.

F2

A. F1 x S

B. F2 x S

C. (F1 – F2) x S

D. (F2 + F1) x S

A body of mass m and speed v has kinetic energy EK. A second body of mass

m moves 2

at speed 2v. The kinetic energy of this second body is A.

10.

EK 2

.

B. EK.

C. 2EK.

D. 4EK.

A spring is compressed by a force F.

F

For a compression e, the force F is given by F = ke. When the compression force is removed, the spring returns to its original length in time t. The best estimate for the power developed by the spring during its expansion is A.

11.

ke . 2t

B.

ke . t

C.

ke 2 . 2t

D.

ke 3 . t

A box of mass m is moved horizontally against a constant frictional force f through a distance s at constant speed v. The work done on the box is A. 0.

B. mgs.

C.

1 2 mv . 2

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D. fs.

12.

An electric motor, with an input power of 250 W, produces 200 W of mechanical power. The efficiency of the motor is A. 20%.

13.

B. 25%.

C. 55%.

D. 80%.

An amount Q of energy is supplied to a machine. The machine does useful work W and an amount R of energy is wasted, as illustrated below. energy supplied Q

useful work W machine

wasted energy R

Which one of the following is a correct expression for the efficiency of the machine? A.

14.

W Q

B.

R Q

C.

W +R Q

D.

W −R Q

A force stretches a wire that is fixed at one end. The value of this force increases from zero to a maximum value and then returns to zero. The graph below shows the variation with force F of the extension x of the wire. x

Which area, or areas, represents the net work done on the wire by the force?

area R

A. Area P C. Area R

area Q

D. Area Q and area R

area P

0 0

15.

B. Area Q

F

A brother and sister take the same time to run up a set of steps. The sister has a greater mass than her brother. Which of the following is correct? Has done the most work

Has developed the greatest power

A.

brother

brother

B.

brother

sister

C.

sister

brother

D.

sister

sister

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16.

The graph shows the variation with force F of the extension s of a spring.

The work done in changing the extension of the spring from 3.0 cm to 6.0 cm is

17.

A.

15 N cm.

B.

30 N cm.

C.

45 N cm.

D.

60 N cm.

A constant force acts on a mass that is initially at rest. Which of the following graphs best shows how the kinetic energy EK of the mass changes with the work W done on the mass? Friction is negligible.

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18. A ball is thrown vertically upwards and comes down again. Air resistance is negligible. Which of the following graphs shows how the gravitational potential energy EP varies with time t?

19.

A pump extracts water from a well of depth h at a constant rate of R kg s–1. What is the power required to raise the water? A.

20.

R gh

B. Rgh

C.

Rg h

D.

hg R

A railway engine of mass m moves along a horizontal track with uniform speed v. The total resistive force acting on the engine is F.

Which of the following is the power of the engine?

A.

F mv

B. Fv

C.

mv F

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D.

v F

KEY FOR PROBLEMS 1.

(a) 25 J

2.

(a) 128 J (b) 2 N

3.

(a) 96 J

4.

4 kg

5.

3s

6.

6 m/s

7.

4N

8.

320 J

9.

18 m

10.

(a) 20J

11.

V=10m/s

12.

V=4 m/s

13.

30 m/s

14.

V=25 m/s ; 3 m

15.

21 j

16.

0.2 m

17.

3.5 N

18.

(a) (b)

(b) 5 m/s

(b) 24 j

(c) 6m/s

(b) 0.25

F cos 25 = 470; (i)

(ii)

520 N;

2

work done – 470 × 2500; 1.2 MJ; Award [1 max] for power of 10 error.

2

1.2  10 6 ; 270 W; Allow correct solution from power = F × v. 1.2  60  60

(c) work still done against friction ; work done raising load vertically / increase in gravitational potential energy; 19.

(a) Calculation of gravitational potential energy: Use of m = dV; Use of E = mgh [m=8.1 10xkg]; E = 64J (b) Explanation: (Some) Water has moved up Why g.p.e. is less:

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2 [8]

3

1

Water has less mass; Wtae has lower density/ moved up same distance/ where the block was. OR Some energy is dissipiated/ lost to surroundings/ converted to other forms; KE/ Internaş energy/ Heat/ Sound. OR Mechanism: via friction or drag/ because the block or water accelerates/ as block hits the bottom.

2

[6] 20.

(a) Energy can not be created or destroyed; It only be transferred/ changed/ converted/ from one form to another. ‘Transformed’can be taken to mean transferred from one form to another. (b) (i)(using EK=1/2 mv2) → 2.2 = ½ x 0.4 x v2 → V = 3.3ms-1. Ignore errors in 3 sig fig. Answer only can gain mark. (ii) (F=ma)=0.4 x 4.59 = 1.84N (iii)(work done in moving 0.2m) = 1.8 x 0.2 = 0.36J; allow ecf (bii) x 0.2 Total work done = 2.2 + 0.36 = 2.6 (1) Joule/ J Same answer is achieved if F = 2N (iv) (use of energy = ½Fx) → 2.6 = ½Fmax x 0.2 → Fmax = 26N allow ecf 10 x (biii) (v) 26 = k x 0.2 K = 130 N/m

2 1

1

2

1

2 [9]

21. (a) Note: for part (i) and (ii) the answers in brackets are those arrived at if 19.3 is used as the value for the height. (i)

height raised = 30 sin 40 = 19 m; gain in PE = mgh = 700 × 19 = 1.3 × 104 J (1.4 × 104 J);

2

(ii)

48 × 1.3 × 104 J = 6.2 × 105 J (6.7 × 105 J);

1

(iii)

the people stand still / don’t walk up the escalator / their average weight is 700 N / ignore any gain in KE of the people;

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1 max

(b) (i)

𝐸𝑓𝑓 =

power required =

𝑃𝑜𝑢𝑡 𝑃𝑖𝑛

𝑃𝑖𝑛 =

6.2  10 5 = 10 kW (11 kW); 60

𝑃𝑜𝑢𝑡 𝐸𝑓𝑓

Pin = 14 kW (16 kW); (ii)

3

the escalator can in theory return to the ground under the action of gravity / OWTTE;

(c) power will be lost due to friction in the escalator / OWTTE;

1 1

The location of the friction must be given to obtain the mark. [9] 22. Mechanical power (a) the rate of working / work per time; If equation is given, then symbols must be defined. (b) P =

(c)

d W F d = ; v = therefore, P = Fv ; t t t

(i) t =

4800 d = 300 s ; ;= 16 v

(ii) W = mgh = 1.2 x 104 x 300 = 3.6 x 106 J;

1

2

2

1

(iii) work done against friction = 4.8 x 103 x 5.0 x 102; total work done = 2.4 x 106 + 3.6 x 106; total work done = P x t = 6.0 x 106; to give P =

(c)

(i) sinθ =

6.0 10 6 = 20 kW ; 300

4

0.30 = 0.047 ; 6.4

weight down the plane = W sin = 1.2 104 0.047 = 5.6 102N; net force on car F = 5.6 102 5.0 102 = 60N;

a=

60 F = 5.0 10 − 2 ms − 2 ; ; 3 m 1.2 10

(ii) v2 = 2as = 2 x 5.0 x 10-2 x 6.4 103; to give v = 25ms-1; (d)

5.6 x 102 N;

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5

2 1 [18]

23. (a) decreases fort he first four seconds. Zero for the remaining six seconds.

2

(b) EK = ½ x 1.4 x 103 x 162 EK = 1.8 x 105J. Except v = 15ms-1 from misleading graph and EK = 1.6 x 105J

2

(c) (use of P = Fv gives) 24 x 103 = F x 16 F = 1500N.

2

KEY for multiple choice questions 1. C

2. D

3. B

4. C

5. B

6. C

7. A

8. C

9. C

10. C

11. D

12. D

13. A

14. B

15. D

16. C

17. B

18. D

19. D

20. B

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166 |TED Ankara College Foundation High School Physics Department

IMPULSE AND MOMENTUM

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PROBLEMS 1. Variation of net force applied to the body of mass 4kg is given. F(N)

(a) What was the impulse in 6 sec?

10 t(sec) 2

4

6

(b) What is the velocity of the body at t=6sec given that its initial velocity is zero?

-10

2. The F-t graph of an object having mass of 14kg and initial velocity of 3 m/s is given. F(N)

(a) What is the impulse in 10 sec?

10

t(s) 4

8

(b) What is the velocity at t=10 sec?

10

3. A ball with mass m=3kg bounces from the surface as shown. If the contact of the ball with the ground is in 0.15sec, find the average force exerted by the surface to the ball? m=3kg

V2=5m/s

V1=5m/s 60

4.

60

An object with mass m=2kg bounces back from the spring on a frictionless surface. If the interaction time is 0.8 sec, find the average force exerted by the spring on the object? 10m/s m=2kg

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5. An object with mass m=3kg is thrown horizontally from h=3.2m. Prove that the change in momentum is equal to the impulse. m=3kg

V=6m/s

h=3.2m

Vf=?

6. Two objects make a perfectly elastic collision, find their final velocities? -

m1=1kg

+ V1=6m/s

V2=0 m2=2kg

7. Two objects make a perfectly elastic collision, find their final velocities? V1=6m/s m1=6kg

+ V2=4m/s m2=2kg

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8. Two objects make a perfectly elastic collision, find their final velocities? V1=4m/s

+ V2=6m/s

m1=3kg

m2=2kg

9. Two objects make an inelastic collision. What is the energy wasted to heat during collision? V1= 8m/s

V2= 2m/s

m1=1kg

m2=3kg

10. A bullet mass of m is fired into a wooden block mass of 4m with velocity of 10 m/s as in the figure. Find the horizontal distance x which the system hit to the ground? Vb = 10 m/s

m2=4m

mb=m

h=80m

x=?

11. A bullet with mass of 20g is fired into a block at rest with mass of 1980g. If the block and bullet can reach 20 cm as shown, find initial velocity of bullet?

V= ? 20cm m=20g

170 |TED Ankara College Foundation High School Physics Department

12. A boy of mass 20 kg. is on a trolley of mass 60 kg. Velocity of the trolley is 10 m/s. If the boy jumps to the ground with a velocity of 2 m/s. with respect to the ground, from the back of the trolley, what happens to the speed of the trolley? V = 10 m/s

13. Objects m1 and m2 collide inelastically at point O. What is their final velocity? V1=2m/s O m1=3kg V2=4m/s m2=2kg

14. Two objects make an elastic collision. If the velocity of one of them is given, find the velocity of the other? V1=15m/s

m1

V1=30m/s

V2=0 60 30

m1=2kg

m2=3kg m2

Before collision

V2=?

After collision

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15. Linear momentum (a)

Define (i)

linear momentum; (1)

(ii)

impulse. (1)

(b)

Explain whether momentum and impulse are scalar or vector quantities. (1)

(c)

By reference to Newton’s laws of motion, deduce that when two particles collide, momentum is conserved. (5)

speed before = 20 m s –1

A rubber ball of mass 50 g is thrown towards a vertical wall. It strikes the wall at a horizontal speed of 20 m s–1 and bounces back with a horizontal speed of 18 m s–1 as shown below. The ball is in contact with the wall for 0.080 s.

speed after =18 m s

(d)

(i)

–1

Calculate the change in momentum of the ball. (2)

(ii)

Calculate the average force exerted by the ball on the wall. (2)

(iii)

Suggest, in terms of Newton’s laws of motion, why a steel ball of the same mass and the same initial horizontal speed exerts a greater force on the wall. (3) (Total 15 marks)

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16. Momentum (a)

State the law of conservation of momentum. (2)

(b)

An ice hockey puck collides with the wall of an ice rink. The puck is sliding along a line that makes an angle of 450 to the wall. wall 45

45

ice rink

direction of puck before collision

direction of puck after collision

The collision between the wall and the puck is perfectly elastic. (i)

State what is meant by an elastic collision. (1)

(ii)

Discuss how the law of conservation of momentum applies to this situation. (2)

(c)

The diagram below is a scale diagram that shows the vector representing the momentum of the puck before collision. Scale: 1.0 cm = 0.10 N s

By adding appropriate vectors to the diagram, deduce that the magnitude of the change in momentum of the puck as a result of the collision is 0.71 N s. (4)

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(d)

The sketch-graph below shows the variation with time t of the force F exerted by the wall on the puck. The total contact time is 12 ms. Estimate, explaining your reasoning, the maximum force exerted by the wall on the puck.

F

(3) (Total 12 marks)

0

0

t

17. This question is about conservation of momentum and conservation of energy. (a)

State Newton’s third law. (1)

(b)

State the law of conservation of momentum. (2)

The diagram below shows two identical balls A and B on a horizontal surface. Ball B is at rest and ball A is moving with speed V along a line joining the centres of the balls. The mass of each ball is M. v Before collision

B

A

During the collision of the balls, the magnitude of the force that ball A exerts on ball B is FAB and the magnitude of the force that ball B exerts on ball A is FBA.

(c)

On the diagram below, add labelled arrows to represent the magnitude and direction of the forces FAB and FBA.

During the collision

A

B

(3)

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The balls are in contact for a time Δt. After the collision, the speed of ball A is +vA and the speed of ball B is +vB in the directions shown. vA After the collision

vB A

B

As a result of the collision, there is a change in momentum of ball A and of ball B. (d)

Use Newton’s second law of motion to deduce an expression relating the forces acting during the collision to the change in momentum of (i)

ball B. (2)

(ii)

ball A. (2)

(e)

Apply Newton’s third law and your answers to (d), to deduce that the change in momentum of the system (ball A and ball B) as a result of this collision, is zero.

(f)

Deduce, that if kinetic energy is conserved in the collision, then after the collision, ball A will come to rest and ball B will move with speed V. (3) (Total 17 marks)

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18. This question is about energy and momentum. 6.0m s–1

A train carriage A of mass 500 kg is moving horizontally at 6.0 m s–1. It collides with another

train carriage A 500kg

train carriage B of mass 700 kg that is initially at rest, as shown in the diagram below.

train carriage B 700kg

The graph below shows the variation with time t of the velocities of the two train carriages before, during and after the collision. v / ms–1 6.0 train carriage B

5.0 4.0 3.0 2.0 1.0 0.0 1.0

2.0

3.0

4.0

5.0

–1.0

6.0

7.0

8.0

9.0

10.0 t / s

train carriage A

–2.0

(a)

Use the graph to deduce that (i)

the total momentum of the system is conserved in the collision; (2)

(ii)

the collision is elastic. (2)

(b)

Calculate the magnitude of the average force experienced by train carriage B. (3) (Total 7 marks)

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19. This question is about momentum and energy. (a)

Define impulse of a force and state the relation between impulse and momentum. (2) definition: relation:

(b)

By applying Newton’s laws of motion to the collision of two particles, deduce that momentum is conserved in the collision. (5)

(c)

In an experiment to measure the speed of a bullet, the bullet is fired into a piece of plasticine suspended from a rigid support by a light thread.

bullet

The speed of the bullet on impact with the plasticine is V. As a result of the impact, the bullet embeds itself in the plasticine and the plasticine is displaced vertically through a height of 24cm 24 cm. The mass of the bullet is 5.2×10–3 kg and the mass

speed V

of the plasticine is 0.38 kg. plasticine

(i)

Ignoring the mass of the bullet, calculate the speed of the plasticine immediately after the impact. (2)

(ii)

Deduce that the speed V with which the bullet strikes the plasticine is about 160 m s–1. (2) (Total 11 marks)

177 |TED Ankara College Foundation High School Physics Department

20. This question is about momentum. (a)

Define (i)

linear momentum. (1)

(ii)

impulse. (1)

(b)

In a ride in a pleasure park, a carriage of mass 450 kg is travelling horizontally at a speed of 18 m s–1. It passes through a shallow tank containing stationary water. The tank is of length 9.3 m. The carriage leaves the tank at a speed of 13 m s–1. 18 m s–1

water-tank

13 m s–1

carriage, mass 450 kg

9.3m

As the carriage passes through the tank, the carriage loses momentum and causes some water to be pushed forwards with a speed of 19 m s–1 in the direction of motion of the carriage. (i)

For the carriage passing through the water-tank, deduce that the magnitude of its total change in momentum is 2250N s. (1)

(ii)

Use the answer in (b)(i) to deduce that the mass of water moved in the direction of motion of the carriage is approximately 120 kg. (2)

(iii)

Calculate the mean value of the magnitude of the acceleration of the carriage in the water. (3)

(c)

For the carriage in (b) passing through the water-tank, determine (i)

its total loss in kinetic energy.

(3)

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(ii)

the gain in kinetic energy of the water that is moved in the direction of motion of the carriage.

(1) (d)

By reference to the principles of conservation of momentum and of energy, explain your answers in (c). (3) (Total 15 marks)

21. This question is about the collision between two railway trucks (carts). (a)

Define linear momentum. (1)

In the diagram below, railway truck A is moving along a horizontal track. It collides with a stationary truck B and on collision, the two join together. Immediately before the collision, truck A is moving with speed 5.0 ms–1. Immediately after collision, the speed of the trucks is v. 5.0 ms –1 B

A

Immediately before collision

v A

B

Immediately after collision

The mass of truck A is 800 kg and the mass of truck B is 1200 kg. (b)

(i)

Calculate the speed v immediately after the collision. (3)

179 |TED Ankara College Foundation High School Physics Department

(ii)

Calculate the total kinetic energy lost during the collision. (2)

(c)

Suggest what has happened to the lost kinetic energy. (2) (Total 8 marks)

22. This question is about driving a metal bar into the ground. Large metal bars can be driven into the ground using a heavy falling object. In the situation shown, the object has a mass 2.0 × 103 kg

object mass = 2.0×103 kg

and the metal bar has a mass of 400 kg. bar mass = 400 kg

The object strikes the bar at a speed of 6.0 m s–1. It comes to rest on the bar without bouncing. As a result of the collision, the bar is driven into the ground to a depth of 0.75 m.

(a)

Determine the speed of the bar immediately after the object strikes it. (4)

(b)

Determine the average frictional force exerted by the ground on the bar. (3) (Total 7 marks)

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23. Collisions A large metal ball is hung from a crane by means of a cable of length 5.8 m as shown below. cable crane 5.8 m wall metal ball

In order to knock down a wall, the metal ball of mass 350 kg is pulled away from the wall and then released. The crane does not move. The graph below shows the variation with time t of the speed v of the ball after release. 3.0

v / m s–1

2.0

1.0

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

t/s

The ball makes contact with the wall when the cable from the crane is vertical. (a)

Use the graph to determine the distance moved by the ball after coming into contact with the wall. (2)

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(b)

For the collision between the ball and the wall, calculate (i)

the total change in momentum of the ball; (2)

(ii)

the average force exerted by the ball on the wall. (2)

(c)

(i)

State the law of conservation of momentum. (2)

(ii)

The metal ball has lost momentum. Discuss whether the law applies to this situation. (2)

(d)

During the impact of the ball with the wall, 12J of the total kinetic energy of the ball is converted into thermal energy in the ball. The metal of the ball has specific heat capacity 450 J kg–1 K–1. Determine the average rise in temperature of the ball as a result of colliding with the wall. (4) (Total 14 marks)

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KEY FOR PROBLEMS

1.

(a) 10Ns

(b) 2.5m/s

2.

(a) 70 Ns (b) 8m/s

3.

1003 N

4.

50 N

5.

F.t = m.V = 24 N.s

6.

-2m/s and +4m/s

7.

1m/s and +11m/s

8.

-4m/s and +6m/s

9.

37.5 j

10. 8 m 11. 200 m/s 12. 14 m/s 13. 2 m/s 14. 103 m/s 15.

(a)

(i)

product of mass and velocity / OWTTE;

1

(ii)

change of momentum / OWTTE; Accept product of force and time taken / OWTTE.

1

(b)

they are vectors because they have magnitude and direction; Answer needs some form of explanation to receive the mark but it can be simple.

(c)

appropriate reference / naming of Newton III; to give forces equal and opposite; time of collision the same for each particle; appropriate reference / naming of Newton II; impulse / change in momentum equal and opposite;

1

5

change of momentum = 0.05 x ((- 18) - 20); = 1.9 kgms-1; Award [1 max] for forgetting vector nature ie 0.1 kg ms-1.

2

(ii)

force = answer to (i) / 0.08; = 23.75N = 24N;

2

(iii)

shorter contact time / greater rebound speed; so rate of change in momentum larger / OWTTE; appropriate reference to Newton’s laws;

(d) (i)

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3 [15]

16.

Momentum (a)

(b)

if the total (or net) external force acting on a system is zero / for an isolated system; the momentum of the system is constant / momentum before collision equals momentum after collision; Award [1] for “momentum before (collision) = momentum after (collision)”. (i)

(ii)

(a collision in which) kinetic energy is not lost / kinetic energy is conserved;

2

1

the momentum of the puck is not conserved since a force acts on it during collision / OWTTE; the rink is attached to the Earth and momentum is given to the Earth such that the change in momentum of the puck is equal to the change in momentum of the Earth / OWTTE; Or the momentum of the Earth and puck are conserved / OWTTE; the change in momentum of the puck is equal and opposite to the change in momentum of the Earth;

(c)

2

vector 5.0 cm long; at right angles to initial vector as shown; By eye is sufficient. resultant vector as shown; stated length = 7.1(±0.2) cm equivalent to 0.71(±0.2)Ns; Length should be checked. Or Second vector at right angles to first; And of equal length; Difference shown as a vertical vector; Of magnitude = 0.71N s

0.5 2 + 0.5 2 ;

4

Caution: Many students are obtaining instead the sum of the two momenta rather than the difference. In this case the numerical answer is the same for the magnitude so watch out. (d)

p 0.71 = = 59 N ; t 12 10 −3 this is the average force and from the graph it can be seen that F = 2Fav; therefore, F = 120N; Or 1 2  0.71 =120 N ; 3 area under graph is p = 0.71N s; area is Fmax t ; and so Fmax = 2 12 10 −3 [12] F=

184 |TED Ankara College Foundation High School Physics Department

17.

(a)

when two bodies A and B interact, the force that A exerts on B is equal and opposite to the force that B exerts on A; or when a force acts on a body an equal and opposite force acts on another body somewhere in the universe;

1 max

Award [0] for “action and reaction are equal and opposite” unless they explain what is meant by the terms. (b)

if the net external force acting on a system is zero; then the total momentum of the system is constant (or in any one direction, is constant);

2

To achieve [2] answers should mention forces and should show what is meant by conserved. Award [1 max] for a definition such as “for a system of colliding bodies, the momentum is constant” and [0] for “a system of colliding bodies, momentum is conserved”. (c) FBA

A

B

FAB

arrows of equal length; acting through centre of spheres; correct labelling consistent with correct direction; (d)

(i)

(ii)

(e)

Ball B: change in momentum = MvB; hence FAB∆t = MvB;

2

Ball A: change in momentum = M (vA –V); hence from Newton 2, FBA∆t = M(vA – V);

2

from Newton 3, FAB + FBA = 0, or FAB = –FBA; therefore –M(vA – V) = MvB; therefore MV = MvB + MvA; that is, momentum before equals momentum after collision such that the net change in momentum is zero (unchanged) / OWTTE; Some statement is required to get the fourth mark ie an interpretation of the maths result.

(f)

3

from conservation of momentum V = vB + vA; from conservation of energy V2 = vB2 + vA2; if vA = 0, then both these show that vB = V; or from conservation of momentum V = vB + vA; from conservation of energy V2 = vB2 + vA2;

185 |TED Ankara College Foundation High School Physics Department

4

so, V2 = (vB + vA)2 = vB2 + vA2 + 2vAvB therefore vA has to be zero;

3 max

Answers must show that effectively, the only way that both momentum and energy conservation can be satisfied is that ball A comes to rest and ball B moves off with speed V. [17] 18.

(a)

(i)

initial momentum = 500 x 6 = 3000Ns; final momentum = 500 x (-1) + 700 x 5 = 3000Ns;

2

(working must be shown to award marks) Allow approach that shows equal and opposite momentum changes. (ii)

1 2

initial kinetic energy =

500 36 = 9000J;

final kinetic energy = 500x 1 + 12 700x 25 = 9000J; (working must be shown to award marks) 1 2

(b)

2

impulse = change of momentum = 700 x 5 = 3500Ns; duration of collision = 2.0s;

3500 =1800 N; to give F = 2.0

3

Accept force in the range 1700N to 1800N even with three significant figures. [7] 19.

(a)

(b)

(impulse =) force × time for which force acts; impulse (Fxt) = change in momentum (∆p);

2

The following points are needed for maximum marks. from Newton 3; when objects are in contact, the forces exerted by the objects on each other are equal and opposite; from Newton 2 / collision time is the same; impulses are equal and opposite; therefore changes in momentum are equal and opposite / total change in momentum is zero; or Accept algebraic solution. from Newton 3; FAB = –FBA ; from Newton 2; FAB x t = mA x vA; = –mB x vB;

(c)

(i)

v=

2 gh ;

to give v = 2.2 ms–1; Award full marks for bald correct answer. (ii)

5

from conservation of momentum / V × 5.2 × 10–3 = 0.38 × 2.2 0.38  2.2 V= 5.2  10 −3 to give V = 160 m s–1

186 |TED Ankara College Foundation High School Physics Department

2

2 [11]

20.

(a)

(b)

(i)

momentum is mass × velocity;

1

(ii)

impulse is force × time / change in momentum; In each case allow an equation, with symbols explained.

1

(i)

p = 450 (18 – 13); = 2250 kg m s–1

1

(ii)

idea of equating p to change in momentum of water; 2250 m= = 118 kg (= 120kg); 19

(iii)

time of trolley in tank = a=

(18 − 13) 0.60

a = 8.3 m s–2

(c)

(d)

21.

3

Or

13 2 − 18 2 v2 = u+2 as a = ; 2  9.3

(i)

EK =

1 1 mv 2 ; = × 450 × (182 – 132); = 35000 J; 2 2

3

(ii)

EK =

1 × 118 × 19 2 = 21000 J; (allow 22 000 J for use of m=120 kg) 2

1

a = 8.3 m s–2;

some water will be thrown “sideways”; this will account for the difference in the kinetic energies; this will not have any momentum in the forward direction / equal masses of water to left and right;

(a)

mass × velocity;

(b)

(i)

(ii)

(c)

or

9 .3 = 0.60 s; 15 .5 2250 force = (= 3750 N); 0.60 3750 a= = 8.3 m s–2; 450

2

3 [15]

1

momentum before = 800 × 5 = 4 000 N s; momentum after = 2 000v; conservation of momentum gives v = 2.0 m s–1;

3

KE before = 400 × 25 = 10 000 J KE after = 1 000 × 4 = 4 000 J; loss in KE = 6 000 J;

2

transformed / changed into; heat (internal energy) (and sound);

2

Do not accept “deformation of trucks”. [8]

187 |TED Ankara College Foundation High School Physics Department

22.

(a)

momentum of object = 2 × 103 × 6.0; momentum after collision = 2.4 × 103 × v; use conservation of momentum, 2 × 103 × 6.0 = 2.4 × 103 × v; to get v = 5.0 m s–1;

(b)

4

KE of object and bar + change in PE = 1.2 × 103 × 25 + 2.4 × 103 × 10 × 0.7533; use E = Fd, 4.8 × 104 = F × 0.75; to give F = 64 kN; Award [2 max] if PE missed F = 40 kN. or a=

v2 ; F – mg = ma; to give F = 64kN; 2s

3

Award [2 max] if mg missed. [7] 23.

Collisions (a)

(b)

(c)

idea of use of area under graph / appropriate equation; distance = 12 x 0.15 x 2.6 (allow 0.14 - 0.15 s for the time) = 0.195m; (allow 0.20m, not 0.2m) (i)

idea of momentum as mv; total change (= 2.6x350) = 910Ns;

p  910  ; force  =  = 6100 N ; t  0.15 

(ii)

idea of average force as

(i)

for isolated / closed system; total momentum remains constant;

(ii)

external force acts on ball; so law does not apply to the ball;

2

2 2 2

or system is ball + wall / Earth; momentum loss of ball = momentum gain of wall / Earth; (d)

EK =

1 2

2

x 350 x 2.62; thermal energy = 350  450  ;

idea of 0.12  EK = mc and  = 9.0  10−4 K;

188 |TED Ankara College Foundation High School Physics Department

4 [14]