Computational Physics Lab

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Mechanics & Thermodynamics Alexander R. Vaucher

Computational Physics experiments suitable for classroom or online laboratory

Table of Contents 1. Probability & Statistics: Coin Toss 2. Projectile Motion 3. Uniform Acceleration & Newton’s 2nd Law 4. Free Fall & Drag Force 5. Circular Motion 6. Conservation of Momentum 7. Archimedes Principle 8. Elasticity & Young’s Modulus 9. Simple Pendulum 10. Damped Harmonic Oscillator 11. Speed of Sound 12. Specific Heat 13. Charles Law & Entropy 14. Appendix: Statistical Error & Error Propagation

Copyright © 2020 by Alexander R. Vaucher All rights reserved. No part of this publication may be reproduced, distributed, or transmitted in any form or by any means, including photocopying, recording, or other electronic or mechanical methods, without the prior written permission of the publisher, except in the case of brief quotations embodied in critical reviews and certain other noncommercial uses permitted by copyright law. For permission requests, please contact the author, subject line “Attention: Permissions Coordinator,” at the email address [email protected]. Printed in the United States of America

Preface This lab manual was originally developed by necessity to provide online lab instruction due to the COVID-19 pandemic. The idea was borne out of the High Energy Physics model, where a few physicists conduct the experiments and share the data with others who conduct their independent analysis, then reach a consensus through collaboration. In the present manifestation, students are provided with data, and are encouraged to collaborate on the analysis, but they are still required to submit individual reports for pedagogical and grading reasons. The manual is in two parts: Mechanics & Thermodynamics; and Electromagnetism & Modern Physics. Its intended for a two or three semester calculus based course for students in Physics, Engineering, Bio-Physics, Bio-Engineering, and Life Sciences.

Alexander R. Vaucher, PhD High Energy Physics Pasadena City College Pasadena, CA August, 2020

Computational Lab

Toss of Coin

A. Vaucher

Toss of Coin 1.0

Introduction All scientific activities involve measurement or collection of data at one point or another. Repeated measurements of any quantity will yield many values, how do we know which value to use? Measured data tends to exhibit clustering in some well-known distributions, with properties that allows us to obtain meaningful information. A most commonly occurring distribution is the Gaussian, or Normal Distribution. In this lab, will use a coin toss to collect data and construct a probability distribution.

2.0

Procedure 2.1 Collect 16 pennies in a cup. Shake the cup and throw the pennies on the table, and count the number of heads. This is one trial. Repeat for a total of 30 trials. Use an excel file to record the data in a table as shown below. Trial 1 2 3 … … … … … 29 30

Number of Heads

Below the number of heads column, use excel functions to calculate the Mean (average), Median, Mode, and Standard Deviation. 2.3 Make another set of columns with all the possible number of heads from each trials, Probability of Occurrence, and Gaussian Distribution. This will look like this: A B C D # of Possible Heads # of Trials With this Probability of Gaussian Distribution in each trials Many Heads occurrence (value in Column B/30) 1 2 3 4 5 6 7 8 Page 1 of 60

Computational Lab

Toss of Coin

A. Vaucher

9 10 11 12 13 14 15 16 To fill out the last column, use excel NORM.DIST function. This will require you to enter the mean and standard deviation values obtained in previous section. 2.4 Plot the data in Column B in the table above in a bar-plot. It should look something like this:

# of trials (Occurance)

Frequency of Occurance (# trials with this many heads) 8 6 4 # trials with this many heads

2 0 1 2 3 4 5 6 7 8 9 10111213141516 Number of Heads

2.5 Do the same for Columns C. Label the plots appropriately. It should look similar to this:

Probability of Occurance

Probability of occurance (#heads/possible 16) 0.4 0.3 0.2 Probability of occurance

0.1 0 1 2 3 4 5 6 7 8 9 10111213141516 Number of Heads

Page 2 of 60

Computational Lab

Toss of Coin

A. Vaucher

2.6 Plot the data in Column D in a scatter-continuous plot and label appropriately. it should look similar to this:

Probability Density

Gaussian Dist. 24 trials, 16 pennies each 0.2 0.15 0.1 Gaussian Dist

0.05 0 0

5

10

15

20

Number of Heads

2.7 Combine the last two plots to obtain something similar to this:

Gaussian Dist. 24 trials, 16 pennies each Probability Density

0.4 0.3 0.2

Probability of occurance

0.1

Gaussian Dist

0 1 2 3 4 5 6 7 8 9 10111213141516 Number of Heads

Calculate the % difference between the mean value you obtained for the number of heads with the theoretical value of 8. What happens if you increase the number of trials?

Page 3 of 60

Computational Lab

Projectile Motion

A. Vaucher

Projectile Motion 1.0

Introduction Any object moving in the earth’s gravitational field follows a trajectory whose coordinates are given by kinematical formulas. In this lab we will investigate how the range of a projectile varies with the launch angle. We’ll also calculate the maximum range and compare with the experimental result.



0 

yi

y0

x Figure 1. A projectile is fired from a cannon. The cannon muzzle is displaced from the pivot point by a distance  . The platform where the cannon sits is at a height y0 . the initial height of the cannon ball when it leaves the muzzle is

yi  y0   sin 

The geometry of the launch is shown in figure 1. The kinematical formulas giving the final horizontal position (range) and the height yf at any moment in time are:

x f   xi t 1 y f  y i  yit  ayt 2

2

(1)

Here, the subscripts “i” and “f” refer to “initial” and “final” values respectively. . 2.0

Measuring the initial velocity magnitude: The apparatus is setup so the cannon ball is launched horizontally as shown in figure 2.

Page 4 of 60

Computational Lab

Projectile Motion

Figure 2. The cannon muzzle is placed in a horizontal position, with height

A. Vaucher

  0.

Thus, the initial

yi  y0 . The entire initial velocity vector is in the x direction, and therefore its

magnitude can be easily determined as described in the text.

In this configuration, the horizontal and vertical distances traveled are given by the formulas:

x0  0t

1 y0  at 2 2 Eliminating t, and solving for the velocity, we obtain the initial velocity v0

0  x0

a 2 y0

(2)

The distances x0 and y0, were measured for 25 trials for the same initial velocity of the projectile. The data obtained is in the format { x0, y0}, and is given here: {0.2, 0.00734918}, {0.4, 0.0108879}, {0.6, 0.0820537}, {0.8, 0.174586}, {1., 0.185494}, {1.2, 0.333048}, {1.4, 0.298559}, {1.6, 0.458682}, {1.8, 0.671499}, {2., 0.677133}, {2.2, 0.946827}, {2.4, 0.910803}, {2.6, 1.41813}, {2.8, 1.43822}, {3., 1.71362}, {3.2, 1.8736}, {3.4, 2.07437}, {3.6, 2.34807}, {3.8, 2.88103}, {4., 2.86282}, {4.2, 3.49134}, {4.4, 3.23579}, {4.6, 3.84638}, {4.8, 4.4828}, {5., 4.67825} Use this data in equation (2) to calculate values for the initial velocity v0. Use a = 9.8 m/s2. Obtain the average initial velocity; record your value here:

0  __________________________

m/s

We will use this value later to calculate the maximum range.

Page 5 of 60

Computational Lab 3.0

Projectile Motion

A. Vaucher

Measuring Range vs. Launch Angle: Orient the launch mechanism as shown in the top figure for the various angles in the table below. For each angle, record the range of the ball in the appropriate column. Data Various Launch Angles

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

Launch Angle, θ (Deg.) 10 15 20 25 30 35 40 45 50 55 60 65 70 75

Range, x (meters) 5.49613 5.65346 5.75923 5.80544 5.78484 5.69138 5.52051 5.26951 4.93771 4.52665 4.04009 3.48399 2.86639 2.19716

Use Excel to plot the range x, on the vertical axis, and the launch angle θ on the horizontal axis. Do not connect the points on the plot. Find the best polynomial fit to the points. By inspection, find the angle where the value of x reaches maximum. Include a copy of the plot with your report. Record your value for θmax and the corresponding xmax here:

 max  ____________________________ (Degrees)

xmax  _____________________________ (meters)

4.0

Calculating the Maximum Range: Using the value of θmax calculate the components of the initial velocity using the magnitude of 0 from section 2.0 and the following formulas:

 xi  0 cos(max )  yi  0 sin( max) Page 6 of 60

(3)

Computational Lab

Projectile Motion

A. Vaucher

Record your values here:

 xi  ______________________________ (m/s)  yi  _______________________________ (m/s) The maximum range can be obtained by two steps. From the second in equations (1), solving for the time t, we obtain the total time of flight for the maximum angle:

t

 yi   yi2  2ay( y f  yi ) ay

(4)

Where:

( y f  yi )  H  max  ( y 0   sin max ) ay   g

(5)

Making these substitutions, we obtain:

t

 yi   yi2  2 g H max g

(6)

This is the time of flight when the ball is launched at the angle that produces maximum range. Use this formula to calculate t, Record your value here:

t = _________________________________ (s) Next, use the first of equations (1) to calculate the maximum range:

x f  xi t Use you value for  xi obtained from equations (3) above.

Record your value of x f here:

x f : ___________________________

Page 7 of 60

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Computational Lab

Projectile Motion

A. Vaucher

Calculate the percent difference between the experimental value for the maximum range xmax and the calculated value x f :

% Diff . 

xmax  x f xf

100%

Record your % Diff. below:

% Diff. = : ___________________________

Discuss possible causes for the difference obtained.

5.0

Additional Questions: 1. The maximum range for projectiles on the surface of the earth occurs at 45 Deg. In your experiment was  max equal to 45 Deg.? Why or why not?

2. A ball is thrown upward with an initial velocity of 10 m/s. How high will the ball reach?

Page 8 of 60

Computational Lab

Uniform Acceleration, Newton’s 2nd Law

A. Vaucher

Uniform Acceleration – Newton’s 2nd Law 1.0

Introduction 1.1 Consider a cart of mass M on a track as shown. The cart is attached to another mass, m, by a rope, which passes over a pulley. Ignoring friction in this system, the system will accelerate to the right as shown. Calculate the acceleration, a, of the system. M Show that the acceleration of the system is given by:

a

 m  a  g , this will be your theoretical value for a: mM  (Show your calculation below, and include a copy in your report)

1.2 Consider the following masses: Cart M, and the mass m: M = 520 gm m = 200 gm Use these values to obtain a value for the theoretical value of the acceleration using the expression obtained above: aTh. = _________________ m/s2 Page 9 of 60

m

Computational Lab

2.0

Uniform Acceleration, Newton’s 2nd Law

A. Vaucher

Procedure 2.1 A virtual experiment using the above arrangement is run for twenty trials. The following data was obtained for the times it took to accelerate the cart for a distance of approximately 94 cm. Using the kinematical formula:

1 x  at 2

2

Obtain the acceleration of the cart for each trial (use excel). Enter your results in the table below, or use excel cut/paste in your report. Trial

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Measured Times (seconds)

Experimental Acceleration of Cart (m/s2)

0.840986 0.820821 0.830916 0.810997 0.846371 0.821074 0.8209 0.861719 0.814069 0.823327 0.857834 0.811378 0.885859 0.829633 0.909435 0.865335 0.820333 0.831816 0.825769 0.776121 Average 

a  (STD) 2.2 Use excel functions to calculate the Mean (average), and Standard Deviation for the acceleration as shown in the table above. 2.3 Determine the error in your mean value of the acceleration from the formula:

Error in a 

Page 10 of 60

a

N

Computational Lab

Uniform Acceleration, Newton’s 2nd Law

A. Vaucher

Where N is the total number of trials in the table above. Report you experimental value for a using the format:

a  3.0

a N

: _________________ m/s2

Analysis: 3.1 Compare the average value of your measured acceleration with the theoretical value calculated in section 1.2:

% diff. in a 

aTh  aexp aTh

 100%

Record The % difference: _________________________________ 3.2 Discuss source of error in your experiment. What could be done to make the result better? 4.0

Determining the coefficient of kinetic friction: As the cart moves on the track, it is subject to frictional force, there is also friction in the pulley as well. Ignoring the friction in the pulley, the system is non-conservative due to the presence of the frictional force, thus energy is not conserved. In a non-conservative system, the work done by the frictional force is equal to the change in the total energy of the system, thus we have for the cart-mass system:

Wnc  E f  Ei

(1)

Wnc  (KE f U f )  (KEi Ui )

(2)

Or:

For the system under consideration this becomes:

1 1 1  1    K Mg x   M  2f  m 2f mgx    M i2  m i2  0  2 2 2  2 

(3)

Re-writing this formula:

  2f  i2    2f   i2    K Mg x  M  m mgx  2   2     

(4)

Rearranging 2 2 1   f  i K   g  2 x

 m    M

 2f  2i   2x

1 m   g M

(5)

Recognizing the term in parenthesis as the acceleration of the cart from the kinematical 2 2 formula a  ( f  i ) 2 x , we obtain for the coefficient of kinetic friction:

Page 11 of 60

Computational Lab

Uniform Acceleration, Newton’s 2nd Law

K 

a m 1  g M

A. Vaucher

 m   M

(6)

Use the average value of the acceleration obtained above in this formula to calculate the coefficient of kinetic friction for this system. Use the methods outlined in the first lab on Statistics & Propagation of Errors to calculate the error in  K , show all your work. Finally, apply Newton’s law, as in section 1 above, but include the effect of friction, and show that the acceleration is given by:

 m  K M a  mM

 g 

Show that this solution in equation (7) agrees with equation (6), obtained by applying conservation of energy. Show your work in the space below:

Page 12 of 60

(7)

Computational Lab

Free Fall & Drag Force

A. Vaucher

Free Fall & Drag Force 1.0

Introduction We have learned that Galileo was able to measure the acceleration of gravity near the surface of the earth, and concluded that absent air resistance, this acceleration is constant for objects independent of mass. Today’s accepted mean value at sea level is

g accepted = 9.8 m/s2

(1)

In this experiment, we will determine the acceleration of gravity for a falling ball.

2.0

Measuring the Acceleration of Gravity Near Earth’s Surface: An apparatus consisting of two wires separated by a space where a steel ball is allowed to fall. The wires are connected to a high voltage spark generator, which produces pulses with a period of 1/30 second. A ribbon of paper is placed in between the falling ball and one of the wires such that it is in contact with the wire as shown in figure 1. When the spark generator is turned and the ball is released, the ball closes the circuit causing the sparks to pass through the paper ribbon leaving a dot on the wire. The dots mark the positions of the ball separated by the time interval of the sparks which is

t  1 30 of a second. Figure1. Depiction of spark timer apparatus. A high voltage pulse with a period of 1/30 of a second is generated between two wires. A paper ribbon placed in contact with one of the ribbons becomes marked with burn mark when the spark passes through it. The spark is produced by a falling steel ball which closes the circuit between the wires allowing the current to pass. As the ball accelerates downwards, a trail of dots separated by increasing distances is formed on the paper ribbon. The time interval between each dot is the same t  1 30 second.

y0 y1 y2 g

y3

V

t 0

V

The running time and distance of fall is given in the set of data below in the format: {tk, yk}, where t is in seconds, and y is in meters: Page 13 of 60

Computational Lab

Free Fall & Drag Force

A. Vaucher

{{0, 0.000434949}, {0.0333333, 0.00495339}, {0.0666667, 0.0241379}, {0.1, 0.0449709}, {0.133333, 0.0913918}, {0.166667, 0.132482}, {0.2, 0.196526}, {0.233333, 0.268654}, {0.266667, 0.342669}, {0.3, 0.442608}, {0.333333, 0.543986}, {0.366667, 0.659036}, {0.4, 0.781937}, {0.433333, 0.916108}, {0.466667, 1.07391}, {0.5, 1.22408}, {0.533333, 1.39637}, {0.566667, 1.57087}, {0.6, 1.75725}, {0.633333, 1.97154}, {0.666667, 2.17914}} Enter this data in an excel file, use the format outlined in the table below. Running time tk

Vertical position yk

Velocity

k 

yk 1  yk t

In excel, plot the velocity on the vertical axis and time on the horizontal axis. Ask excel to do a best fit line, and display the equation of the line on the plot. The equation is essentially the kinematical equation   a t  0 , where “a” is the slope of the line and corresponds to your measured value of the acceleration of gravity. 0 is the y intercept, and should be zero, or some very small number corresponding to zero initial velocity. Calculate the percent difference between the value you obtained for g and the accepted value. Record your result here:

% 

g measured  g accepted g accepted

100% :__________________________________________

Include a copy of your plot, and show your calculation in your report.

3.0

Drag Force on a Falling Ball: An object falling through the air experiences a resistance force due to friction with the air, its called the “drag” force. The drag force is given by:

FD  CD

1 2 A 2

Where:

Page 14 of 60

(2)

Computational Lab

Free Fall & Drag Force

A. Vaucher

FD  Drag Force CD  Coefficientof Drag

  Densityof surrounding fluid Air ( )   Velocityof object A  Frontalcross sectionalareaof object Let’s consider a metallic sphere of radius R, and mass m falling through the air. The net acceleration of this sphere is:

a  g  aD

(3)

Where aD is the acceleration due to the drag force

aD 

FD m

(4)

Substituting equation (4) into (3), the net acceleration becomes:

ag

FD m

(5)

Substituting equation (2) into (5), we obtain:

a  g C D

1  2A 2m

(6)

Finally, we observe that the velocity of the ball is related to the acceleration by

y

 2  02 2a

, where the initial velocity is zero, thus we have:  2  2 a y . Making this

substitution in equation (6), we obtain for the acceleration:

a

g  CD  y A  1   m  

(7)

With constant, a is expressed as a function of y:

a( y) 

g  CD  y A  1   m  

(9)

Equation (9) gives the acceleration of a falling ball of mass m, cross sectional area A as a function of falling distance y through the air. Using the following parameter for the falling ball:

Page 15 of 60

Computational Lab

Free Fall & Drag Force

A. Vaucher

m  0.120Kg R  0.00564 meters A   R2

(10)

o   1.204Kg/m 3(Densityof airat20 C)

g  9.8 m/s 2 Substitute the data from equation (10) into equation (9), obtain a reduced a simplified formula for the acceleration as a function of falling distance y. Plot this formula for distances on the intervals 0 < y < 1.5 meters. How does the acceleration vary as the ball falls through the air? Include a copy of your plot, and explain in your report. A typical passenger jet flies at about 40,000 ft, or 12,000 meters. Using the same procedure as above, plot equation (9) for intervals of 1000 meters, thus y = {0, 1000, 2000, …, 12,000}. Ask excel to do a best polynomial fit. From the plot, what is the acceleration of the ball after it has fallen 12,000 meters? Include a copy of the plot and your answer in the report. 4.0

Additional Questions: 1. Ideally, the terminal velocity of a falling object is defined to be the velocity when the drag force on an object is equal to its weight, thus its acceleration is zero. For the ball considered in section 3 above, if this condition is met, what would be the terminal velocity? 2. From equation (2), the drag force varies with velocity, and the velocity in turn is a function of the acceleration and distance of fall:  2  2 a y , using this in the equation (2), the drag force becomes:

FD C D a yA

(11)

Substituting for the acceleration from equation (7), equation (11) becomes:

FD ( y ) 

CD  g y A  CD  y A  1   m  

(12)

This gives the drag force as a function of the fall distance y. Plot this function on the interval {0 < y < 12,000 meters}. Plot also the weight of the ball vs. fall distance (constant horizontal line). Combine both plots. Is terminal velocity achieved for this falling ball? Explain in your report.

Page 16 of 60

Computational Lab

Circular Motion

A. Vaucher

Circular Motion 1.0

Introduction Consider an object represented by the red dot, constrained to move in a circle as shown in Figure1 below. Refer to this link for the animation: https://www.compadre.org/Physlets/mechanics/ex10_2.cfm Figure1. An object, red dot, in circular motion about a fixed point. Refer to above link for the animation (Physlet Ch. 10, Exploration 10.2).

The angular position of the object is given by the kinematical formula:

1 2

   0  0 t   t 2

(1)

Where  0 is the initial position; 0 is the initial angular velocity; and  is the angular acceleration.

2.0

Procedure In the animation, set the initial values:

0  0

0 15 [ rad / s]

 5[ rad / s]2

Run the animation to see how it works. Restart the run, stopping after approximately one revolution, record the time shown and the corresponding angular position (note that the dot does not stop in exactly the same position each time, but advances slightly. Estimate the angular position the best you can. Hint: make the estimate in degrees, then convert to radians). Record your data in an excel file using the format in the table below.

Page 17 of 60

Computational Lab

Circular Motion

Approximate Revolution

Measured Times (seconds)

A. Vaucher

Approximate Angular Position of Dot (rad)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

3.0

Analysis: 3.1 In excel, use a scatter plot to plot the angular position on the vertical axis vs time on the horizontal axis from the data in the table above. Do not connect the dots. This is your experimental position vs time. In excel, generate another plot for the same times recorded in the table, but calculate the position using the kinematical formula, equation (1):

1 2

   0  0 t   t 2 For this plot, use a smooth line with a different color. This is your theoretical position vs time. Use excel feature to combine both plots, and show error bars of 5% on the experimental data. How do your experimental and theoretical data compare?

3.2 The angular velocity is given by the kinematical formula: Page 18 of 60

Computational Lab

Circular Motion

A. Vaucher

  0   t

(2)

Using the initial values for 0 and  , calculate the angular velocity after 5 seconds from equation (2). Record your value for  here:

 (t = 5 s) = ____________________________ (rad/s) The angular velocity, is also the angular frequency of the rotation of the dot. It is related to the linear frequency v (Rev/s) by:

  2 v

(3)

Calculate the linear frequency after 5 seconds for this dot, include in your report. The tangential velocity and acceleration are related to the angular velocity and acceleration by the following formulas:

 R a  R

(4) (5)

Where R is the radius of the circle in the animation. Using a scale of one square per cm, estimate a value for R, then convert to meters. Calculate the linear acceleration, and the velocity after 5 seconds. Discuss the results in your report.

4.0

Additional Questions: 1. Calculate the Earth’s linear velocity, angular velocity, and radial acceleration in its orbit around the sun. The Earth’s orbital radius, and period are:

rE  1.5 10 11 [ m] TE  3.154 10 7 [ s] 2. Repeat the same calculation for the planet Mercury. Mercury’s orbital radius and orbital period are:

rMe  5.14 10 10 [ m] TMe  7.6 10 6 [ s] Calculate the ratios of velocities, angular velocities, and radial acceleration for Mercury and Earth. What do you conclude, and why?

Page 19 of 60

Computational Lab

Conservation of Momentum

A. Vaucher

Conservation of Momentum 1.0

Introduction In collisions, the condition for conservation of momentum is that there are no external forces present. This condition is almost always satisfied, since collision time is very short, and only internal forces act. There are two types of collisions: Elastic collisions, where kinetic energy is also conserved; and non-Elastic collisions, where kinetic energy is not conserved. There are also events where energy is released from within a system. These events are known as “recoils”. A recoil event can be treated as a collision, where momentum is conserved, but kinetic energy is not. In this lab, we will investigate four different types of events, two elastic collisions; one non-elastic collision; and a recoil.

2.0

Elastic Collision With One Mass at Rest Consider the collision shown in figure1. Mass m2 is at rest. Mass m1 has momentum p1i and collides with m2. Both masses leave the collision region. If the following data is known:

1  23

2 27

p1 f 12 [kgm/s]

m1 2.7  [kg]

Use conservation of momentum and kinetic energy to obtain p2f, p1i, and m2. p1 f

m1

1 m1

m2

2 p1i

p2 i  0

m2 p2 f Figure1. An elastic collision where one mass, m 2, is initially at rest. After the collision, both masses leave the collision as shown.

Page 20 of 60

Computational Lab 3.0

Conservation of Momentum

A. Vaucher

Head-On Elastic Collision Consider the collision shown in figure 2. Two masses traveling in opposite directions collide. Given the following data:

1  7.3

2  8.9

p1 i  14 [kgm/s]

p2 i11.2  [kgm/s]

Use conservation of momentum and kinetic energy to determine the final momenta p1f , p2f, and the ratio m1/m2 .

p1 f

m1

p1i

1 m2

m1

2

p2i

m2 p2 f Figure 2. Head-On elastic collision between two masses.

4.0

Non – Elastic Collision Figure 3 shows a collision between two masses, where m2 is initially at rest. After the collision both masses combine and continue moving in the same direction as the incident mass m1. Given the following data:

p1  36 [kgm/s]

m1  8.3 [kg]

m 24.2  [kg]

Determine  2 for the combined masses after the collision. Calculate the ratio KE2 KE1 , is kinetic energy conserved in this collision?

Page 21 of 60

Computational Lab

Conservation of Momentum

A. Vaucher

m1

m2

m1  m2

p1

p0

p2

Figure 3. Mass m2 is initially at rest. Mass m1 collides with m2, the two combine, and continue moving the original direction of m1.

5.0

Recoil Event Consider the event in figure 4. An object of mass M breaks apart into two masses m1 and m2. The two masses move in opposite directions. If the initial mass of the object, and the mass and momentum of one of the components are known:

M  72 [kg]

m1 47.2 [kg]

p 1198.5  [kgm/s]

What is the momentum, and velocity of m2? And what is the total kinetic energy after the recoil? Is kinetic energy conserved in this recoil? Is total energy conserved?

m1

M

m2

p1

p0

p2

Figure 4. Recoil event. Mass M breaks into two masses m 1 and m2. Both masses leave traveling in opposite directions.

Page 22 of 60

Computational Lab

Archimedes Principle

A. Vaucher

Archimedes Principle 1.0

Introduction Archimedes determined that objects immersed in a fluid experience an upward force, called the “Buoyant Force”. The buoyant force acting on large ships is what keeps them afloat in oceans or large bodies of water. If an object is immersed in water, the buoyant force is given by the difference of the weight of the object in air and in water:

FB  Wair  Wwater

(1)

Archimedes Principle also gives the buoyant force as the weight of the displaced fluid by the submerged object: FB   V g (2) Where ρ is the density of the fluid, in this case water; V is the submerged volume of the object, and g is the acceleration of gravity. 2.0

Calculating the Buoyant Force: Given three objects: Rectangular Solid; Sphere; and Cylinder, determine the buoyant force on each by calculating the volume using the dimensions given below:

Object; Material Rectangular Solid (Iron) Sphere (Gold) Cylinder (Aluminum)

Object Data Density

[Kg/m3]   7,860   19,320   2,705

Length: 5 cm Radius: 5 cm Radius: 3 cm

Width:3cm

Height:2 cm

Height:12 cm

Calculate the volume and weight of each object; enter your values in the table below: Object; Material

Density

Volume, V

3

[Kg/m ]

3

[m ]

Mass

V

[kg] Rectangular Solid (Iron) Sphere (Gold) Cylinder (Aluminum)

  7,860   19,320   2,705

Page 23 of 60

Wair  m g [N]

Computational Lab

Archimedes Principle

A. Vaucher

Calculate the Buoyant force on each object when submerged in water, using

 water  1000 [kg/m 3] Object; Material

Density Water

Volume, V 3

[m ]

[Kg/m3]

Buoyant Force

 water V g [N]

Rectangular Solid (Iron) Sphere (Gold) Cylinder (Aluminum)

1000 1000 1000

From equation (1), we can calculate the weight in water for each object:

Wwater  Wair  FB

(3)

Summarize your results in the table below: Weight and Buoyant Force Data

Wair

Wwater

FB

(N)

(N)

(N)

Rectangular Solid (Iron) Sphere (Gold) Cylinder (Aluminum)

3.0

Specific Gravity: Specific Gravity of a material is defined to be the ratio of the density of the material and the density of water:

S

 material  water

(4)

Multiplying the numerator and denominator by the volume of the material and g:

S

 material Vmaterial g m g Wair    water Vmaterial g FB FB

(5)

Using the definition of the buoyant force, equation (2) becomes:

S

Wair Wair  Wwater

(6)

Use the data from the table in section 2.0 above to calculate the specific gravity for the three objects used. Enter your results in the table below. Note that S has no units. Specific Gravity Page 24 of 60

Computational Lab

Archimedes Principle S

A. Vaucher

% 

S  S acc S acc

 100%

Rectangular Solid (Iron) Sphere (Gold) Cylinder (Aluminum)

Use the table below to calculate the percent difference in the last column on the right in the above table Sacc, Specific Gravity of Common Materials Water 1.0 Aluminum 2.7 Copper 8.96 Lead 11.35 Iron (cast) 7.13 Gold 19.32 Mercury 13.56 Steel (carbon) 7.8

4.0

Additional Questions: 1. A Blimp is filled with Helium, and is spherical in shape. The radius of the Blimp is 15 meters. If a cargo totaling 12,470 Kg, is attached to the Blimp, what is the buoyant force acting on it? Assume the density of air is 1.225 Kg/m3, and the density of Helium is 0.179 Kg/m3.

2. What is the net force acting on the Blimp? What is its acceleration? Which way will it move?

Page 25 of 60

Computational Lab

Elasticity & Young’s Modulus

A. Vaucher

Elasticity & Young’s Modulus 1.0

Introduction This lab is a virtual adaptation of Experiment 9 on Elasticity in the Physics Lab Manual by J. Quan. See Figure1. Figure1. Experimental setup, from Fig.3 of the lab manual experiment on Elasticity. As the wire is stretched by the weight, the mirror tilts. Correspondingly, the reflected laser beam moves up on the screen a distance b above the un-tilted position.

A wire is stretched by subjecting it to the force of weights. A mirror mounted on a tripod, with one leg resting on a support attached to the wire, will tilt upwards as the wire is stretched. A laser beam reflected from the mirror will move upwards on the screen a distance b above the un-tilted position, as the wire is stretched. From the geometry shown in figure1,

tan  

L b ; tan2   a x

For small angle  ; tan    , and similarly tan2  2 give:

L 

(1)

 , eliminating  , equations (1)

a b 2x

(2)

The stretching of the wire is also related to its original length L, by Young’s Modulus:

F L Y A L

(3)

Where F is the force of the weight; A is the wire cross sectional area (assume circular); L is the original length of the wire, and Y is Young’s Modulus. Since F = mg , substituting this into equation (3), and solving for ΔL we obtain:

 Lg  L   m Y A Page 26 of 60

(4)

Computational Lab

Elasticity & Young’s Modulus

A. Vaucher

Where, ΔL can be calculated from equation (2), by measuring b, and x from the experiment for each value of the mass m. Next, plotting ΔL vs m, will give a line whose slope is the coefficient of m in equation (4). And using the cross sectional area and initial length of the wire, Young’s modulus can be calculated.

2.0

Procedure In the experiment described above, see figure1, the values for the wire were as follows:

r  0.3[mm] L 53[ cm ] And the values for the mirror and support were: a  6 [c m]

x 3.2 [m]

(5)

(6)

For the mass values shown in the table below, the measured values for the deflection b; were also recorded: Vertical Deflection of Laser Beam at different Masses Trial Mass Vertical Deflection b [Kg] [m] 1 1.5 0.0149 2 2.0 0.0192 3 2.5 0.0245 4 3.0 0.0299 5 3.5 0.0341 6 4.0 0.0395 7 4.5 0.0448 8 5.0 0.0491 9 5.5 0.0544 10 6.0 0.0587 11 6.5 0.0639 12 7.0 0.0689

3.0

Analysis: Using the values of b from the table of data, and the input configuration values from equation (6), calculate the stretching of the wire using equation (2). In excel, create a table of values for ΔL‘s obtained and the corresponding mass values from the data table. Plot ΔL vs. m in a scatter plot. The plot you obtain is essentially a plot of equation (4). In excel show the equation of the best fit line on this plot. The slope of the line will then correspond to the coefficient of m in equation (4):

 Lg  Slope    Y A Page 27 of 60

(7)

Computational Lab

Elasticity & Young’s Modulus

A. Vaucher

From equation (7), and data in equations (5), obtain a value for Young’s Modulus Y. This will be your experimental value for Y, record it here: Yexp = _________________________________________________ [Pa]

If the accepted value for Young’s Modulus for this wire is Y = 200 GPa, Calculate the percent difference:

% 

Yexp  Y Y

100% :_____________________________________________

Look up a table of values of Young’s Modulus for various materials; can you identify the type of material this wire is made of? Include your finding, as well as all plots and calculations in your report.

Page 28 of 60

Computational Lab

Simple Pendulum

A. Vaucher

Simple Pendulum 1.0

Introduction The pendulum is one of the simplest devices in physics, yet, we can learn some very profound principles from it. It consists of a string fixed from a support, and a weight, commonly a metal sphere (bob) attached to the other end. See figure below. When the bob is displaced slightly from the vertical, this angular displacement is called the amplitude. Applying Newton’s laws to the pendulum leads to the differential equation:

d 2   2 sin  0 2 dt

(1)

For small angle oscillation sin    and equation (1) reduces to

d 2   2  0 dt2

(2)

This is the standard form for a simple harmonic oscillator, where   g l is the angular frequency; in general   2 T , where T is the period of oscillation. For the pendulum, this period is given by:

T  2

l g

(3)

Where l is the length of the pendulum from pivot point to center of the bob, and g is the acceleration of gravity, see figure below. The solution to equation (2) is:

 (t)  0cos(  t   ) Where  0 is the initial maximum amplitude;  is the frequency; and  is an arbitrary phase.

d

θ

2R Page 29 of 60

l dR

(3.1)

Computational Lab

Simple Pendulum

A. Vaucher

Equation (3) was obtained with the assumption of small amplitude oscillation, and we see that the period is independent of the amplitude. In this lab we will explore how the period of the pendulum changes with increase in the amplitude. . 2.0

Measuring the Period for Differing Amplitudes: As we see from the formula above, the period of the pendulum depends only on the length of the pendulum and the acceleration of gravity. This, however, is only an approximation. To learn how good this approximation is, we measure the period of the pendulum for different amplitudes as outlined in the table below. A pendulum of length l =1.0 m is set to oscillate for 20 cycles for each angle (one cycle is one round trip of the pendulum bob). A stop watch is used to record the total time for the 20 oscillations. The data is recorded in the table below. Calculate the period for one oscillation at each amplitude for the two runs as indicated in the table. Calculate the average period for each amplitude and record in the last column. Data for Simple Pendulum

1 2 3 4 5 6 7 8 9 10 11 12

3.0

Amplitude Angle,  0 (Deg.) 5 10 15 20 25 30 35 40 45 50 55 60

Time for 20 Oscillation, t1 (sec)

Time for 20 Oscillation, t2 (sec)

40.1569 40.1652 40.4529 40.5008 40.4242 40.6507 41.2307 41.1619 41.337 41.9282 42.3212 42.4351

40.1456 40.224 40.3213 40.403 40.6295 40.8639 41.099 41.3442 41.6381 42.0414 42.1064 42.5347

Period T1 for Trial 1 (t1 /20)

Period T2 for Trial 2 (t2 /20)

Average Period T=(T1+T2)/2

Analysis for Period & Amplitude: Using excel, do a scatter plot (points) of the average period T vs. the amplitude  0 . Do a best polynomial fit (to second order should be sufficient), and display the equation of the polynomial on the plot. If the period depends on the amplitude, a measurable slope should appear. Do you detect a slope? What is your conclusion? Discuss your analysis and include a copy of your plot in your report.

3.1

Comparing Measured Period with Theoretical Expectation: The solution to equation (1) using slightly more advanced tools gives the following approximate expression for the period of the pendulum: Page 30 of 60

Computational Lab

Simple Pendulum

T  2

A. Vaucher

1 2 0  1  sin  g 4 2

l 

(4)

We see here that, the period shows a dependence on the amplitude  0 . Using excel, plot this formula for  0 values in the table above (use a smooth curve). Combine this plot with the experimental data plot form section 2 without the best fit curve, so it is easier to compare the theoretical curve with the experimental data. Add 5% error bars to your experimental data. How does the theoretical calculation of the period compare with your measurement? Include copies of all the plots in your report.

4.0

Measuring the Acceleration of Gravity: The pendulum can also be used to measure the acceleration of gravity. Using your conclusion from the analysis in part 3 above, select the most appropriate amplitude from the table above, and calculate the value of g from the formula for the period:

g exp 

4 2 l T2

Show your calculation below. This is your experimental value for g.

Record your value of g here: gexp : ___________________________  g :________________________

Calculate the error Δg using error propagation formulas discussed in class, and appearing in the appendix. Show all your work. Error in Calculation, Δg :

Page 31 of 60

Computational Lab

Simple Pendulum

A. Vaucher

Calculate the percent error between the experimental and accepted value for g. Use gacc= 9.8 m/s2, show your calculations below. Error between measured and accepted values:

% Error 

g exp  g acc g acc

Page 32 of 60

100%

Computational Lab

Simple Pendulum

A. Vaucher

% Error : ___________________________ 5.0

Additional Questions: 1. Suppose you were on the planet Mars, and using a pendulum of length 1 meter, you measured a period of 3.26 seconds. What is the gravitational acceleration on Mars?

2. A pendulum can be used to construct a clock. If the period of oscillation of the clock is 1 second, what should the length of the pendulum be?

Page 33 of 60

Computational Lab

Damped Harmonic Oscillator

A. Vaucher

Damped Harmonic Oscillator 1.0

Introduction A mass “m”, on a spring system without energy dissipation exhibits simple harmonic motion. In real systems, however, dissipation is present. This dissipation is usually proportional to the velocity of the oscillator, and exerts a dissipative force given by:

F  b Where  is the velocity, and “b” is called the damping constant. When damping is present, the oscillator satisfies the differential equation:

d 2 y b dy k   y0 dt 2 m dt m

(1)

(2)

The coefficient of y is recognized as the angular frequency of the oscillator when there is no damping:

02 

k m

(3)

The general solution for the amplitude is:

y (t)  Ae Where A is the amplitude, and oscillator,



 2bm t

cos( t  )

(4)

is an arbitrary phase. The frequency of the damped

 , in equation (4) is given by: k b2   m 4m 2

In this experiment, we will determine how the frequency of oscillation varies with the dissipation constant for a damped oscillator using the simulation at: https://www.compadre.org/Physlets/waves/ex16_3.cfm , See figure 1.

Figure 1. Damped harmonic oscillator, adapted from the simulation cited above. The damping is produced by varying the damping constant as an input to the simulation. The maximum amplitude “A” is a user determined input. Its value is read from the plot generated by the simulation.

Page 34 of 60

(5)

Computational Lab 2.0

Damped Harmonic Oscillator

A. Vaucher

Measuring the mass on the spring: When there is no damping, the frequency of oscillation is given by equation (3) above. Solving this equation for the mass give:

kT 2 m 4 2

(6)

Setting b = 0, and k = 2, 4 respectively, run the simulation, measure the period for each case, and determine the mass on the spring using equation (6). Record your data in the table below. Average the two values of the mass, and use the average in subsequent sections. To determine the period, let the simulation to run long enough to accumulate several peaks on the plot. The time between two consecutive peaks is the period. To obtain an average value for the period, count the number of peaks n, then divide the total time by the number of intervals (or number of peaks minus one):

 t  Tmeasured   n   n 1 Trial 1 2

b [Ns/m] 0 0

K [N/m]

T [s]

(7) m [kg] (equation 6)

2 4 Average mass

Include copies of the plots in your report. 3.0

Measuring the Frequency of Oscillation: Set the input value for spring constant at k = 4, run the simulation for the values of the damping constant b, shown in the table below. For each run, measure the period of oscillation as in section 2 above. Use this value of determine the frequency  . This is your experimental value of  . Next, use equation (5) to calculate the theoretical value of  . Enter both values in the table. Trial

K [N/m]

b [Ns/m]

Tmeasured [s] (equation 7)

exp 

2

th

Tmeasured

(equation 5) [1/s]

[1/s] 1 2 3 4 5 6 7

4 4 4 4 4 4 4

0.25 0.5 1.0 1.25 1.5 1.75 2.0

Include a copy of your plot showing the velocity for trial 1 only. Using excel, plot

exp

vs b, and

th

vs b on separate plots, then combine the plots.

Page 35 of 60

Computational Lab

Damped Harmonic Oscillator

A. Vaucher

Add 5% error bars to your experimental plot. What is the general trend of frequency as the damping constant increases? How do the experimental results agree with the theoretical calculation? Explain in your report. Include the combined plot only in your report. 4.0

Calculating Energy Loss After Maximum Velocity is Attained: In a damped system like the one under consideration, the damping begins immediately as velocity increases. The maximum velocity is obtained by differentiating equation (4) with respect to time, and setting

  0 , and t  T 4 .

See the figure 2.

Max amplitude at t = 0 Figure 2. Damped oscillator amplitude and velocity. Maximum velocity is reached after the first ¼ period. After that, both the amplitude and velocity begin to diminish under the influence of the damping force.

Max velocity at ¼ period

The maximum velocity is given by:

max   Ae Since

 2bm t t

T 4

(8)

T  2  , then:

t

T 2   t 4 4 2

(9)

Using in equation (8), the maximum velocity becomes:

max    Ae

b

4m

(10)

The maximum kinetic energy of the system is obtained from the maximum velocity:

1 2 KEmax  m max 2

(11)

For Trial 1: Using equations (10) and (11), calculate the maximum kinetic energy. The maximum potential energy in the system is in the spring at t = 0. From the plot, read the maximum value for the amplitude A , and calculate the maximum potential energy:

Page 36 of 60

Computational Lab

Damped Harmonic Oscillator

A. Vaucher

1 2 PEmax  k Amax 2

(12)

Enter the results of your calculations in the table below. Determine the energy loss after ¼ cycle, record that value in the table below. Energy Loss after ¼ cycle Trial

PEmax

KEmax

% 

KEmax  PE max  100% PEmax

1 Is energy conserved for the early part of the oscillation? Explain in your report

5.0

Calculating the Average Power Loss Over One Period: The average dissipative power in the oscillator is given by:

P  b 2

(13)

Where the average square velocity is given by:

A2 e2T1  ( 1 e 2T1 )(2  2  T12 )2   2T13 2

(14)

Where the term  is given by:

  And,

b 2m

(15)

T1 is the period of oscillation for trial 1, measured in the table from section 3 above.

Calculate the average dissipated power. Record your value in the space below:

P  __________________________________________ watts We can estimate the number of cycles it takes before the oscillator nearly stops:

nstop 

PEmax P

How many cycles before the oscillator nearly stops? Compare with plot from the simulation. Discuss the effect of this dissipation in your report.

Page 37 of 60

Computational Lab

Speed of Sound

A. Vaucher

Speed of Sound 1.0

Introduction When an object travels at speeds that exceed the speed of sound in air, the resulting sound waves form a cone as shown in figure 1.

Figure 1. Cone formed by sound wave fronts produced by an object traveling faster than the speed of sound in air. The speed of the object (source) is expressed as



a ratio to the speed of sound

 

c:

c , appearing in the lower

right corner. (From Physlet simulation: https://www.compadre.org/Physlets /waves/ex18_4.cfm

From figure 1, we see that the angle of the cone is related to velocity of the object and speed of sound by:

sin  

c

In this lab we will measure the angle formed for various values of determine how



(1)



   c , and

varies with the angle. Then use the Doppler Effect to calculate the

speed of sound in air.

2.0

Relation Between Cone Angle and Speed: Using the simulation at: https://www.compadre.org/Physlets/waves/ex18_4.cfm Click “Restart”, and set the ratio of the speed of the object to the speed of sound to the values shown in the table below. For each value of  in the table below, take a snap shot of the image and paste it in a WORD file. Print all the images, and on the paper,

Page 38 of 60

Computational Lab

Speed of Sound

A. Vaucher

measure each of the conic angles formed for that value of the speed. Record your data in the table below. Trial

Set Value of  (approximate)

1 2 3 4 5 6 7 8

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

Cone Angle,  [Deg.] (Measured)

Using excel, plot the measured values of



1 sin  (Measured Value of  )

vs the measured values of



from the table

above. Apply a best fit curve. How does cone angle vary with the speed of the source? Include copies of the figures and plot in your report, discuss the results. 3.0

Measuring the Speed of sound: Typically, the source of the sound in this experiment is an aircraft traveling with the speed  . As it passes overhead at some altitude, the frequency of the sound heard by an observer vary as it approaches, and recedes according to the Doppler formula:

f f Writing the left side of equation (2) as:



2 1 

(2)

2

 f  f f

, equation (2) becomes:

 f  2  2   f  0 If the observed values of

f

(3)

are given in the table below, solve the quadratic equation

(3) for each value of  . Enter your solution in the table as shown. Trial 1 2 3 4 5 6 7 8

f



c  

(Doppler)

-6.04 -4.04 -2.98 -2.69 -2.30 -2.08 -1.89 -1.71 Average Speed of Sound:

Page 39 of 60

c

Computational Lab

Speed of Sound

A. Vaucher

If the object is an airplane traveling with a speed   448 m / s , calculate your value for the speed of sound:

c    , enter your values in the last column in the table above.

Calculate the average for the speed of sound. Enter the average in the table as shown. Finally, calculate the percent difference between the average value obtained and the speed of sound at a typical altitude of 10,000 meters

c10K  299 m / s , using the

formula:

% 

c  c10K  100% c10K

 %  ___________________________________________________

Page 40 of 60

Computational Lab

Specific Heat

A. Vaucher

Specific Heat 1.0

Introduction When two substance at different temperatures come into contact, heat is transferred from the hot object to the cold object. This heat transfer will continue until the temperature of both objects reaches the same value. At this point, no more heat will transfer between the objects, see figure 1. This last condition is known as equilibrium. Thus when two objects reach thermal equilibrium, they are at the same temperature. Figure 1. The natural flow of heat is from a hot body to a colder body. At thermal equilibrium, both objects reach the same temperature, and heat stops flowing between them.

The amount of heat energy required to raise the temperature of a substance by one degree Celsius per unit mass, is a property of the material, and is known as the “Specific Heat Capacity” or “Specific Heat” for short. It is given by:

c Where

1 Q m T

(1)

Q is the heat gained or lost, and T is the temperature difference the object is

subjected to. This formula can be written in terms of the change in temperature as:

Q  mc (T f T i ) Where

Ti , T f

(2)

are the initial and final temperatures of the object respectively.

In this lab we will measure the specific heat of 3 blocks, all made of the same material, then obtain an average for the specific heat value. Finally, we will compare the value obtained with a known value. 2.0

Procedure 2.1 Using the animation at this link: https://www.compadre.org/Physlets/thermodynamics/ex19_3.cfm Set the value of the initial temperatures of each mass to the values in the table below. When the mass is lowered into the water, heat transfers until thermal equilibrium is reached. Record the initial temperatures of both the mass and the water. After equilibrium is reached, record the final temperature of the combination. Page 41 of 60

Computational Lab

Specific Heat

A. Vaucher

(Note, the final temperature appears in a yellow box to the lower right of the animation window. Note also that the two thermometers should show the same final equilibrium temperature, but they do not, an apparent “bug” in the animation) Trial

Mass of Block [Kg]

Mass of Water

Initial Block Temperature Block

( Ti ) [Kelvin]

Initial Water Temperature Water

( Ti ) [Kelvin]

Final Equilibrium Temperature (Tf ) [Kelvin]

1 2 3

3.0

1 2 3

10 10 10

Specific Heat of the Block ( cBlock , equation (4))

500 600 700

Analysis: When equilibrium is reached, the heat lost by the block is equal to the heat gained by the water, from equation (2), we can write the equilibrium condition as: Block Water QLost  QLost

Or:

mBlock cBlock (Ti Block  T f )  mWater cWater(T f T i Water)

(3)

Solving equation (3) for the specific heat of the block, we obtain:

cBlock  The specific heat of water is

mWater cWater (T f Ti Water ) mBlock (Ti Block  T f )

4.186  103 [JKg -1K -1]

(4)

, use this value in equation (4),

and data from the table above to calculate the specific heats of the blocks. Record your values in the table as well. Calculate the average of the specific heats obtained, and record your value here:

cBlock  :______________________________________ [JKg -1 K -1]

Finally compare your obtained average value with the expected value:

cexp ected  385 [JKg -1K -1]

, using the formula:

Page 42 of 60

Computational Lab

Specific Heat

% Diff 

cBlock  cexpected cexp ected

A. Vaucher

100%

Record your value here:

%Diff  :__________________________________________

Do a search and determine the material of the block? Include your finding in your report.

4.0

Additional Questions: 1. In an industrial plant, 2000 Kg of water at 100 oC has to be cooled and frozen to ice. How much heat must be removed to freeze this water? Express your answer in Joules. (1 cal. = 4.184 Joules; Heat of fusion for water = 80 cal/gm).

.

2. If a 15000 watt compressor cooling system is used remove all this heat, how long will It take to freeze the water?

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Computational Lab

Charles Law & Entropy

A. Vaucher

Charles Law & Entropy 1.0

Introduction We have learned that for an ideal gas, the state of the gas is given by its pressure, volume and temperature. The relation between these parameters is given in the ideal gas law: (1) PV  RT Where, P is the pressure, V is the volume, and T is the temperature of the gas respectively. μ is the amount of gas in moles, and R is the ideal gas constant. This law applies also for real gases where the interaction between the gas molecules is considered weak. As a gas transits from one state to another, equation (1) allows us to relate the two states: PV PV 1 1  2 2 (2) T1 T2 If the gas undergoes a constant pressure transition from state 1 to state 2, equation (2) reduces to:

V1 V2  T1 T2

(3)

Equation (3) is known as Charles Law.

2.0

Verification of Charles Law for Air:

Fig.1. Apparatus for Charles Law. The piston cylinder must be horizontal to keep the pressure constant.

The apparatus consists of an aluminum can, and the cylinder-piston apparatus as shown in figure 1. The inside volume of the aluminum can, ambient pressure and temperature are given below:

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Computational Lab

Charles Law & Entropy

A. Vaucher

Vcan  130[ml ] T  26.3 [C ] P  98.6 10 3 [ Pa] Using this data, calculate the number of moles of the air in the can by using the ideal gas law:



PV RT

(4)

Record your value of μ here: μ = ___________________________________________________ (mole)

The can is placed in the water bath. After connecting the pressure sensor and the temperature sensor to Pasco Capstone software. Data collection is set to record both table and plot. The plot has the pressure on the vertical axis and temperature on the horizontal axis. The heat is turned on, and heat the air in the can until the water boils. The gas has transitioned from state 1 at room temperature to state 2 at about 100 deg. Celsius. With data still recording, the heat source is turned off, and add ice to the hot water bath to remove heat and allow the temperature to drop back to room temperature. The gas is returning to its initial state. The recording is stopped. On the column with the pressure data, a calculation is entered to convert the pressure data to volume data. Since this is a constant pressure process, the pressure is fixed at atmospheric pressure. Thus the volume is obtained by:

V

 RT P

(5)

  number of moles obtained earlier Where:

R  8.31[J mol-1 K -1] T  temperature of the gas from the column of data P  Pressure value from the column of data

The plot will change from Pressure vs Temperature to Volume vs Temperature. A copy of the plot appears in figure 2.

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Computational Lab

Charles Law & Entropy

A. Vaucher

Figure 2. Plot obtained with experimental apparatus in figure 1. The volume on the vertical axis is calculated from real time measurement of the temperature and pressure in the can. The gas (air) starts at state (V 1, T1), heat is added, until it reaches state (V2, T2). The gas is then cooled to return back to its original state, labeled (V 3, T3), because due to experimental fluctuations, is not exactly identical to the initial state.

From plot record the initial values of the volume and temperature for state 1 (Cold - room temperature) and state 2 (Hot - Boiling), and state 3 (Cold – Back to room Temperature again). State 1 (Cold) State 2 (Hot) State 3 (Cold) Volume [m3] V1  V2  V3  Temperature [K]

T1 

T2 

T3 

Using the data in the above table, verify Charles Law. Enter your calculations in the table below:

r1 

V1 T1

r2 

V2 T2

Is Charles Law satisfied? Page 46 of 60

% 

r1  r2 100%  r1  r2     2 

Computational Lab 3.0

Charles Law & Entropy

A. Vaucher

Measuring the Entropy of the Gas: The gas in this experiment undergoes a reversible process. It starts out at state 1 (cold); transitions to state 2 (hot); then returns again to state 3 (cold) which should coincide with state 1. Due to experimental uncertainties, however, state 3 will differ slightly from state 1. As we have learned, for reversible processes the change in entropy is zero. We will use the data obtained for the above 3 states to calculate the entropy in going from state 1 to 2: ΔS12 and returning back from state 2 to 3: ΔS31. The entropy for these processes is given by the formula:

 Tf   Vf  Sif   CV ln     R ln    Ti   Vi 

(6)

Where CV is the specific heat at constant volume for air, it has a value of:

CV (Air) 20.87 [J mol 1 K ]1 Using equation (6), calculate the entropies for this reversible process. The second Law of Thermodynamics states that for reversible processes such as this, the total entropy of the system is zero:

S  S 12  S 21  0

(7)

Enter the results of your calculations in the table below: Entropy of Reversible Process

S12

S23

[J/K]

[J/K]

S  S 12  S

23

[J/K]

Is the second law of thermodynamics satisfied?

4.0

Additional Questions: 1. For a constant pressure process, the entropy can be calculated with the following formula:

 Tf  Si f   Cp log    Ti  Where Cp is the specific heat of air at constant pressure. The specific heat at constant pressure differs slightly with temperature, it has the value 29.2 [J mol-1 K-1] for temperature range: 300K < T < 360K which is sufficient for this experiment. Calculate the entropy ΔS12 using this formula, and compare with your earlier result. .

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Computational Lab

Charles Law & Entropy

A. Vaucher

2. Consider 3.2 moles of Nitrogen gas at 600 0K, going through a sudden irreversible process where it expands from a sphere radius 10 cm to a final state with a sphere of radius of 15 meters. The expansion is considered fast enough that there is no time for heat to enter or leave the system, and over the time scale of the event, can be considered adiabatic. What is the final temperature of the gas? and; What is the change in entropy for this process? (Hint: Consider a parallel reversible process, assume an ideal gas approximation,  = 1.4 for Nitrogen).

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Computational Lab

Appendix: Error Analysis

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A. Vaucher

Computational Lab

Appendix: Error Analysis

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A. Vaucher

Computational Lab

Appendix: Error Analysis

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Computational Lab

Appendix: Error Analysis

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Computational Lab

Appendix: Error Analysis

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Computational Lab

Appendix: Error Analysis

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Computational Lab

Appendix: Error Analysis

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A. Vaucher

Computational Lab

Appendix: Error Analysis

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A. Vaucher

Computational Lab

Appendix: Error Analysis

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A. Vaucher