COLLEGE PHYSICS II Fall 2012spiff.rit.edu/classes/phys212.w2012/phys212_winter_2012.pdf · 2014. 2....

COLLEGE PHYSICS II

1017-212

Activities Manual

School of Physics and Astronomy

Rochester Institute of Technology

(Last Revised Fall 2012-1)

Preface

Colleagues: Like any lab manual this is merely a punctuated equilibrium in the evolution

of a Lab Manual. Many before us have contributed to the present form, and we are deeply

grateful for the hard work that obviously went into developing many of the activities

appearing in this incarnation of the manual. We have included several activities that

involve many of the concepts and techniques introduced and developed in College

Physics II. There are more activities included here than needed to occupy all of the

workshop time allotted to a ten week quarter. And, by this we mean to say that there are

not enough. In other words, there are hosts of other activities that we envision as relevant

and useful that we have not yet had the time to develop for inclusion here. Perhaps you

will help us? At the risk of overstating it, the basic philosophy we follow here is to assure

that the students who really want to learn never run short of meaningful challenges. Most

of the activities vary generally between easy and intermediate levels of difficulty

(however one measures that …), and some are meant to be challenging to even the top

students. In all cases we have, to the best of our abilities and within the constraints of

finite resources (especially time), selected and developed activities that strive to exercise

the ability of the student to think about the principles of physics in relation to the physical

world they purport to describe. In other words, we don’t much care for memorization or

for plug and chug tasks. Instead, we wish to stress the larger view that there are very

good reasons why folks intending to work in a very wide range of fields study physics.

Students: This manual is now yours to complete. The only way to make sure you get all

you can out of it is to get all that you can out of it. It is your responsibility to dedicate the

effort and time needed to work through the activities included. That level of commitment

will differ from student to student. Further, it is not enough to simply answer the

questions and complete the tasks. At every step of the way you must think hard about the

meanings of the concepts you are learning and using. There is a real value to be had here

for those who will do what it takes to have it. There is a world of critical thinking and

scientific reasoning that will open to you in proportion to your exercise of the thought

processes that underpin repeatable science. After all, you now own an entire manual full

of problems to which you can practice applying the most powerful analytical tool

heretofore discovered by humans, the human mind. Well, the mind, like the body,

benefits and grows stronger as a result of healthy exercise. Naturally your instructor will

set a course though these and possibly other activities. But, that is all a function of time

allocation and assessment – you are welcome to work on all of the activities whether or

not they are “assigned” in any particular class. The real deal is the experience, and that is

really largely what you make it. So, make much.

Thomas W. Herring

Edwin E. Hach III

Rochester, New York

November 2009

Table of Contents

Pressure ………………………………………………………………………………..... 1

Pressure Problems ……………………………………………………………………... 5

J-Tube Problem …………………………………………………………………………7

Buoyant Force ……………………………………………………………………….…. 9

Polar Bear (Ursus maritimus) Problem …………………………………………...…. 17

Bernoulli Problems …………………………………………………………………… 19

Viscosity Problems ……………………………………………………………….…… 21

Absolute Zero …………………………………………………………………………. 23

Thermal Expansion Problem ……………………………………………………...…. 27

The Ideal Gas “Law”.………………………………………………………….……… 29

Ideal Gas Process Problem …………………………………………………………… 33

Cyclic Processes and the 2nd

Law of Thermodynamics …………………………….. 37

Heat Engines …………………………………………………………………...……… 41

P–T Phase Diagrams ………………………………………………………………..… 45

Calorimetry ………………………………………………………………………..….. 47

Heat Transfer Questions ………………………………………………………...…… 55

Graphing – Homework ……………………………………………………………..… 57

Newton’s “Law” of Cooling …………………………………………………..……… 61

Infrared Radiation ……………………………………………………………………. 65

Simple Harmonic Motion …………………………………………………………..… 67

Forces and Energy in Simple Harmonic Motion ………………………..………….. 73

The Simple Pendulum …………………………………………………………..……. 79

Waves ……………………………………………………………………………..…… 89

Sound – Frequency Spectrum ………………………………………………….…….. 95

Sound Level and Intensity Problems ………………………………………..………. 99

Supplementary Notes: Doppler Effect with Sound …………………...………….... 101

Doppler Effect …………………………………………………………………..…… 109

Standing Waves on a String ………………………………………………………… 111

Sound Interference Problems ………………………………………………….…… 121

Interference …………………………………………………………………….……. 123

Interference and Diffraction ………………………………………………………... 129

Spectroscopy (An application for diffraction gratings) …………………...………. 137

Thin Film Interference Practice Problems …………………………………..……. 141

Snell’s Law ………………………………………………………………………..…. 145

Reflection of Light ……………………………………………………………...…… 151

Spherical Mirror ………………………………………………………………....….. 155

Thin Lens Equation …………………………………………………………….…… 163

Polarization ……………………………………………………………………......… 169

Thin Convex Lenses ……………………………………………………………….... 173

Appendix A: Introduction: Uncertainties, Error Propagation, and Graphing … A-1

Appendix B: O-Haus Dial-O-Gram Balance………………………….…………... A-12

1

P = 1 atm

P = 0

Your Name (Print): ____________________________________________ Date: ____

Group Members: ______________________________________________ Group: ___

______________________________________________

Pressure

1. A cylinder is fitted with an air-tight circular cap (shaded in the

diagram) of mass m that is supported from within the cylinder by

a metal rod. The interior of the cylinder is completely evacuated,

and the exterior is exposed to the atmosphere. Atmospheric

pressure exerts a force on the top cap.

(a) Give an algebraic expression for the net force on the

top cap. Define all terms in your equation.

(b) Draw vectors representing the force exerted by the atmosphere on each of

the three randomly-selected segments of equal area on the top cap. Draw

the length of the force vector in proportion to the magnitude of the force.

(c) Write an expression for the magnitude of the force due to the air on the top

cap in terms of the exterior pressure p and the radius of the cylinder R.

(d) If the cylinder has a radius of 15.0 cm, what is the magnitude of the force

on the top cap due to atmospheric pressure? Write the result in SI units.

2

2. Otto von Guericke is credited with inventing the vacuum pump

in the mid 1600s. He demonstrated the force exerted by

atmospheric pressure by joining two hemispheres to form a

sphere and then evacuating the interior space using his vacuum

pump. Two teams of horses were unable to pull the

hemispheres apart; but, when air was let into the interior space,

the hemispheres were easily separated.

PROBLEM: Magdeburg plates are similar to von Guericke’s sphere. Two

flat circular plates, each with a 12.0 cm diameter, are separated by an

O-ring. A small syringe is used to reduce the pressure between the

plates to 0.600 atm. Calculate the force (in Newtons) required to

separate the plates.

O-ring

Plexiglass plates

12 cm

RFr

Fr

3

CHALLENGE QUESTION:

The two vessels shown below have different shapes but the same circular base area

(circle of radius R). The vessels weigh the same when empty. Each vessel is filled with

water to the same depth d (so that there is more water in vessel A than in vessel B).

d

Vessel BVessel A

(a) Is the water pressure at the bottom of Vessel A greater than, less than, or equal to

the water pressure at the bottom of Vessel B? Explain your answer.

(b) When filled with the same depth of water, which vessel weighs more? Explain

your answer.

(c) Each vessel supports the water inside by exerting an upward force equal in

magnitude to the weight of the water. According to (a), the water exerts the same

downward force on the circular base of each vessel, yet according to (b), the weight

of the water in each vessel is different. Explain this apparent paradox. (I.e., what

supports the extra weight in Vessel A?)

5



______________________________________________

Pressure Problems

The pictured U-tube is open to the atmosphere at both ends. Assume that the mystery

fluid does not mix with the mercury.

(a) Let p0 represent atmospheric pressure, ρHg represent the density of mercury, and ρf

represent the density of the mystery fluid. Obtain an algebraic expression for the

density of the mystery fluid in terms of some or all of the following: h1, h2, h3, and

ρHg.

(b) For the situation shown in the figure, which is true, ρf < ρHg or ρf > ρHg? Explain.

Mercury (Hg)

Mystery Liquid h1

h2

A

h3

6

(c) If h1= 2.55 cm and h2 = 25.00 cm (clearly the figure is not to scale), then what is

the most likely identity of the mystery fluid? On what scientific evidence did you

base your choice?

(d) If the absolute pressure at Point A is 114,200 Pa, what is the height h3? Use the

numerical values from Part (c).

(e) Using previous numerical values, determine the gauge pressure at Point A. Take

atmospheric pressure to be p0 = 101,300 Pa.

7



______________________________________________

J-Tube Problem

A J-tube having a uniform cross sectional

diameter of 3.00 cm is filled as shown in the

figure. The mass on the left arm is

symmetrically placed on a frictionless piston

that floats on the mercury in the arm on the

left. You may neglect the mass of the piston

and the mass of gas in the top of the arm on the

right.

(a) Determine the heights h1 through h4.

(The figure is not exactly to scale)

(b) If the mercury level is as shown in the

figure when the gauge pressure of the

gas in the right arm is 98200 Pa, how

much mass, M, sits atop the piston on

top of the left arm?

Show your work on separate pages.

200 ml of

H2O

280 ml of

Glycerin

120 ml of

Oil

300 ml of

Ethyl

Alcohol

M Frictionless

Piston

Gas pga = 98200 Pa

Pressure

Gauge

h1

h2

h3

h4

Hg

9



______________________________________________

Buoyant Force

Refer to the Introduction: Uncertainties, Error Propagation, and Graphing for a review

of uncertainties and uncertainty propagation.

If you immerse an air-filled ball under water, you will find that an upward force is

exerted on the ball. This force is called the buoyant force B

Fr

, and is always exerted on

any object immersed in any fluid (liquid or gas). The buoyant force is always directed

vertically upward (opposite the weight force). Archimedes’ Principle states that an object

immersed in a fluid experiences an upward buoyant force BF whose magnitude equals the

weight of the displaced fluid:

( )

( )displaced fluid

displaced fluid

Weight of the displaced fluid

B

B

B

F

F

F

=

=

=

m g

V gfρ

where fρρρρ is the density of the fluid.

1. Displaced Volume

(a) A vessel is completely filled with a fluid of density fρ . An object of mass 0m and

volume 0V is then completely submerged in the fluid. Give an algebraic

expression for the volume of water that spills over the edge of the vessel.

( )spilled =V _________________________

(b) The spilled fluid is the displaced fluid – fluid that would have occupied the

volume if the object had not been submerged. When an object is completely

submerged, the displaced volume is identical to the volume of the object. Write an

expression for the weight of the displaced fluid in terms of the density of the fluid

ρf and the displaced volume 0V (equal to the volume of the object since the object

is completely submerged).

10

2. Determination of a Buoyant Force from Archimedes’ Principle

(a) The volume of a solid metal specimen is Vo = (8.4 ± 0.4) cm3. Calculate the

buoyant force exerted on the specimen when it is completely submerged in fresh

water. Write the answer in SI units.

BF = ____________________________

b) Calculate the uncertainty in this buoyant force. Write the answer in SI units.

SHOW ALL WORK.

BFδδδδ = ____________________________

c) Write this buoyant force in “proper form”.

11

MEASUREMENTS

All measured quantities must have units and associated uncertainties.

Calculated values should also have units and uncertainties, but only report

uncertainties for calculated values when requested.

The Ohaus Dial-O-Gram balance has a precision of 0.01 g. All mass

measurements must be recorded with this precision. Refer to the back of this

manual for a brief overview of the Ohaus balance.

3 Determination of the buoyant force from Newton’s second law

(a) An object having mass mo is suspended at rest from a scale using a

light string. Draw a free body force diagram for the object (use Tr

to represent the

tension in the string and OWr

for the weight of the object). Using Newton’s

Second Law, write an expression for the magnitude of the tension in terms of the

magnitude of the weight of the object.

(b) Suppose the same solid specimen is completely submerged in a fluid of

densityfρ . Draw a free body force diagram for the solid (use 'T

r

to represent the

tension in the string while the solid is submerged in the fluid). Using Newton’s

Second Law, write an expression for the magnitude of the tension 'T in terms of

the magnitudes of the buoyant force (FB) and the weight of the solid (WO).

(c) Write the relationship between the spring scale readings (T and T ′ ) and the

magnitude of the buoyant force )( BF .

Scale

fρ

12

4 Measurement of the buoyant force

(d) Suspend a metal specimen from the stirrup from which the scale pan hangs and

record the scale reading (the mass of the solid). Calculate the magnitude of the

tension in the string T (i.e., the object’s weight in air) and its uncertainty δT.

(e) Completely submerge the solid specimen in water; making sure that the specimen

does not touch any part of the container. Record the scale reading and use this

“apparent mass” to calculate the magnitude of the tension in the string 'T and its

uncertainty 'Tδ .

(f) Use your result from Part (c) to calculate the magnitude of the buoyant force FB acting on the metal solid. Calculate its uncertainty in δFB. Write the magnitude of the buoyant force in proper form.

13

5. Density Calculation

(a) Calculate the volume 0V of the metal specimen from the buoyant force.

0V = ____________________________ ± ____________________

(b) Calculate the density of the metal from the volume 0V and the weight of the

specimen in air.

Mρρρρ = ____________________________ ± ____________________

14

6. Exercises about Floating Objects

(a) An object (mass 0m and volume 0V ) floats partially submerged. Is the volume of

the displaced fluid )( dispV greater than, less than or equal to the volume of the

object )( 0V ? Explain your answer

(b) When an object floats, the displaced fluid is the volume of the object that is below

the fluid level. Suppose an object (mass 0m and volume 0V ) floats partially

submerged with a volume dispV in the fluid (density fρ ). Write an expression for

the weight of the displaced fluid in terms of the density of the fluid and the

displaced volume. Explain your answer.

(c) An object floats at rest (partially submerged). Draw a free body force diagram for

the object. Is the weight of the displaced fluid )( dispW greater than, less than, or

equal to the weight of the object )( 00 gmW = ? Explain your answer.

15

(d) Derive an expression for the volume fraction of the object that is partially

submerged submerged 0V V in terms of the density of the object ρo and the density of

the fluid ρf.

(e) The density of steel is 7860 kg/m3. The density of sea water is 1025 kg/m

3. How

is it possible for steel ships to float?

(f) The average density of a human is 980 kg/m3. Why do swimmers float higher in

salt water than in fresh water?

16

CHALLENGE QUESTIONS

(1) A lead sphere and a steel sphere of equal mass are completely submerged in

fresh water. Which sphere experiences the larger buoyant force? Use correct

physics principles to justify the correct answer. Explain your answer.

(2) A fisherman throws the iron anchor from his floating boat into the lake. Does the

level of the lake rise, fall, or stay unchanged? Explain your answer.

17



______________________________________________

Polar bear (Ursus maritimus) Problem

A polar bear having a mass Mbear steps onto an ice sheet having a thickness h and having

two large, parallel faces each with area A. The ice sheet, being made of frozen,

compressed fresh water snow, has a density of ρice and it floats in sea water having

density ρsea. The ice sheet is shown schematically below.

(a) Before the bear steps onto the sheet, what fraction d/h of the thickness of the sheet

is submerged? (Here, d is distance from the bottom of the ice sheet to the

waterline as shown above.) Express your answer algebraically using the

symbols introduced in the problem; do not insert numerical values yet!

(b) What is the minimum possible area, A, such that the ice sheet remains afloat

supporting the bear once it steps onto the sheet? Express your answer

algebraically using the symbols introduced in the problem; do not insert

numerical values yet!

Close-up for part (a):

d

Ice sheet

Area = A

ρice ≈ 920 kg/m3

h = 12 cm

ρsea water ≈ 1025 kg/m3

h

18

Adult male polar bears range in mass from 400 kg to 680 kg; typical adult female polar

bears are about half the mass of typical adult males. Consider Isbjøm, a large adult male

(680 kg), and Nanook, a small adult female (200 kg).

(c) Assuming that the ice sheets in the region where this population of polar bears

lives have approximately uniform thicknesses of about 12 cm, estimate the

smallest area of an ice sheet that will just barely support Isbjøm. Use the densities

for ice and sea water that are given in the figure.

(d) Making the same assumptions about the ice sheet as you did in Part (c), estimate

the smallest area of an ice sheet that will just barely support Nanook.

(e) Compute the mass of the ice sheet needed to support Isbjøm. How does this

compare with the actual mass of Isbjøm?

(f) If Nanook steps on the ice sheet that just barely supports Isbjøm, how deep will

the bottom of the ice sheet be submerged when the system is in static equilibrium?

Assume that Isbjøm is not on this ice sheet when Nanook steps onto it.

(g) What will happen if Isbjøm steps onto the ice sheet that barely supports Nanook?

Assume Nanook is not on this ice sheet when Isbjøm steps onto it. Explain using

robust principles of physics.

19



______________________________________________

Bernoulli Problems

1. A pipe of radius 2.5 cm carries fresh water through the basement of a two story

home at a speed of 0.90 m/s and a gauge pressure of 0.9 atm. A bathroom faucet

on the second floor, a height of 8.2 m from the basement pipe, is turned on. What

is the speed of the water as it flows from the bathroom faucet?

2. A beaver dam forms a pond with a surface level 2.0 m above the bottom of the

dam. The dam has developed a leak at the very bottom through a circular hole of

radius 3.0 cm. You can safely assume that the cross-sectional area of the surface

of the pond is much, much larger than the cross-sectional area of the hole.

(a) At what speed does water exit the hole?

(b) If the pond holds 2.0×104 L of water, and assuming that the speed you found in (a) remains constant as the pond drains, estimate how long it

takes to drain the pond completely?

(c) Is the time that you found in (b) an underestimate or an overestimate of the

time actually required to drain the pond? Explain.

3. Water flows steadily from an open tank as shown in the figure below. The

elevation of the water’s surface in the tank (point 1) is 15.0 m. The elevation of

the exit pipe (points 2 and 3) is 4.00 m. The drain pipe narrows from a radius of

12.0 cm at point 2 to a radius of 7.25 cm at point 3. The cross sectional area of

the tank is much larger than the cross sectional area of the drain pipe.

(a) What is the volume flow rate in m3/s of water draining from the tank?

(b) What is the gauge pressure at point 2?

15.0 m

4.00 m 2 3

1

20

4. Water in a river flows at 5.0 m/s. One end of a large hose is inserted just into the

water as shown while the other end runs up to an open container on the riverbank.

What is the maximum height above the level of the river that the top of the

container can be placed such that it will still be filled by water from the river?

5. Water is pumped up a hill and expelled into the atmosphere via a pipeline as

shown in the figure. If the speed of effusivity of the water is 18 m/s, what must be

the absolute pressure maintained by the pump? Treat the water as an ideal fluid.

6. Water flows at a speed of 1.7 m/s through a horizontal pipe having a circular cross

section 5 cm in diameter. The pipe widens to a diameter of 14 cm. Treat the water

as an ideal fluid.

(a) What is the speed with which the water flows through the wider section of

the pipe?

(b) What is the difference in pressure between a point within the fluid in the

narrower section and a point within the fluid in the wider section?

(c) In which section of the pipe is the pressure higher?

Choose one: narrow wide

7. Consider a square plate of density ρp = 2700 kg/m3, side length L = 2.0 m, and

unknown thickness t lying in a parking lot. There is a wind with velocity v = 30

m/s blowing over the plate. What is the maximum thickness the plate can have

and still flutter in the wind? (Treat the air as an ideal flowing fluid and use as its

density 1.20 kg/m3.)

18 m/s

5.5 m

D1 = 5.0 m

D2 = 1.5 m

Pump

5.0 m/s

21


Group Members: ____________________________________________Group: ___

______________________________________________

Viscosity Problems

1) If he uses a hose having a uniform diameter of 6 cm and a length of 20 m,

Poindexter can fill his swimming pool with 1000 m3 of water at 40°C in 14 hours.

What is the gauge pressure maintained by the pump at the intake end of the hose?

2) An unknown fluid flows through a 2.0 m long cylindrical pipe of diameter 3.0 cm.

The pressure difference needed to maintain a flow rate of 4.0 L/s is 4025 Pa.

(a) What is the viscosity of the liquid?

(b) What might the liquid be (try the table in your book)?

22

3) Motor oil at 40°C (η = 0.07 Pa⋅s, ρ = 900 kg/m3) flows steadily from an open tank as shown in the figure below. The elevation of the surface of the oil in the tank

(point 1) is 15.0 m. The elevation of the exit pipe (points 2 and 3) is 6.00 m. The

drain pipe has radius 2.0 cm and is 3.00 m long. The cross sectional area of the

tank is much larger than the cross sectional area of the drain pipe. Assume that

the oil behaves as an ideal fluid until it reaches the horizontal drain pipe (between

points 2 and 3) where it should be considered to have a non-zero viscosity.

What is the average velocity of the fluid flowing from the drain pipe?

Hint: You should write down one equation involving vave in the drain pipe and

another involving v2 in the tank; both equations will also involve p2. You can

then assume that vavg = v2 (which it must for the volume flow rates at point 2 and

within the drain pipe to be equal). Combining these equations together will allow

you to eliminate the unknown pressure p2. The result will be a quadratic equation

of the form Avavg2 + Bvavg + C = 0, which you can solve to get vavg.

15.0 m

6.00 m

1

2 3

23



______________________________________________

Absolute Zero

You will determine the temperature at which the pressure of an ideal gas

extrapolates to zero. You will use the Celsius scale, defined by water freezing at 0ºC, and

boiling at 100ºC at standard atmospheric pressure.

It is impossible to attain temperatures anywhere close to absolute zero in the

workshop. Therefore, you will extrapolate from the data measured by the Instructor. He

or she will use a fixed amount of a gas and measure the pressure at different Celsius

temperatures.

The pressure will be measured at five temperatures, those of: (i) hot water; (ii)

room temperature; (iii) an ice and water mixture in equilibrium; (iv) dry ice (solid CO2);

(v) liquid nitrogen. A digital thermometer is used to measure the temperature of the bath.

A metal sphere filled with a fixed volume of helium gas is used to determine the pressure

at each temperature.

MEASUREMENT

Record the units and the uncertainty in the digital thermometer and the pressure gauge in

the column headings below. Record the values for P and T for the five different baths as

they are measured by your instructor.

Substance

Pressure

Temperature

Hot water

Room temperature

Ice/water mixture

Dry ice

Liquid nitrogen

24

1. Plot the measured Pressure (in psi) versus Temperature (in °C) on the supplied

graph paper. The temperature scale should extend from −360°C to +140°C. The pressure scale should extend from 0 to 19 psi. Make sure to include horizontal

and vertical error bars when possible. Most of the grade is reserved for good

graphing techniques.

2. Draw a best fit straight line through the data. Determine and record the horizontal intercept of the best fit line directly from your graph. Be sure to include units.

Horizontal intercept:

3. Estimate the uncertainty in the horizontal intercept, given by

intercept

maximum intercept - minimum interceptδ =

2

Be sure to include units

interceptδ =

4. Absolute zero is the temperature at which the pressure P = 0; that is, the horizontal intercept. Write your value for absolute zero in Celsius degrees in

proper form:

Absolute zero (°C):

5. Should the value of the horizontal intercept depend on whether you plot the

pressure in lb/in2 or in Pa? State your reason(s).

6. Using your graph, what is the pressure of the gas in the bulb at the boiling point of

helium (−269°C)? Show this pressure on the graph.

Pressure at −269°C:

27



______________________________________________

Thermal Expansion Problem

Initially a concrete cylindrical shell having inner diameter 11.900 cm and outer diameter

12.000 cm surrounds a solid aluminum cylinder having diameter 11.845 cm. The initial

length of each cylinder is 150.00 m. The system is initially configured so that at room

temperature (23ºC) the cylinders are concentric and their ends coincide (as shown). The

figures below show end-on and side views of the system. Note that the figures are not to

scale; for example, the thickness of the concrete shell and the gap between the aluminum

cylinder and the concrete shell have both been greatly exaggerated for clarity.

Li

dconc,outer

dconc,inner

daluminum

28

(a) At what temperature of the overall system will the aluminum core just barely

completely fill the cross sectional area of the hollow interior of the concrete

casing? [Hint: Does the “hole” in the concrete expand too?]

(b) What is the difference in the lengths of the two samples at the temperature you

found in Part (a)? Which sample is longer?

(c) First, calculate the initial volume of the concrete cylindrical shell using the initial

radii and length.

Then calculate the final volume using the final radii and length.

Finally, use these results to determine the percentage change in the volume.

(d) Calculate the percentage change in volume of the concrete cylindrical shell directly using the formula for volume expansion: ∆V = β Vi ∆T. Does your answer here agree with Part (c)?

(e) How would it affect the answer to Part (a) if the outer shell were made of glass?

(f) How would it affect the answer to Part (a) if the inner core were made of iron?

29



______________________________________________

The Ideal Gas Law

In this activity you will explore the concepts behind the kinetic theory of gases, make

predictions from the Ideal Gas Law, and then computer simulate the behavior of an ideal

gas using the Atoms in Motion software.

PREDICTION ACTIVITY

Predict the results of the following three processes for an ideal gas that has N

atoms, pressure p, volume V, temperature T, and total thermal energy Eth. Explain your

predictions in terms of the Ideal Gas Equation.

Process 1: Double the kinetic energy to 2Eth but leave N and V unchanged.

Circle your predictions.

Pressure Doubles Stays the same Reduced by half

Temperature Doubles Stays the same Reduced by half

Explain the reason(s) for your predictions.

30

Process 2: Double the number of atoms to 2N but leave Eth and V unchanged.

Circle your predictions.




Process 3: Double the volume to 2V but leave Eth and N unchanged.

Circle your predictions




31

SIMULATIONS

To test your predictions, you will use the computer to simulate the behavior of an

ideal gas. Open the Atoms in Motion software by double clicking on the Atoms in Motion

icon on the desktop. Open the file lab_air.am. The simulation will begin as soon as the

file is opened.

(1) Click on the DISP icon and make sure the box to display the kinetic energy is checked. (This is useful because the way that you change the energy, by adding to

the original energy, makes it easy to inadvertently use the wrong value.)

(2) Check that the following values are in the simulation:

(a) Atom Properties: Click on the ATOM icon on the toolbar. Name nitrogen oxygen argon 4

th atom

Number 78 21 1 0

Diameter 3 3 3 −

Mass 4.65 5.31 6.63 −

Color cyan red yellow −

(The dashes in the last column mean that you can enter any values, since there are

zero atoms of this type used in the simulation.)

(b) Box: Click on the BOX icon on the toolbar and

ensure that box width is 161 × 10−10 m.

(c) KE: Ensure that total kinetic energy is 621.2 × 10–21 J. If it is not, click on the KE icon on the toolbar to change it.

(KE in the simulation is the same as Eth in the textbook.)

(3) Let the simulation continue for 1 minute and then begin averaging the simulation

values by clicking the AVG button. Allow the averaging to run for 2 minutes, and

then stop the simulation by clicking the STOP button. Enter the simulated pressure

into the “Origin Simulation” column of Table 1.

Process 1:

Double the energy of the system by clicking on the KE icon on the tool bar and

entering 2.621 in the box Kinetic energy to add to system. The new value should then be

1242 × 10–21 J. Click OK. Turn off averaging by clicking the AVG button. RUN the simulation for 2 minutes, and then begin averaging by clicking the AVG button again.

Allow the simulation to run for 2 minutes. Then click STOP and record the values for p,

V, T, N and Eth in the appropriate column of Table 1.

Do the simulated values for p and T match your predictions? Explain any

difference(s).

32

Process 2:

Restore the system to its original values by first exiting and

then restarting the Atoms in Motion program and reopening

the file lab_air.am.

Click on the Atom icon on the toolbar and double the number of atoms in the

simulation by doubling the number of each type of atom. Click OK. Turn off averaging

by clicking the AVG button. Click RUN, wait 1 minute, begin averaging. Average for 2

minutes, then STOP the simulation. Record the values for p, V, T, N and Eth in the

appropriate column of Table 1.


difference(s).

Process 3:

Restore the system to its original values by first exiting and

then restarting the Atoms in Motion program and reopening

the file lab_air.am.

Double the volume of the box in the simulation by clicking the Box icon on the

toolbar and entering 202.8 for the Box width. Click OK. Turn off averaging by clicking

the AVG button, then click RUN, wait for 1 minute, and begin averaging. Average for 2

minutes, then STOP the simulation. Record the values for p, V, T, N and Eth in the

appropriate column of Table 1.


difference(s).

TABLE 1: Simulation Results

Variable Original

Simulation

Process 1

Double Kinetic

Energy

Process 2 Double N

Process 3 Double Box Size

p

V 24 34.173 10 m−×

T 300 K

N 100

Eth (U) 21621.2 10 J−×

33

5.00 × 105

1.00 × 105

p (Pa)

V (m3) 2.00 6.00

A

B

C

The Herring Cycle


Group Members: ____________________________________________ Group:___

______________________________________________

Ideal Gas Process Problem

A monatomic ideal gas is run through the cycle shown starting in state A. The

temperature of the gas in state A is TA = 300 K. The cycle happens within a sealed

chamber outfitted with a piston as necessary.

The cycle is composed of three processes, A → B, B → C, and C → A.

1) For each individual process. . .

(a) Name (if possible) the process by type, and enter this name in Table 2.

(b) Compute the initial and final value of each of its thermodynamic state

variables, p, V, n, T, and Eth, and enter them in Table 1.

(c) Compute the work W done by the gas, and enter it in Table 2.

(d) Compute the change in internal energy ∆Eth of the gas, and enter it in Table 2.

(e) Use the First Law of Thermodynamics to compute the amount of heat

transferred Q, and enter it in Table 2. Is the heat absorbed or emitted by

the system?

34

Table 1: States and state variables

State

variable State A State B State C

p

V

T

n

Eth

Table 2: Processes and process variables

Process

variable A→→→→B B→→→→C C→→→→A

name

W

∆Eth

Q

2) For the entire cycle . . .

(a) Compute the net change in the thermal energy of the system.

(b) (i) How much heat is absorbed by the system? (total energy input)

(ii) How much heat is exhausted by the system? (“useless” output)

(iii) Compute the net amount of heat transferred from the environment

to the system.

(c) Compute the net amount of work done by the system on the environment

(“useful” output). How does this compare to the net heat transferred

from the environment to the system?

(d) Determine the overall efficiency of the cycle. (This is the ratio of the

“useful” output to the total energy input.)

35

Molar Specific Heat

The amount of heat transferred per mole to (or from) a gas is proportional to the change

in temperature of the gas as a result of the transfer. The constant of proportionality

depends upon the type of process used to accomplish the heat transfer. For example, if

the heat is transferred at constant volume to a system containing n moles of an ideal gas,

the relationship is

Q = nCV∆T

where CV is the molar specific heat of the gas at constant volume. CV is a property of

the gas. On the other hand, if the heat is transferred at constant pressure to a system

containing n moles of an ideal gas, the relationship is

Q = nCP∆T

where CP is the molar specific heat of the gas at constant pressure. CP is also a property

of the gas.

3) Referring to the cycle we are studying, in each case where it is applicable,

Compare Q as computed using a molar specific heat to that computed above in

Problem 1(e).

37


Group Members: ____________________________________________ Group:___

______________________________________________

Cyclic Processes and the 2nd

Law of Thermodynamics

1) Field Marshal von Weisenheimer claims that he has invented a heat engine that

can lift a mass of 1 kg by a height of at least 10 meters using a single cycle of the

process represented in the figure. [You may recall that the acceleration due to

gravity near the surface of Earth is approximately g = 9.8 m/s2.]

Can the Field Marshal’s claim be true? In order to thoroughly complete your

answer, you must support it with robust physical principles and numerical

evidence. For example, if you say yes, explicitly demonstrate numerically the fact

that the laws of thermodynamics are obeyed by the process and that the lifting

task is accomplished. If you say no, what law(s) of physics is(are) violated?

Demonstrate any such violation(s) numerically.

TH = 525K

TC = 300K

∆Eth = 0 W

QH = 300J

QC = 200J

38

2) A 0.04 mol sample of an ideal gas evolves through the process 1→2→3→4→1 represented on the PV diagram shown in the figure.

(a) Determine the actual efficiency of the heat engine based on this cycle.

(b) Compute the actual coefficient of performance of a refrigerator based on

the cycle 4→3→2→1→4.

(c) Compute the actual coefficient of performance of a heat pump based on

the cycle 4→3→2→1→4.

P (atm)

V (cm3) 100 400 500

5.0

8.0 1 2

3 4

39

3) An ideal gas operates as a heat engine by repeating the cyclic process

A→B→C→D→A. The temperature of the gas in State A is 350 K. The processes

B→C and D→A are isothermal.

(a) Compute the efficiency of the heat engine. You will need to use the fact

that the work done by an ideal gas during an isothermal process is given

by

=

i

flnV

VnRTW . (This formula is derived by computing the area

under the curve, just like for other processes, but for this process the

computation requires calculus.)

(b) Compare the efficiency of this heat engine with that of a Carnot cycle

operating between the same two temperature extremes. Does your result

make sense?

P (Pa)

V (m3) 0.02 0.07 0.10

4.0×105

A B

C

D

2.8×105

0.8×105

41



______________________________________________

Heat Engines

(A) Carnot Engine (1824)

The Carnot engine is the most efficient engine possible that operates between two given

temperatures HT and .CT The general expression for efficiency e is

1CH C

H H H

W QQ Qe

Q Q Q

−= = = = −

cycleWork done

Heat input.

The Carnot (i.e., “maximum” or “ideal”) efficiency is given by

1 C H C

H H

T T Te

T T

−= − =Carnot .

Question:

The internal combustion engine in almost all automobiles converts some of the

heat generated by the combustion of gasoline into mechanical work (turning the wheels),

and exhausts the remaining heat into the atmosphere. The maximum temperature

attainable through the combustion of gasoline is approximately 3800 oF. If the exhaust

temperature is 150 oF, what is the maximum theoretical efficiency of this engine? Be

careful about units!

NOTE: The actual efficiency of an internal combustion engine is about 20%.

42

(B) Diesel Engine (Patented in 1892)

Rudolf Diesel tried unsuccessfully to produce an engine based on the Carnot cycle.

However, his diesel engine nearly matches a Carnot engine in efficiency and is one of the

most efficient in use today. The diesel cycle includes two adiabatic processes (2→3 and

4→1). The dashed lines show two isothermal curves that are drawn for comparison.

1. Name all of the processes in the Diesel cycle. Write these names on the diagram.

(For example, 4→1 is an adiabatic compression.)

2. Draw arrows on the PV diagram that indicate all heat flows involved in the cycle.

Label Q12 as the heat that enters or leaves between points 1 and 2, Q23 as the heat

that enters or leaves between points 2 and 3, Q34 as the heat that enters or leaves

between points 3 and 4, and Q41 as the heat that enters or leaves between points 4

and 1. Some of these Q’s may be zero.

P

V

1 2

3

4

43

3. Is W12 positive, negative, or zero? WHY?

Is W23 positive, negative, or zero? WHY?



4. At what point on the PV diagram does the temperature have its maximum value?

Explain your answer. [Hint: To answer this and the next question, it is useful to

compute whether the temperature increases or decreases for each process.]

5. At what point on the PV diagram does the temperature have its minimum value?

Explain your answer.

45



______________________________________________

P-T Phase Diagrams

A P-T phase diagram describes the relationship between the solid, liquid, and gaseous

states of a substance. Three solid lines indicate the values of pressure and temperature at

which two different phases of the substance coexist in equilibrium. A schematic of the P-

T diagram for water is shown below. The axes are not drawn to scale.

1. A thin wire that is attached to two hanging masses is placed across a block of water

ice. After a while, the wire penetrates the ice and is now frozen inside of the ice with

no trace of its path above it. Why does this happen? Use the P−T phase diagram above to explain the observation.

2. Chemical reactions occur more slowly at lower temperatures. Why does it take longer

to hard boil an egg (egg cooked in boiling water until the egg is solid throughout) at

the top of a mountain than it does at sea level. Use the P−T phase diagram to explain this effect.

47



______________________________________________

Calorimetry

Theory:

When subsystems are put into thermal contact with one another, the system as a whole

will eventually reach thermal equilibrium. A measurable signature of thermal equilibrium

is the common temperature of all of the subsystems. Consider two subsystems (in this

case the water and the aluminum cube) that we place into thermal contact with one

another and that we otherwise thermally insulate from the rest of the universe. In such a

case, the two subsystems exchange thermal energy via heat transfer until they reach a

state of thermal equilibrium. Colloquially, the “hot” one “cools down” and the “cool” one

“heats up” until they reach the same temperature. If we assume for the moment that the

thermal insulation separating the system (i.e. the two subsystems as a whole) from the

surroundings (the rest of the universe), then we can express mathematically the condition

for thermal equilibrium in terms of the heat transfers that lead there using

021net =+= QQQ

This is an example of a detailed balance equation. They are actually pretty common in

physics and engineering (and accounting). Recall that a heat transfer is positive if heat is

transferred to the subsystem and negative if heat is transferred from the subsystem. So, in

this simple case with only two subsystems, we expect that in reaching thermal

equilibrium one heat transfer is positive and the other is negative. In this way it is clear

how the net heat transfer is zero.

In order to compute any such heat transfer, we must consider two possibilities. (1) In the

case in which no phase change occurs for a subsystem, the heat transferred to (or from)

it is directly related to the temperature change, T∆ , of the subsystem via TMcQ ∆= . In

this equation, M is the mass of the subsystem and c is its specific heat (also sometimes

called “heat capacity”), a material property. It is apparent that, in the absence of a phase

change, heat transferred to a subsystem raises its temperature ( 00 >∆⇒> TQ ) and that

heat transferred from a subsystem lowers its temperature ( 00

48

of fusion and vaporization, respectively. Lf and Lv are the specific latent heats of fusion

and vaporization, respectively, and each is a material property.

When subsystems, whether two or many, in mutual thermal contact seek thermal

equilibrium, we must follow the history of each and include every heat transfer for each

subsystem in the detailed balance equation that expresses the thermal equilibrium

condition. For instance, if one such subsystem is an ice cube having mass M, and if that

ice cube completely melts during the approach to thermal equilibrium, then we must

include three heat transfers due to it:

(1) the heat transfer necessary to raise the temperature of the ice cube

from its initial value up to the freezing point of the subsystem

(2) the heat transfer necessary to change the subsystem completely

from solid ice at its freezing point into liquid water at its freezing

point

(3) the heat transfer necessary to raise the temperature of the liquid

water that used to be the ice cube (still with mass M but now

having the specific heat of liquid water) from the freezing point to

the equilibrium temperature

If we wish to consider a thermally isolated system in which n subsystems reach thermal

equilibrium, we must include all of the heat transfers that occur in achieving the thermal

detailed balance,

Qnet = Q1 + Q2 + Q3 + … + Qn = 0

Calorimetry is an experimental technique that allows us to make use of the detailed

balance condition for thermal equilibrium in order to measure material properties,

namely specific heats of samples. A calorimeter is the thermally insulated chamber in

which subsystems are placed in thermal contact for the purpose of performing a

calorimetry experiment. Really, a calorimeter is a lot like a thermos jug.

Practice the Concepts:

NB. Your instructor will recommend the amount of class time to spend on these questions

before you begin the experiment; you should finish them outside of class.

1) Subsystem 1 has a mass M1, a specific heat c1 and is initially at temperature T1i

when it is placed in thermal contact with Subsystem 2 having a mass M2, specific

heat c2 and initial temperature T2i. Assume that system comprised of the two

subsystems is thermally isolated from the rest of the universe and that neither

subsystem experiences a phase change as they reach thermal equilibrium with one

another. The system eventually reaches an equilibrium temperature of Teq. Derive

an algebraic expression for the specific heat, c2, of Subsystem 2 in terms of the

other quantities. [You will want to use a slightly more complicated version of this

result, valid for three different objects, in the Analysis part of the lab.]

49

2) A 25 gram sample of copper is initially at a temperature of 72ºC when it is added

to a 45 gram sample of water. The initial temperature of the water is 4 ºC. Assume

that the system is thermally isolated from the environment and determine its

equilibrium temperature.

3) Subsystem 1 has a mass M1, and is initially a solid having a specific heat c1s. It is

initially at temperature T1i < T1fp, where T1fp is the freezing point temperature of

Subsystem 1. When it is placed in thermal contact with Subsystem 2 having a

mass M2, specific heat c2 and initial temperature T2i, Subsystem 1 melts

completely into its liquid phase having a specific heat of c1l. The system

eventually reaches an equilibrium temperature of Teq. The specific latent heat of

fusion of Subsystem 1 is Lf1.Assume that the system comprised of the two

subsystems is thermally isolated from the rest of the universe. Derive an algebraic

expression for the equilibrium temperature, Teq, of the system.

4) 70 grams of ethyl alcohol and 30 grams of water, each initially at room

temperature (23ºC) are poured into a thermally isolated calorimeter with a 28

gram ice cube initially at 0 ºC. What fraction of the ice cube melts? What is the

equilibrium temperature of the contents of the calorimeter?

5) A 50 gram sample of lead at 42ºC, a 70 gram sample of copper at 81ºC, a 62 gram

sample of aluminum at 60ºC, and 125 gram sample of iron at 15ºC are all placed

into a thermally isolated calorimeter that contains 0.12 L of water initially at 1ºC.

Determine the equilibrium temperature of the contents of the calorimeter.

6) A 200 gram sample of ice initially at 0ºC is sealed in a thermally isolated

calorimeter with 30 grams of steam (water vapor) initially at 100ºC. What is the

equilibrium temperature of the system?

Procedure for Experiment:

This experiment requires some careful choreography in order to assure the efficient

performance of the essential measurements. Read all of the instructions carefully as a

group before starting the experiment. The text boxes are intended to emphasize the tasks

that are to be done, to the extent possible, simultaneously.

Determine the mass of the empty, dry aluminum cup of the calorimeter.

Determine the mass of the dry, aluminum sample.

Assemble the calorimeter and place a thermometer through a hole provided in the lid of

the calorimeter. There are two such holes in the lid of the calorimeter; one of them is for

the stirring mechanism.

50

Team members 1 and 2:

Remove the cork and pour 75 ml of cold water into the calorimeter. Then, replace the

cork to seal the calorimeter. Take the cold water from the cold water bath in the sink at

the front of the lab. Make sure that you do not include any ice with your water sample!

Obtain the cold water thermal drift data. This is the first 180 seconds worth of

temperature versus time data. We must do this to set a baseline against which to

accurately determine the actual temperature change that occurs when the aluminum

sample and the initially cold bath come to thermal equilibrium. This data allows us to use

a graphical technique to “subtract” out the background warming due to imperfections in

the thermal insulation of the calorimeter. In doing this step you will start to fill out the

pre-fabricated temperature time series data table. Continuously agitate the contents of

the calorimeter while taking all temperature time series data. Mixing allows the system

to more quickly and thoroughly attain thermal equilibrium.

Team member 3: Immerse the aluminum sample in the double Styrofoam cup containing newly poured hot

water. The hot water is stored in a coffee urn somewhere in the lab, probably on the front

table. Cover the cup with a plastic lid. Insert through the holes in the plastic lid a

thermometer and a plastic stirring rod. Use the stirring rod to agitate the contents so that

the aluminum sample will come to thermal equilibrium with the hot water. You should

start this step when your lab partners fill the calorimeter with cold water. This, aided by

constant agitation with the stirring rod, assures that the sample has time to reach

equilibrium with the hot water. This equilibrium is what sets the initial (high) temperature

of the sample, TH.

Team member 3: As the 180 second (3 minute) mark approaches, read the temperature, TH, of the hot bath

containing the aluminum sample. Remove the sample from the hot water (being careful

not to burn yourself). Quickly drop the sample onto a paper towel, roughly dry the

sample, remove the cork in the lid of the calorimeter, deposit the warm sample in the

calorimeter and replace the cork, again sealing the calorimeter.

Team members 1 and 2:

Once the sample has been added, it is especially important that you continue to agitate

the contents of the calorimeter in order to keep the temperature of the water as uniform

as possible!!! Continue to obtain temperature time series data in 20 second intervals.

Use your judgment as to when the temperature of the mixture has sufficiently settled into

the equilibrium thermal drift pattern, though you should take at least 500 seconds worth

of data.

Determine the mass of the aluminum cup of the calorimeter with the sample and the

water. Along with your dry mass measurements, this will allow you to determine the

mass of the water in the calorimeter.

51

Procedure for Analysis:

There are three important temperatures that must be determined: TC, TH, and Teq. The

most common error in this lab is using incorrect values for these temperatures.

• TC is the temperature of the water in the calorimeter just before adding the aluminum sample (at t = 180 seconds in most cases, but it may be different if you

added your sample before taking the t = 180 s measurement or if you added it

sometime after t = 200 s). Note that TC is not the temperature at t = 0 s since we

are interested only in the temperature rise due to the sample being added, not the

rise due to heat leaking into the calorimeter from the outside.

• TH is the temperature of the sample just before adding it to the calorimeter. This should be the temperature of the hot water that the sample was immersed in, with

that temperature taken just before the sample was removed from the hot water.

• Teq is the temperature of the water in the calorimeter after the sample has been added and the system has come to equilibrium. If the calorimeter provided

perfect isolation from the environment, this would be straightforward to measure.

However, because the temperature of the system continues to drift up due to heat

leaking into the calorimeter from the outside, we will use the graphical procedure

discussed below to try to isolate the temperature rise due to the sample being

added from the temperature rise due to the imperfection of the calorimeter.

Carefully graph the temperature inside the calorimeter versus time. You should clearly

see the trends that represent the thermal drifts before the sample is added and after the

system reaches thermal equilibrium. These two gently sloping thermal drift trends are

connected by a decidedly steeper transient behavior that occurs while the sample and the

water seek thermal equilibrium. Extrapolate the equilibrium drift portion of the curve

back to t = 180 seconds (the time just before the sample was placed in the calorimeter). A

best fit line based on the thermal drift after the sample and the cold water reach

equilibrium should be sufficient for doing this extrapolation. Read the temperature value

of the extrapolated equilibrium temperature trend line at t = 180 sec. Use this value for

the equilibrium temperature of the contents of the calorimeter. This is a quick and dirty

way of subtracting the background effects due to thermal drift (which, in turn, is due to

unwanted yet unavoidable heat transfer into the system through the calorimeter).

Draw another “plausible” fit trend line through the equilibrium thermal drift data points,

guided, of course, by the error bars on those points. Draw the second line in such a way

that it produces the largest plausible overestimate of Teq at 180 seconds based on your

data. Read this overestimate from your graph and label it (Teq)max. Compute and record

the experimental uncertainty in δTeq=(Teq)max - Teq.

Use your data to compute the specific heat of your aluminum sample and the uncertainty

in its specific heat as discussed further under “Analysis”. Display your final result in

proper form, cAl±δcAl .

52

Data:

Temperature Time Series Data

Time

(sec)

Temperature

(ºC)

Time

(sec)

Temperature

(ºC)

0 320

20 340

40 360

60 380

80 400

100 420

120 440

140 460

160 480

180 500

200 520

220 540

240 560

260 580

280 600

300 620

Uncertainty in time readings (reaction time) δt = ±_______________

Uncertainty in Temperature readings (least count) δT = ±_______________

MAl cup = _________________±__________

MAl sample = _________________±__________

TC = _________________±__________

TH = _________________±__________

Teq = _________________±__________

MAl cup+sample+water = _________________±__________

Mwater = _________________±__________

cwater = (4190±10) J/(kg·K)

Result: cAl =________±_________

53

Analysis:

Using the quantities you measured in the lab, compute the central value, cAl, for the

specific heat of your aluminum sample. Show all of the steps of your calculation. (This

means that it is not sufficient to simply type it all into the big-screen calculator. Even if

that is, in fact, what you do, write down a hard copy record in the space provided of the

steps you follow.)

Note that there are three objects to consider:

(1) The aluminum sample, whose temperature is changing from TH to Teq. (2) The water, whose temperature is changing from TC to Teq. (3) The aluminum cup, whose temperature is also changing from TC to Teq.

The specific heat of the aluminum sample and cup are both unknown, but since they are

made out of the same material you can assume that their specific heats have the same

value (which we are calling cAl).

54

Using the quantities you measured in the lab and the associated uncertainties in them,

compute the overestimated value, (cAl)max, for the specific heat of your sample. Then, use

the overestimate and the central value to determine the uncertainty, δcAl, in your result for the specific heat. Show all of the steps of your calculation. (This means that it is not

sufficient to simply type it all into the big-screen calculator. Even if that is, in fact, what

you do, write down a hard copy record in the space provided of the steps you follow.)

55



______________________________________________

Heat Transfer Questions

1) A silver rod having a length of 83 cm and a cross-sectional diameter of 2.4 cm is

used to conduct heat from a reservoir at a temperature of 540ºC into an otherwise

completely thermally insulated chamber that contains 1.43 kg of ice at 0ºC. How

much time is required for the ice to melt completely?

2) The surface temperature of the Sun is about 5800 K. Estimate the rate at which

the Sun radiates heat into space assuming that the emissivity of the Sun is 1.

Note: The radius of the sun is rsun ≈ 6.96 × 108 m.

3) A small, spherical glass bead having a radius of 1.5 mm is removed from a blast

furnace at a temperature of 195ºC and is placed in an environment at room

temperature, 27ºC. The net amount of heat radiated from the bead in the first 5

seconds after removing it from the furnace is 86 mJ. Estimate the emissivity of

the bead.

Silver

540ºC ice

56

4) Parallel Heat Flow: Express the steady-state rate of heat conduction from the

reservoir at temperature TH to the reservoir at temperature TC algebraically in

terms of quantities labeled in the figure. (Do not be fooled by the graphics: Path 2

has a uniform cross-sectional area.)

5) Series Heat Flow: Express the steady-state rate of heat conduction from the

reservoir at temperature TH to the reservoir at temperature TC algebraically in

terms of quantities labeled in the figure. (Note: The temperature TM at the

interface between the conductors will be useful in setting up your equations;

however, its value is not independently under your control and thus your end

result for the rate of heat conduction should not depend on it.)

TH TC

k1, L1, A1

k2, L2, A2

k1

L1 L2

k2 A1 A2

TH TC TM

57



______________________________________________

Graphing - Homework

The variables D and t (time) are related by the function KtoDD-e= , where oD and K are

constants. oD is the initial value of D, that is, the value of D at time t = 0.

What are the units of K?

The quantity )1( K is called the time constant. The time constant refers to the time

required for the value of D to decrease to 37% of its initial value. Thus, at time t = 1/K,

the variable D is given by

)368.0(=

e=

e=

1-

)(1/-

o

o

KK

o

DD

DD

DD

In other words, D has lost 63% of its initial value.

A set of data for D versus t is shown in the table below.

t (s) D (ºC)

0.0 20.0

0.5 16.7

1.0 13.9

1.5 11.6

2.0 9.7

2.5 8.1

3.0 6.7

3.5 5.6

4.0 4.7

4.5 3.9

5.0 3.2

5.5 2.7

6.0 2.3

6.5 1.9

7.0 1.6

7.5 1.3

8.0 1.1

58

1. What is the value of oD ?

2. Plot D versus t on linear graph paper. Part of the grade is reserved for good graphing techniques.

3. Fit the data with a curve (not a line!) and from that curve graphically determine the time constant (1/K). Clearly mark this time on your graph! Record the value

below.

4. Calculate the value of the K.

5. If the value of K was larger than you calculated, how would your graph change? Be specific.

6. What would be different about the graph if oD is increased? Be specific.

7. Now plot by hand D versus t on semi-log graph paper. When plotting on semi-log graph paper you still plot the original data (D on the logarithm axis and t on the linear

axis). What type of graph results on the semi-log graph paper in this instance; in other

words what type of curve fits the data best?

With semi-log graph paper the “y” axis is logarithmic (to the base 10) while the “x”

axis is still linear. Notice that the divisions on the “y” scale are not uniformly spaced.

The graph you obtain would be the same if you calculated the logarithm of the “y”

data and plotted that against “x” on linear graph paper.

61



______________________________________________

Newton’s “Law” of Cooling

Note: Special care should be taken when dealing with liquids that could be spilled

on or near electrical outlets or with liquids that are at high temperatures.

This experiment will explore the variables that affect the transfer of heat between an

object and its surroundings.

INTRODUCTION AND THEORY An object (or a liquid) at a temperature T in thermal contact with the room air at

temperature TR experiences a transfer of heat to its surroundings as long as T > TR. Heat

is transferred from the hotter object to the cooler object until thermal equilibrium

between the object and the room is established; in other words, until T = TR.

The rate at which the temperature of an object changes is governed by Newton’s Law of

Cooling. This law states that the rate at which an object cools is directly proportional to

the temperature difference between the object and its surroundings. Thus, the larger the

temperature difference with the surroundings, the faster the cooling rate will be. If we

define D as the temperature difference between the object and the surroundings to be D,

i.e.,

RTTD −= , (1)

then the rate of change in D during an interval of time t∆ is given by

DKt

D−=

∆

∆ (2)

where K is the cooling constant. The cooling constant depends on a number of factors,

such as the insulating properties of the container, the amount of substance that is being

cooled, etc., but it is constant for any one experimental situation. Therefore, the value of

K determines the rate at which the temperature difference decreases for a given

experimental set-up. Increasing the value of K causes the temperature difference to

decrease more rapidly. Thus, a poorly insulated container will result in a larger

value of K.

As time goes on, the difference in temperature between the surroundings and the object,

D, becomes smaller; that is, that is, 0→D as ∞→t . Thus, using calculus, Eq. (2) can be shown to have the solution

62

Kt

oDD-e= (3)

where t is time and 0D is the initial )0( =t temperature difference between the object

and its surroundings. Dividing both sides of Eq. (3) by 0D and then taking the natural

logarithm we find

tKD

D−=

0

ln . (4)

Finally, solving for K we have

Kt

D

D

−=

0

ln

(5)

which will allow us to find K. Note that “ln” represents the natural log (that is, log to the

base e).

PREDICTION

1. Imagine two cups of water, one at 30°C and one at 40°C. The cups are identical and

room temperature is 23°C. If the rate at which the object cools depends upon its temperature difference from its surroundings, which has cooled more after one

second? Which is hotter after one second? Assume that K is 0.5 s-1

. Support your

answer with numerical calculations (HINT: use Equation 3).

2. Given a glass beaker and a Styrofoam cup containing the same amount of warm water

at the same temperature, which do you expect will give a higher value of K? Justify

your response using correct physics principles.

63

MEASUREMENT You will make measurements to determine which factor(s) have the greatest effect on the

rate of heat transfer between a warm fluid and the surrounding air. Everyone will use the

open glass beaker; your second measurement will be determined by your instructor. From

the data collected by the class, you will draw conclusions about the factor(s) which have

the greatest effect on the rate of heat transfer. Factors that may be relevant are:

a) A shiny container which is open at the top (i.e., aluminum foil wrapped around the glass beaker).

b) A container with better side insulation (a Styrofoam cup). c) A larger amount of liquid in a glass beaker. d) A cover over a glass beaker. e) Other? Include a description.

Describe which factor you are assigned and the heat transfer mechanism(s) you expect

this factor to change.

Open the Student Shares folder on the desktop, navigate to the LoggerPro file “2xx

College Physics Students\212 College Physics II\LoggerPro\Cooling Activity.cmbl”, and

copy this file to the desktop. Plug the temperature probes into Ch1 and Ch2 on the

LabPro unit and open the file Cooling Activity.cmbl experiment file.

Measure and record the room air temperature using a digital thermometer. Take a number

of readings in the vicinity of your experiment and calculate the average value.

Troom =

Fill the glass beaker and the “other” cup with hot water from the faucet. Make sure that

the water is as hot as possible. Unless otherwise directed, both containers should have the

same amount of water (100 mL). Place temperature probe 1 (Ch 1) in the glass beaker

and temperature probe 2 in the “other” cup. After the temperature probes have had a

chance to come to equilibrium with the water in each container (i.e., the temperature is no

longer rising) begin to Collect data. After beginning data collection, adjust the additional

columns as directed below while the data is being acquired. Do not disturb the

temperature probes once data collection has begun.

The initial temperature for each container minus the room temperature is 0D for that

container. Double click on the column header D Glass. Replace the zero in the Equation

box with the average room temperature. Click Done and repeat the procedure for the

“other” cup.

Double click on the column header ln DGlass. Replace the zero in the Equation box with

64

the value for 0DG (the first numerical value in the D Glass column). Click Done and

repeat the procedure for the “other” cup.

Use the plot to find the cooling constant K for the beaker and for the “other” cup. Find

the slope in each case by doing a linear fit on the two plots. Make sure that you select

both data sets when the Select Columns dialog box comes up. Print out the graph using

the Print Graph option and attach it to this report. Make sure the curve fitting parameters

are included. Write the results below in “proper form”.

Kglass =

Kother =

Which cooled faster, the hot water in the glass beaker or the water in the “other” cup?

From the results obtained by the rest of the class, what appears to be the dominant heat

transfer mechanism(s) in the cooling process? Justify your conclusion.

65



______________________________________________

Infrared Radiation

1) A shiny stainless steel cup has a small piece of black adhesive tape attached to the outer surface. Does the shiny portion of the cup have a higher, lower, or same

emissivity than the black tape? Explain your answer.

2) Suppose that the temperature of the cup’s surface is measured as 89ºC everywhere when probed directly using a digital thermometer. Will the temperature of the black

tape, as measured using an infrared thermometer, also be 89ºC? If it is different,

is it likely to be very different? Explain your answer. Assume the black tape is a

perfect absorber. NOTE: The infrared thermometer is preset to measure the

temperature of objects with an assumed emissivity of 0.95 and in a room

temperature environment.

66

3) Will the temperature of the shiny cup, as determined by the infrared thermometer, be close to 89 ºC or significantly different? Explain your answer.

4) An infrared thermometer reads 32ºC when pointed at the shiny portion of the cup. What is the emissivity of the shiny cup? Assume room temperature is 20

oC.

67



______________________________________________

Simple Harmonic Motion

You will study mechanical systems that perform simple harmonic motion.

Simple harmonic motion (SHM) occurs when a system oscillates around a point of stable

equilibrium under the influence of a net force that is proportional to the displacement

from the equilibrium position and oppositely directed. Application of Newton’s Second

Law to such a system results in an acceleration (a) given by

( )22 2a ω x f xπ= − = − , (1)

where x is the displacement from the equilibrium position, and ω is a constant (the

angular frequency) that depends only on the physical properties of the oscillating

system.

The position as a function of time for a system undergoing SHM is

cosx A ω t= , (2)

where A is the amplitude (maximum magnitude of the displacement) of the oscillation.

Equation (2) is appropriate when at time 0t = the system is passing through the maximum displacement position, and is instantaneously at rest.

The period (T) of the oscillation is the time required to complete one oscillation. For

SHM, the period also depends only on the mechanical properties of the system:

2 1

Tω f

π= = . (4)

Example: The displacement of an oscillating system is found to be

rad

(14.7 cm) cos (5.11 )s

x t

= . (5)

(a) Is the system executing simple harmonic motion? Justify your answer.

68

(b) What is the numerical value of the amplitude of the motion?

(c) What is the numerical value of the period of the motion? Show all work.

PREDICTION: MASS/SPRING SYSTEM

A mass m is attached to a spring that has a spring constant k. The mass is displaced from

the equilibrium position and released from rest at time t = 0. Application of Newton’s

Second Law results in

xkam −= (6)

Write ω in terms of physical properties of the system (in this case, m and k). HINT:

Compare Equation (6) with Equation (1).

Write the period in terms of the physical properties of the system (mass and spring

constant).

69

1. If the amount of suspended mass is increased, do you expect the period to increase,

decrease or remain the same? Explain your answer.

2. If the amplitude of the motion is increased, do you expect the period to increase,

decrease or remain the same? Explain your answer.

70

3. A particular system has an equilibrium position at 0y = , and the +y direction is

vertically upward. The mass is pulled down from its equilibrium position and

released from rest at time 0t = . Sketch the displacement of the mass as a function of time on the grid below. Include at least two complete cycles.

Po

siti

on

(y)

Time

4. Using the graph of displacement versus time above, sketch the corresponding

velocity versus time graph on the grid below. Include at least two complete cycles.

Velo

cit

y (

v)

Time

5. Using the letters A, B and C, label three consecutive times on your velocity-vs.-time

graph when the mass is passing through its equilibrium position.

71

ACTIVITY: MASS/SPRING SYSTEM

You will now measure the period of oscillation of a vibrating mass/spring system

and experimentally determine the dependence of the period on the amount of suspended

mass and the amplitude of the oscillation.

1. Determine the period of oscillation of the coiled brass spring with 250 g suspended

from it. NOTE: The brass holder has a mass of 50 g. Measure and record the

amplitude. Include all measurements and their uncertainty. Also include a brief

description of the experimental technique. NOTE: Making one measurement of the

time to complete one cycle is terrible experimental technique.

2. Experimentally determine whether the period depends on the amount of suspended

mass. Show all data and calculations below to support your conclusion. Include a

brief description of the experimental design you used to answer this question.

Are your results consistent with your prediction (Prediction 1)? Explain any

discrepancies.

72

3. Experimentally determine whether the period depends on the amplitude of the

oscillation. Show all data and calculations below to support your conclusion.

Include a brief description of the experimental design you used to answer this

question.

Are your results consistent with your prediction (Prediction 2)? Explain any

discrepancies.

73



______________________________________________

Forces and Energy in Simple Harmonic Motion A mass m is attached to a spring. The spring is stretched and then released from rest at

time t = 0. The diagram below shows the position of the mass at equal intervals of time.

Frame A corresponds to t = 0. The magnitude of the displacement in frames A and E are

equal. Answer the questions on the following pages of this handout using this diagram.

m

m

m

m

m

m

m

A

B

C

D

E

F

G

x = 0

74

1. Clearly indicate on the diagram the force, acceleration, and velocity vectors in each

frame. Write 0=F , 0=a , and v = 0 to indicate zero magnitudes.

2. In what frames is the magnitude of the displacement maximum? Explain your

answer.

3. In what frames is the magnitude of the force maximum? Explain your answer.

4. In what frames is the magnitude of the acceleration maximum? Explain your answer.

5. In what frames is the magnitude of the velocity maximum? Explain your answer.

6. What multiple or fraction of a period is represented by frames A through E? Explain

your answer.

75

7. The displacement is written as a function of time as

( ) ( )cos cos 2x A t A f tω π= = . Write the velocity as a function of time:

8. Using your expressions for x and v as functions of time, write the kinetic energy K,

elastic potential energy U, and the total mechanical energy E as functions of time.

9. Show that the total mechanical energy is constant.

10. In what frames is the potential energy zero? Explain your answer.

76

11. In what frames is the potential energy maximum? Explain your answer.

12. Rank the frames in order of decreasing elastic potential energy. Clearly indicate any

frames where the potential energy is the same. Justify your answer.

13. In what frames is the kinetic energy zero? Explain your answer.

14. In what frames is the kinetic energy maximum? Explain your answer.

15. Rank the frames in o

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