Exam 3 F13 Phys 1220__________

your name

Each problem is of equal value.

You can skip four problems. You must work at least one of the two electromagnetism problems, 7 and 8. You must work at least one of the three problems: 3, 5, and 6. Finally, you must work at least one of problems 1, 2, and 4.

If you work more problems than required, we will count the best grades according to the above rules.

Tips for better exam grades :

Read all problems right away and ask questions as early as possible.

Make sure that you give at least a basic relevant equation or figure for each sub-problem.

Make use of the entire exam time.

Show your work for full credit. The answer ‘42’ only earns you any credit IF ‘42’ is the right answer. We reserve points for ‘steps in between’, figures, units, etc. If all you give us is a numeric answer you may receive only a C grade for the problem and in some cases even less.

No credit given for illegible handwriting or flawed logic in an argument.

All multiple choice questions may have more than one correct answer. For full credit, you need to mark all correct answers and mark no incorrect answer, and, where required, explain your reasoning.

1. ThermodynamicsA perfectly insulated beaker of negligible mass contains 1[kg] water at 85°C. The volume of the beaker is cylindrical with cross sectional area 6.24 [cm2] and height 10 [cm]. The system is being cooled to a final temperature of 35°C by adding ice (for details see b)).

a) Which statements about this overall process are true? Explain briefly your reasoning why a statement is true or false.

A- There is no energy exchange between the system and the environment.

B- The internal energy, U, of the system water and ice is constant as the water cools.

C- In order to calculate the total heat in b), four different terms have to be considered to address all heat terms.

D- There is no latent heat release involved in the process.

E- The volume of the beaker plays no role in the calculation in b).

F- Mechanical work is done on the ice.

G- The initially liquid water takes during the overall process an adiabatic path in the p-V diagram.

b) How many [kg] ice at T = -5°C must be added to make the system temperature Tfinal= +35°C?

2. First Law of ThermodynamicsAn engine takes 3.25 mole of an ideal He gas through the cycle shown in the figure. Note that the temperature of the gas does not change during process c-a.

Hints: CP = CV + R. Monatomic and diatomic gases have a different number of degrees of freedom at intermediate temperatures as those in this problem.1[Pa] = 1 [N/m2] is a unit of pressure.

a) Find the pressure at point a.b) Find the temperature at points a, b, and c.c) How much heat went in or out of the gas during segments a-b, b-c, and

c-a? In each case explain whether heat entered or left the gas.

3. Thermodynamics

a) You were assigned to watch a number of videos regarding thermodynamic processes that occur. Consider these two examples:

• evaporative freezing of water

Explain the experimental setup, the process through which the water cools, and the reasons why one can exclude other forms of cooling.

• making water boil by taking heat away

Explain the experimental setup and the process(es) through which the boiling occurs.

b) Draw for each experiment the path the water takes in the p-V and in the p-T diagram. Explain your reasoning!

4. Thermodynamics: conceptual

Two equal size boxes, A and B, contain ideal gases. An inserted thermometer shows TA= 50⁰C and TB= 10⁰C. This is all we know about the gas in the boxes.

Which of the following statements must be true? Could be true? Must be false?Explain your reasoning.

a) The pressure in A is higher than the pressure in B.

b) A and B do not contain the same type of gas.

c) The molecules in A have more average kinetic energy per molecule than those in B.

d) The average speed of the molecules in A is larger than that of the molecules in B.

e) If the molecules have the same mass, the density of the molecules in B is larger than the density in A. Density is defined as mass per volume.

f) The mean free path of the molecules could be the same in both containers.

5. Thermodynamics: conceptual

Compare the adiabatic and isothermal process with each other. Include a discussion of the first law, the total heat, and total work for each done on or by the working fluid in an adiabatic versus an isothermal compression.

6. Thermodynamics: Lab

Consider the lab about specific heat.

a) Explain how the experiment is set up and how measurements of specific heat are indirectly obtained. Explain how the experiment works.

b) Let the data for the measured temperature change in independently repeated measurements be 12, 8, 16 degrees centigrade, respectively, for a mass of 0.1±0.005[kg] in 0.2[kg] of water. Use error propagation to determine the error in the specific heat.

Hint: Note that it takes 1[cal] of heat to increase the temperature of 1[g] of water by 1[⁰C].

c) Discuss two leading errors of the experiment. One error should be a random error and the other one a systematic error. State clearly which error is random and which is systematic. What could one change in the experimental setup to minimize the systematic error?

7. Inductance and Inductors

Consider an ideal RL- circuit.

a) Derive from the Kirchhoff Rules the differential equation that governs the time dependence of the circuit.

b) Find the solution for that differential equation and determine the expression for the angular frequency, .

c) Discuss what kind of energy is stored and how and where it is stored. How does the energy behave as function of time?

8. Circuits with inductors

Find the reading in each ammeter and voltmeter

a) Just after switch S is closed (t=0)b) After S has been closed for a very long time ( t→∞).c) Draw a qualitative time diagram for the dependence of current and voltage

on time for t > 0 (the switch S is closed at t=0) and label all relevant time, current, and voltage values.

Master Equations – Physics 1220

∆ L=α L0∆T∧∆V=βV 0∆T

Q=mc∆T∧Q=±mL

( dQdt )cond

=kATH−TCL

∧( dQdt )rad

=Ae σ ∙ (T 4−T S4 )

p ∙V=n RT∧mtot=nM ,M=N Am

K trans=32kBT∧vrms=√ 3kBTm ∧λ=v ∙tmean=

V4 π √2 r2N

∧CV , diatomic=52R∧CP=C v+R

W=∫V 1

V 2

p dV∧∆U=Q−W

e= WQH

∧eCarnot=1−T C

T H

∆ S=∫1

2 dQT

∧S=k ∙ ln (w )

F= 14 π ε0

∙|q1|∙|q2|r2

∧E=F0q0

ΦE=∫ E ∙d A=Qencl .

ε0

U= 14 π ε0

∑i

q iq0r i

∧V=Uq0

∧E=−(i ∂V∂x + j ∂V∂ y

+ k ∂V∂ z )

C≡QV ab

∧C plate=ε0Ad

∧,∈series , 1Ceq

=∑i

1C i,∧,∈¿ ,C eq=∑

iCi

U=Q2

2C∧u=1

2ε E2

I=dQdt

=n|q|vd A∧J=nq vd

ρ=EJ∧V=IRwith R=ρ L

A

P=V ab ∙ I

Req=∑iRi (series ) 1

Req=∑

i

1R i

(¿)

Kirchhoff Rules∑ I=0 ( junctionrule ) ,∑ V=0 (looprule)

Capacitor charging q=Qf ∙ (1−e−tRC )∧i= ε

R∙ e

−tRC

F=q v× B and ΦB=∮ B ∙ d A=0 and F=I l × B( for straight wires)

B= μ04 π

∙ q v ×rr2

magnetic field created by movingcharge

B=μ04 π

∙ I d l× rr2

Biot−Savart field created by straight wire carrying current

B=μ0 I2πr

and FL

=μ0 I ∙ I

'

2 πrfor 2∥, straight wires

∮ B ∙ d l=μ0 I encl Amper e' sLaw

ε=∮ (v × B ) • dl motional emf

ε=−dΦB

dt=∮ E ∙ d l Faraday ' s Law

ε=−L didtwhere L=

NΦB

i

U=12L I 2∧u= B2

2μ

ω=√ 1LC

for LC circui ts∧ω '=√ 1LC

− R2

4 L2

I rms=I

√2∧i=I ∙ cos (ωt )V accordingly AC current

V R=IR ,V L=I X L ,V C=I X C ,V=IZ

Z=√R2+ (X L−XC )2=√R2+(ωL− 1ωC )

2

Pavg=12VI cos (ϕ )∧tan (ϕ )=

ωL− 1ωC

R

B=μ0ε 0 cE∧c= 1√μ0 ε0

∧v= 1√με

S=1μ0E× B∧I=Savg

Some constants:

R = 8.315 [J/molK]

1 [cal] = 4.186 [J]

α Al2.4 ∙10−5 [K−1]

cH2O= 4190 [J/(kg K)]

cice= 2100 [J/(kg K)]

LF,H2O= 3.34 105 [J/kg]

LV,H20 = 2.256 106 [J/kg]

Thermal conductivity: copper 385 [W/mK], steel 50.2 [W/mK]

Electron charge 1.6 10∙ -19 [C]

14 π ε0

=9∙109[ N m2

C2 ]0 = 4 10∙ -7 [Wb/Am]

Need to take your mind off the exam for a minute? Check these out:

The Annotated Thermometer:

110° F. (43.3° C.) Too hot to think. Canadians evaporate.

100° F.(37.8° C.) Arizonans turn on their swamp coolers. Eggs cook on sidewalks.

90° F. (32.2° C.) Californians turn on their air conditioners80° F. (26.7° C.) Californians go swimming. Minnesotans turn on their fans.

70° F. (21.1° C.) Lower end of comfort zone for Sun Belt Denizens.Canadians consider joining nudist colonies. Cat only comes in for food.

60° F. (15.6° C.) Californians put on sweaters (if they can find one in their wardrobe).

50° F. (10.0° C.) Miami residents turn on the heat. Wisconsinites plant gardens.

40° F. (4.4° C.) You can see your breath. Californians shiver uncontrollably.Minnesotans go swimming.

35° F. (1.7° C.) Italian cars don't start.

32° F. (0° C.) Water freezes. San Franciscans speak of L.A. favorably.

30° F.(-1.1° C.) You plan your vacation to Australia. Minnesotans put on T-shirts. Politicians begin to worry about the homeless. British cars don't start.

25° F. (-3.9° C.) Boston water freezes. Californians weep pitiably. Minnesotans eat ice cream.Canadians go swimming.

20° F. (-6.7° C.) You can hear your breath. Politicians begin to talk about the homeless.New York City water freezes. Miami residents plan vacation further South.

15° F. (-9.4° C.) French cars don't start. Cat insists on sleeping in your bed with you.

10° F. (-12.2° C.) Too cold to ski. You need jumper cables to get the car going.

5° F.(-15° C.) American cars don't start.

0° F.(-17.8° C.) Alaskans put on T-shirts. Too cold to skate.

-10° F.(-23.3° C.) German cars don't start. Eyes freeze shut when you blink.

-15° F. (-26.1° C.) You can cut your breath and use it to build an igloo. Arkansans stick tongue on metal objects. New York landlords turn on heat. Miami residents cease to exist.

-20° F.(-28.9° C.) Cat insists on sleeping in your pajamas with you. Politicians actually do something about the homeless. Minnesotans shovel snow off their roofs. Japanese cars don't start.

-25° F.(-31.7° C.) Too cold to think. You need jumper cables to get the driver going.

-30° F.(-34.4° C.) You plan a two week hot bath. The Mighty Monongahela freezes. Swedish cars don't start.

-40° F. (-40° C.) Californians disappear. Minnesotans button top button. Canadians put on sweaters.Your car helps you plan your trip South.

-50° F.(-45.6° C.) Congressional hot air freezes. The Green Bay Packers practice indoors.Alaskans close the bathroom window.