FINAL EXAM - Purdue University

Spring 2011PHYS 172: Modern Mechanics

Lecture 25 – Entropy Read 12.6 – 12.8

Exam 3 results: Multiple choice: 42.5/70 = 60.7%Handwritten: XXX

FINAL EXAM:1. Comprehensive2. About 20-25 multiple choice questions only.

If you have Final Exam conflict:1. Notify us immediately (before next Monday, Apr 25).

Indicate your exam schedule for Thursday, Friday and Saturday.2. Depending on your input, the makeup exam will be scheduled for Friday or

Saturday.

EXAM 3 REVIEW:hardest problems

3

Exam 3: Hardest problem, 39%A rocket is at rest (not moving) in outer space far away from any massive objects, i.e. the external net force on the rocket is negligible. The rocket turns on its engine throwing out burning gases of rocket fuel to the left, while accelerating the right. Consider a system consisting of the rocket, fuel and burning gases, and describe the direction of motion of the center of mass of this system.

39% A) The center of mass will not move12% B) The center of mass will accelerate to the right at the same rate as the rocket itself 44% C) The center of mass will accelerate to the right at rate that is smaller than that of rocket3% D) The center of mass will accelerate to the left1% E) The answer depends on the chemical energy released during burning

,CM

net ext

dPF

dt=

What will be the velocity of the center of mass vcm ?

LECTURE 16 clicker question: A system is initially at rest and consists of a man with a bottle sitting on ice (ignore friction). The man then throws the bottle away as shown.

Exam 3: Hardest problem, 39%A rocket is at rest (not moving) in outer space far away from any massive objects, i.e. the external net force on the rocket is negligible. The rocket turns on its engine throwing out burning gases of rocket fuel to the left, while accelerating the right. Consider a system consisting of the rocket, fuel and burning gases, and describe the direction of motion of the center of mass of this system.

39% A) The center of mass will not move12% B) The center of mass will accelerate to the right at the same rate as the rocket itself 44% C) The center of mass will accelerate to the right at rate that is smaller than that of rocket3% D) The center of mass will accelerate to the left1% E) The answer depends on the chemical energy released during burning

,CM

net ext

dPF

dt=

5

Exam 3: Hard problem, 42%

Two hockey pucks are pulled using attached weightless strings as shown using the same force. The string in case 2 is wrapped around the puck and can unwind freely. In case 1 it is attached to the center. Which puck will accelerate quicker?

42% A) They accelerate at the same rate46% B) Puck 110% C) Puck 2

1% D) Mass of the puck is needed to answer this question

,CM

net ext

dPF

dt=

6


A string is wrapped around a disk of mass m = 1.6 kg and radius R = 0.09 m. Starting from rest, you pull the string with a constant force F = 6 N along a nearly frictionless surface. At the instant when the center of the disk has moved a distance x = 0.10 m, a length d = 0.28 m of string has unwound off the disk.

26% A) Krot = F.d13% B) Krot = F.x12% C) Krot = F.d + F.x42% D) Krot = F.d - F.x

7


A sphere of mass m is attached to a fixed pivot by a weightless string. Initially, it moves around the circle of radius R with speed v. As it keeps circling the string is pulled into the sphere by a mechanism hidden within the sphere. What will be the linear speed vf of the sphere when it gets two times closer to the axle, i.e. moves around a circle of radius R/2?

A) vf = 2v 42%B) vf = 4v 29C) vf = v 10D) vf = v/2 12E) vf = v/4 7

vR

A AL r p= ×

AA net

dLr F

dt= ×

A A AL r p r mv const= ≈ =

8

Exam 3: Hard problem, 55%A perfectly elastic ball of mass 100 g bounces off the Earth. Which of the statements are correct:

1) The velocity (vector) of the ball experiences larger change during the collision than the velocity of Earth

(not to scale)

2) The momentum of the ball experiences larger change during the collision than the momentum of Earth

3) The position of the center of mass of the system ball+Earthfluctuates in the process

A: TRUE B: FALSE

A: TRUE B: FALSE

A: TRUE B: FALSE

9


A bullet of mass m is fired horizontally and at a distance d above the axis of a cylinder of mass M, radius R. The cylinder is initially at rest and is free to rotate on a massless axle through it's long axis. The bullet adheres to the cylindersurface as it hits it. The cylinder begins to rotate at an angular speed such that it completes one complete revolution in T after catching the bullet. Calculate the initial speed of the bullet prior to impact.

v

m M

initial final

R

1. Which fundamental principle will you use?A) Momentum principleB) Energy principleC) Angular momentum principleD) None of the above

d

10



2. Angular momentum of which system is conserved during collision?A) BulletB) CylinderC) Bullet + cylinderD) None of the above

AA net A

dLr F

dtτ= × =

Approximation: all other interactions during collision are negligible

v

m M

initial final

R

d

11



Initial angular momentum in respect to center axle:

A)B)C)D)

0L mvd= +

v

m M

initial final

R

d

0L mvR= +L mvd MvR= +

2 / 2L mvd Mv R= +

12


A bullet of mass m is fired horizontally and at a distance d above the axis of a cylinder of mass M, radius R. The cylinder is initially at rest and is free to rotate on a massless axle through it's long axis. The bullet adheres to the cylindersurface as it hits it. The cylinder begins to rotate at an angular speed such that it completes one complete revolution in T after catching the bullet. Calculate the initial speed of the bullet prior to impact (Idisk = MR2/2).

v

m M

initial

R

d

( ) 2

2M m R

L ω+

=

fv Rω=

L mvd=

2Tπω =

final

What is the final angular momentum?

2

2 f

MRL mv Rω= +

2

2 f

MRL mv dω= +A) B) C)

13


A bullet of mass m is fired horizontally and at a distance d above the axis of a cylinder of mass M, radius R. The cylinder is initially at rest and is free to rotate on a massless axle through it's long axis. The bullet adheres to the cylindersurface as it hits it. The cylinder begins to rotate at an angular speed such that it completes one complete revolution in T after catching the bullet. Calculate the initial speed of the bullet prior to impact (Idisk = MR2/2).

v

m M

initial

R

d

fv Rω=

2Tπω =

final

2

2 f

MRmvd mv Rω= +

222M R

mvd mT

π = +

Find v

The Boltzmann distribution

What is the probability to find an atom in a solid at certain energy level?

N=100 oscillators, q=4 quanta

Clicker:In how many ways can we distribute 4 quanta in

object “2”? A) 1 wayB) 4 waysC) 4!=1*2*3*4=24 ways

Separate system into 2 objects:Object 1: 99 atomsObject 2: 1atom

Probability to find n quanta on Object 2 is proportional to the number of microstates with (q-n) quanta in Object 1

( )( )

1 !! 1 !

q N

q N

+ −Ω =

−

1 2 1Ω = Ω Ω = ΩTotal number of microstates with n quanta in object 2:


What is the probability to find an atom in a solid at certain energy level?

N=100 oscillators, q=4 quanta


The probability of finding a microscopic system to be in a state with energy E above ground state

is proportional to

( )EkTE e

−Ω

See pages 394-395 for derivation

Boltzmann factor

Number of microstates corresponding to energy E

For large systems!


e-1 =

0.37

e-1 =

0.37

EkTe

−

Clicker:At very high temperature it is possible to have more oscillators in the first excited state than in ground state.

A) TrueB) False

The Boltzmann distribution: hydrogen at RT

Room temperature ~300 K:kT = 0.026 eV

10.2

eV

kT=0.026 eV

EkTe

−

Temperature at which kT = 10.2 eV : T=120,000 K


EkTe

−

Measure energy on a single oscillator many times. Number of measurements at energy E is proportional to:

EkTe

−

Measure energy on each oscillator on a system at once. Number of oscillators at energy E is proportional to:

Boltzmann distribution: distribution of oscillator energies in a large system

FINAL EXAM - Purdue University

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