Classical Mechanics - Rotational Motion

Questions 41-42

A cylinder with moment inertia 4 kgm² about a fixed axis initially rotates at 80 radians per second about this axis. A constant torque is applied to slow it down to 40 radians per second. The kinetic energy lost by the cylinder is

A. 80 J
B. 800 J
C. 4000 J
D. 9600 J
E. 19,200 J
(GR9277 #41) 

Solution:

Rotational Kinetic Energy:  



Answer: D

Classical Mechanics - Rotational Motion

Questions 41-42

If the cylinder takes 10 seconds to reach 40 radian per second, the magnitude of the applied torque is

A. 80 Nm
B. 40 Nm
C. 32 Nm
D. 16 Nm
E. 8 Nm
(GR9277 #42)
Solution:



Answer: D

Classical Mechanics - Lagrangian

If  ∂L/∂qn = 0, where L is the Lagrangian for a conservative system without constraints and qn is a generalized coordinate, then the generalized momentum pn  is

A. An ignorable coordinate
B. Constant
C. Undefined
D. Equal to  
E. Equal to the Hamiltonian for the system
(GR9277 #43)
Solution:

Lagrangian Equation of Motion for the generalized coordinate q:



Momentum,



For the generalized coordinate q:


If   

Answer: B

Classical Mechanics - Lagrangian

A particle of mass m on the Earth’s surface is confined to move on the parabolic curve y = ax², where y is up. Which of the following is a Lagrangian for the particle? 

A.

B. 

C. 

D.

E.
(GR9277 #44)
Solution:

Kinetic Energy, 

Potential Energy,

Lagrangian, 

Given the curve ax², 






Answer: A

Classical Mechanics - Energy

A ball is dropped from a height h. As it bounces off the floor, its speed is 80% of what it was just before it hit the floor. The ball will then rise to a height of most nearly

A. 0.94 h
B. 0.80 h
C. 0.75 h
D. 0.64 h
E. 0.50 h
(GR9277 #45)
Solution:

Conservation of energy before and when the ball hits the ground:




Conservation of energy when and after the ball hits the ground



Answer: D

Thermal Physic - Critical Isotherm

Questions 46-47.


Isotherms and coexistence curves are shown in the PV diagram above for a liquid-gas system. The dashed lines are the boundaries of the labeled regions. Which numbered curve is the critical isotherm? 

A. 1
B. 2
C. 3
D. 4
E. 5
(GR9277 #46)
Solution:

Critical isotherm (critical temperature) → dP/dV = 0
  • the derivative of the curve is zero
  • the tangent to the curve results in a horizontal line
  • the point where the vertical and horizontal dashed lines cross


Answer: B

Thermal Physics - Thermal Equilibrium

See Problem #46

In which region are the liquid and the vapor in equilibrium with each other?

A. A
B. B
C. C
D. D
E. E
(GR9277 #47)
Solution:

Region A: V small  → Liquid
Region E: P large, V small → Liquid
Region D and C: V large →  Gas
Phase equilibrium exists only if the temperature of isotherm lies below critical temperature.

Answer: B

Lab Methods - Uncertainty

The magnitude of the force F on an object can be determined by measuring both the mass m of an object and the magnitude of its acceleration a, where F = ma. Assume that these measurements are uncorrelated and normally distributed. if the standard deviations of the measurements of the mass and acceleration are and respectively, then is

A.

B.

C.

D.

E.
(GR9277 #48)

Solution:




Answer: C

Nuclear & Particle Physics - Muon

Two horizontal scintillation counters are located near the Earth’s surface. One is 3.0 meters directly above the other. Of the following, which is the largest scintillator resolving time that can be used to distinguish downward-going relativistic muons from upward-going relativistic muons using the relative time of the scintillator signals?

A. 1 picosecond
B. 1 nanosecond
C. 1 microsecond
D. 1 millisecond
E. 1 second
(GR9277 #49)
Solution:

Muons can travel with maximum speed near the speed of light  →  = 3 × 10m/s.
s = 3 m
t = s/v = 3/(3 × 108) = 10−8

Micro = 10−6
Nano = 10−9
Pico = 10−12

10−8 second is close to 1 nanosecond.

Answer: B

Quantum Mechanics - Wave Function

The state of a quantum mechanical system is described by a wave function. Consider two physical observables that have discrete eigenvalues: observable A with eigenvalues {α}, and observable B with eigenvalues {β}. Under what circumstances can all wave functions be expanded in a set of basis states, each of which is a simultaneous eigenfunction of both A and B?

A. Only if the values {α} and {β} are nondegenerate
B. Only if A and B commute
C. Only if A commutes with the Hamitonian of the system
D. Only if B commutes with the Hamiltonian of the system
E. Under all circumstances
(GR9277 #50)
Solution:

For two physical quantities to be simultaneously observable, their operator representations must commute, .

Answer: B


Proof:

Wave function,