Classical Mechanics - Conservative Force

Suppose that the gravitational force law between two massive objects were

F₁₂ = r̂₁₂ Gm₁m₂/r₁₂^(2+ɛ) 

where ɛ is a small positive number. Which of the following statements would be FALSE?

1. The total mechanical energy of the planet-Sun system would be conserved.
2. The angular momentum of a single planet moving about the Sun would be conserved.
3. The periods of planets in circular orbits would be proportional to the (3+ɛ)/2 power of their respective orbital radii.
4. A single planet could move in a stationary non circular elliptical orbit about the Sun.
5. A single planet could move in a stationary circular orbit about the Sun.

(GR9677 #23)

Solution:

(A) TRUE.
Gravitational force is a conservative force.
In conservative field, the total mechanical energy is conserved.

(B) TRUE
In conservative field, angular momentum, L is conserved.

(C) TRUE
F_g = F_c
GMm/r^(2+ɛ) = mrω²
GMm/r^(2+ɛ) = mr(2π/T)²
GM/r^(3+ɛ) = 4π²/T²
T² = 4π²r^(3+ɛ)/GM
T ∝ r^(3+ɛ)/2 

(D) FALSE
Central force = centripetal force (F_g = F_c) produces circular orbit.
Non central forces do not produce circular orbit.

(E) TRUE
See (D)

Answer: D

Notes:

Central force:

1. It is a force whose magnitude depends only on the distance between the object and the origin.
2. It is a conservative field, can be expressed as F = − ∇V (the negative gradient of a potential energy).
3. Gravitational force, Coulomb force, and Elastic Force (Harmonic Oscillator) are examples of central (conservative) forces.
4. In conservative field, the net work done by the force is zero, W = ∮_c F ∙ dr = 0 → the total mechanical energy is conserved.
5. Conservative force is irrotional (torque = 0), since curl ∇V or ∇ × ∇V = 0.
6. Torque, τ = dL/dT = 0 → angular momentum, L is conserved
7. Central force = centripetal force (F_g = F_c) produces circular orbit.

Thermal Physics - Conduction Electron

The Fermi temperature of Cu is about 80,000 K. Which of the following is most nearly equal to the average speed of a conduction electron in Cu? 

A. 2 × 10⁻² m/s
B. 2 m/s
C. 2 × 10² m/s
D. 2 × 10⁴ m/s
E. 2 × 10⁶ m/s

(GR9277 #23)

Solution:

T_F = 8 × 10⁴ K
m_e = 9.11 × 10⁻³¹ kg
k = 1.38 × 10⁻²³ J/K

Fermi energy = Kinetic energy



Answer: E

Condensed Matter - Conductor and Semiconductor

Which of the following statements concerning the electrical conductivities at a room temperature of a pure copper sample and a pure silicon sample is NOT true? 

1. The conductivity of the copper sample is many orders of magnitude greater than that of silicon sample.
2. If the temperature of the copper sample is increased, its conductivity will decrease.
3. If the temperature of the silicon sample is increased, its conductivity will increase.
4. The addition of an impurity in the copper sample always decreases its conductivity.
5. The addition of an impurity in the silicon sample always decreases its conductivity.

(GR8677 #23)

Solution:

A. TRUE.
Copper → good conductor
Silicon → semi conductor

B. TRUE.
Conductivity is the inverse of resistivity, 
Copper → good conductor → resistivity, → , decreases.

C. TRUE.
Silicon → semi conductor → resistivity, → , exponentially increases

D. TRUE.
For conductor, doping will decrease conductivity.

E. FALSE.
For semiconductor, doping increases the charge carrier density (electron or electron holes) in the conduction band → increasing conductivity

Answer: E

Classical Mechanics - Non Uniform Circular Motion

A particle constrained to move in a circle with 10-meter radius. At one instant, the particle’s speed is 10 meters per second and is increasing at a rate of 10 meters per second squared. The angle between the particle’s velocity and acceleration vector is

A. 0^o
B. 30^o
C. 45^o
D. 60^o
E. 90^o

(GR0177 #23)

Solution:



 = acceleration vector

The speed is increasing →   ∥ 

Given:
v = 10 m/s
r = 10 m
a_tan = 10 m/s²

a_c = v²/r = (10)²/(10) = 10 m/s²

tan θ = a_tan/a_c = 10/10 = 1 → θ = 45^o

Answer: C