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.