Questions: Effective Potential in Central Force Motion

Question 1

For a particle moving under gravity with nonzero angular momentum, what happens to the radial motion as r → 0?

• A. The particle spirals into the center because gravity is attractive
• B. The centrifugal barrier L²/(2mr²) grows without bound, making small r energetically inaccessible
• C. The particle oscillates through the origin in harmonic motion
• D. Nothing special happens — radial motion continues unchanged toward smaller r

Answer: (B)

The centrifugal barrier term L²/(2mr²) grows as r → 0 (like 1/r²), which is stronger than the gravitational attraction −GMm/r. For any nonzero angular momentum, the effective potential diverges to +∞ at small r, creating an energetically forbidden region near the origin. This is why planets do not spiral into the Sun despite gravity being attractive: angular momentum prevents radial collapse. Only if L = 0 (purely radial motion) does the particle actually reach r = 0.

Question 2

A particle moving in a gravitational potential has total energy E = U_eff(r_min), where r_min is the location of the minimum of the effective potential. What kind of orbit does this describe?

• A. A hyperbolic orbit — the particle has enough energy to escape
• B. An elliptical orbit — the particle oscillates between two turning points
• C. A circular orbit — the particle moves at constant radius
• D. A parabolic orbit — the particle just barely escapes to infinity

Answer: (C)

The minimum of U_eff is the point where dU_eff/dr = 0 — where centrifugal and gravitational forces balance, so the net radial force is zero. If E exactly equals U_eff at this minimum, the horizontal energy line just touches the curve; there is only one radial position the particle can occupy, meaning r is constant. That is a circular orbit. For E slightly above the minimum, there are two turning points and the orbit is elliptical. For E ≥ 0, there is one turning point at large r and the orbit is hyperbolic or parabolic.

Question 3

The effective potential U_eff = U(r) + L²/(2mr²) represents a real physical force acting on the particle in addition to the central force.

• T. True
• F. False

Answer: False

The centrifugal barrier L²/(2mr²) is not a real force — it is a mathematical artifact of using conservation of angular momentum to eliminate the tangential velocity and reduce the problem to one radial dimension. The particle is subject only to the actual central force (e.g., gravity). The effective potential is a computational tool that encodes all r-dependent physics, not a new physical interaction.

Question 4

If angular momentum L doubles while total energy E stays the same, a gravitationally bound orbit will become more tightly bound (smaller semi-major axis).

• T. True
• F. False

Answer: False

Doubling L raises the centrifugal barrier everywhere, lifting the entire effective potential curve and shifting its minimum to a larger radius. If E is held fixed, the inner turning point moves outward (the barrier pushes the particle away from the center), resulting in a more extended orbit, not a more compact one. More angular momentum generally means a wider, less tightly bound orbit shape.

Question 5

Explain how the effective potential reduces a two-dimensional orbital problem to a one-dimensional problem, and what physical insight this reveals about orbit types.

Model answer: Conservation of angular momentum (L = mr²ω) lets us write the tangential kinetic energy ½mr²ω² as L²/(2mr²), a function of r alone. Adding this to U(r) gives U_eff(r) = U(r) + L²/(2mr²). The total energy equation becomes E = ½m(dr/dt)² + U_eff(r) — identical in form to a 1D particle in potential U_eff. Plotting U_eff versus r and drawing a horizontal line at energy E immediately reveals the orbit type: two intersections mean elliptical, one tangent at the minimum means circular, one intersection at large r means hyperbolic.

The key is that angular momentum conservation removes one degree of freedom exactly. The angular coordinate θ is hidden inside the constant L, so only r needs tracking. Orbit classification becomes a graphical exercise rather than a differential-equations problem — a major analytical payoff from identifying the right conserved quantity.