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Central Forces & Effective Potential (IPhO Mechanics)

IPhO mechanics lesson on central-force motion, conserved angular momentum, effective potential, turning points, and orbit stability.

  • International Physics Olympiad preparation

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Central-force problems (gravity, electrostatics, radial springs) look two-dimensional, but they collapse to one-dimensional radial motion in an effective potential. Once you can write E = 1/2 mr dot ² + V_eff(r) and sketch V_eff, you can read off turning points, bound motion, and circular-orbit stability with very little algebra.

Learning objectives

By the end, you should be able to:

  • IPHO-M-CEN-01: eliminate the angular coordinate using conserved angular momentum and obtain the radial energy equation;
  • IPHO-M-CEN-02: classify allowed radii, turning points and circular-orbit stability from V_eff(r); and
  • verify a result through dimensions and the head-on or circular-orbit limit.

The official syllabus includes angular momentum, energy, gravity, gravitational potential and Kepler laws. Effective potential is a Mini Physics organising method for applying that official content; the phrase itself is not an official topic label.

Effective potential, turning points, and stable circular orbitAn effective potential curve rises steeply at small radius, has a minimum at r zero, and approaches zero from below at large radius. A negative energy line intersects at two turning points. The minimum represents a stable circular orbit.rEnergyEr₋r₊r₀angular-momentum barrierstable circular orbitVeff(r)
Scroll diagram horizontally to read all labels.
Allowed radii satisfy E ≥ Veff. Intersections are turning points; the minimum is a stable circular orbit because dVeff/dr = 0 and d²Veff/dr² > 0.
Prerequisites (quick refresh)

1. Definitions

  • Central force: vector F(r) = F(r) r hat (always radial).
  • Central potential: V(r) with vector F = -dV/drr hat.
  • Angular momentum (planar polar coordinates): ℓ = m r² φ dot (conserved for central forces).
  • Effective potential:
    V_eff(r) = V(r) + ℓ²/2mr².
  • Turning point: radius where r dot = 0, so E = V_eff(r).
  • Circular orbit: constant r = r₀ (a stationary point of V_eff).
  • Stable circular orbit: V_eff has a local minimum at r₀.
  • Reduced mass: for two-body motion, the relative coordinate uses μ = m₁m₂/(m₁ + m₂).

2. Key ideas

  • Motion is planar for central forces, so use polar coordinates (r,φ) in that plane.
  • Two conserved quantities usually do most of the work:
    ℓ = mr²φ dot, E = 1/2 m(r dot ² + r²φ dot ²) + V(r).
  • Eliminating φ dot gives the 1D radial energy equation:
    E = 1/2 mr dot ² + V(r) + ℓ²/2mr² = 1/2 mr dot ² + V_eff(r).
  • Turning points and allowed regions are read from intersections of E with V_eff(r).
  • Circular orbits satisfy .dV_eff/dr|_r₀ = 0, and stability comes from the sign of .d²V_eff/dr²|_r₀.
Write these three lines early (then sketch V_eff)
  1. ℓ = mr²φ dot (conserved).
  2. E = 1/2 mr dot ² + V(r) + ℓ²/2mr².
  3. Turning points: r dot = 0 so E = V_eff(r).

3. Detailed explanation

3.1 Why the effective potential works

Start from

E = 1/2 m(r dot ² + r²φ dot ²) + V(r).

With ℓ = mr²φ dot conserved, substitute φ dot = ℓ/(mr²) to get

E = 1/2 mr dot ² + V(r) + ℓ²/2mr².

The extra term ℓ²/(2mr²) is the centrifugal barrier.

3.2 Turning points and orbit types

Allowed radii satisfy E-V_eff(r) nonnegative (since it equals 1/2 mr dot ²). If the allowed region is bounded between two turning points, the motion is radially bounded. If it extends to infinity, the motion is unbound (scattering or escape).

3.3 Circular orbits and stability

Circular motion corresponds to r = r₀ constant. In the effective potential picture, that means r₀ is a stationary point. A local minimum gives a restoring radial force and therefore stable small radial oscillations.

4. Common mistakes

  • Forgetting the centrifugal term ℓ²/(2mr²).
  • Using m instead of the reduced mass μ in two-body problems.
  • Declaring “stable” from dV_eff/dr = 0 without checking the second derivative.
  • Mixing up V(r) and V_eff(r): the latter is a radial-only tool.

5. Problem-solving tips

  1. Sketch V_eff(r) immediately. It guides the entire solution.
  2. Use E = V_eff to find turning points; do not differentiate unless asked for circular orbits.
  3. Check the limits: small ℓ (weak barrier) and large ℓ (strong barrier).
  4. For circular orbits, after finding r₀, compute v using ℓ = mr₀v as a consistency check.

6. Worked examples

1) Attractive inverse-square: circular orbit radius, speed, energy, stability

Let

V(r) = -k/r

for a particle of mass m with angular momentum ℓ.

Effective potential:

V_eff(r) = -k/r + ℓ²/2mr².

Circular orbit condition:

dV_eff/dr = k/r²-ℓ²/mr³ = 0 ⇒ r₀ = ℓ²/mk.

For a circular orbit, ℓ = mr₀v, so

v₀ = ℓ/mr₀ = k/ℓ.

Energy:

E = 1/2 mv₀²-k/r₀ = 1/2 m(k/ℓ)² - k/(ℓ²/(mk)) = -mk²/2ℓ².

Stability: compute the curvature:

d²V_eff/dr² = -2k/r³ + 3ℓ²/mr⁴.

Using ℓ²/(mr₀³) = k/r₀² gives

.d²V_eff/dr²|_r₀ = k/r₀³,

which is positive, so the circular orbit is stable to small radial perturbations.

2) Repulsive inverse-square: distance of closest approach (turning point)

Consider a particle of mass m in the repulsive potential

V(r) = k/r

with energy E and angular momentum ℓ.

The closest approach rₘᵢₙ occurs when r dot = 0:

E = k/rₘᵢₙ + ℓ²/2mrₘᵢₙ².

Multiply by rₘᵢₙ²:

Erₘᵢₙ² - krₘᵢₙ - ℓ²/2m = 0.

The positive root is:

rₘᵢₙ = (k + square root of (k² + 2Eℓ²/m))/2E.

Checks:

  • If ℓ = 0 (head-on), then rₘᵢₙ = k/E.
  • Increasing ℓ increases rₘᵢₙ due to the barrier term.

7. Extension

1) Power-law potentials: when is a circular orbit stable?

Let

V(r) = a rⁿ.

Then

V_eff(r) = a rⁿ + ℓ²/2mr².

Circular orbit condition:

dV_eff/dr = a n rⁿ⁻¹-ℓ²/mr³ = 0 ⇒ ℓ²/m = a n r₀ⁿ⁺².

A physical circular orbit with r₀ > 0 therefore requires an > 0.

Curvature:

d²V_eff/dr² = a n(n-1)rⁿ⁻² + 3ℓ²/mr⁴.

Substitute the circular-orbit relation:

.d²V_eff/dr²|_r₀ = a n(n-1)r₀ⁿ⁻² + 3 a n r₀ⁿ⁻² = a n(n + 2) r₀ⁿ⁻².

Given the existence condition an > 0, stability requires n > -2. The case n = -2 is marginal at this order.

8. Practice and evidence

Practice
  • For one potential, sketch V_eff(r) for two different ℓ values and identify allowed regions and turning points.
  • Do one problem where you must decide stability by checking whether V_eff has a minimum.
  • Treat IPHO-M-CEN-01/02 as passed only if your graph labels axes, energy, turning points and one limiting case.
  • Then continue via the IPhO Mechanics Hub.
Syllabus and review details

No official syllabus alignment is listed for this lesson.

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