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Physics · University

Central forces, angular momentum and effective potential

A central force is directed along the line joining the particle to a fixed centre and has magnitude depending only on radial distance in the usual model.

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Prerequisites

Vectors, conservation laws and derivatives.

Study notes

Central force

A central force is directed along the line joining the particle to a fixed centre and has magnitude depending only on radial distance in the usual model.

Conservation

Its torque about the centre vanishes, so angular momentum about that centre is conserved. The trajectory lies in a plane when angular momentum is nonzero.

Radial energy

For conserved angular momentum L, the effective radial potential is V(r)+L²/(2mr²). The second term describes the angular contribution.

Stability

A circular orbit corresponds to a stationary point of effective potential. A local minimum gives stability against small radial displacements in this model.

Worked examples

Why is torque about the centre zero for a central force?

The cross product r×F vanishes for collinear vectors.

Position and force are parallel or antiparallel

A 2 kg particle has tangential speed 3 m/s at radius 4 m. Find angular-momentum magnitude.

For tangential motion L=mrv=2×4×3.

24 kg m²/s

Check your understanding

  1. Which quantity is conserved about the centre under a central force?

    1. Angular momentum
    2. Radial velocity always
    3. Kinetic energy always
    4. Radius always

  2. For L=6 kg m²/s, m=2 kg and r=3 m, find L²/(2mr²).

  3. Explain how a radial turning point is identified in an effective-potential diagram.

  4. Which local feature suggests a stable circular orbit?

    1. Any local maximum
    2. A local minimum of effective potential
    3. Any point with negative potential
    4. Any zero of radius

Answers and marking guidance

Answers and marking notes
  1. Angular momentum

    Zero torque implies conservation of angular momentum, not constant radius.

  2. 1 J

    The angular contribution is 36/(2×2×9).

  3. At a turning point radial speed is zero, so total energy equals effective potential. Allowed radial regions have total energy at least as large as effective potential.

    Credit equality at the turning point and the inequality for allowed motion.

  4. A local minimum of effective potential

    Near a minimum, small radial displacements produce restoring behaviour.

Further reading and review references

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