# Angular Momentum and Its Conservation

> Angular momentum conservation for NEET Physics: L = Iω, torque, the skater effect, planetary motion and worked rotational numericals.

- Canonical URL: https://prepelephant.com/topics/neet-ug/physics/angular-momentum-conservation
- Exam / course: NEET-UG · Subject: Physics
- Publisher: PrepElephant (https://prepelephant.com) — Prepared and reviewed by the PrepElephant Academic Review Team
- First published: 2026-10-02
- Last updated: 2026-10-02
- How to cite: "Angular Momentum and Its Conservation", PrepElephant, https://prepelephant.com/topics/neet-ug/physics/angular-momentum-conservation

## Direct answer

An ice skater who pulls in her arms spins faster without any external push — the visible face of angular momentum conservation. For a rigid body L = Iω; for a particle L = mvr about the chosen axis, directed along the axis by the right-hand rule. Since τ = dL/dt, angular momentum stays constant whenever the net external torque about that axis is zero: conservation reads I1ω1 = I2ω2, so pulling mass toward the axis lowers I and raises ω, the kinetic energy changing only because the pulling muscles or gravity perform work. Gravity exerts no torque about the sun, which is why each planet conserves L and sweeps out equal areas in equal times.

## What you must remember

- **Two forms:** rigid body L = Iω; particle L = mvr sin θ about the axis; the rotational analogue of p = mv.
- **Rotational Newton's second law:** τ = dL/dt; if τ_ext = 0 about an axis, L about that axis is conserved even while I and ω individually change.
- **The skater effect:** arms pulled in decrease I, so ω increases by the inverse factor; kinetic energy (1/2) Iω^2 rises, supplied by muscular work done against the centrifugal tendency.
- **Planetary link:** gravity is central, so L = mvr is constant; the areal velocity dA/dt = L/2m, which is Kepler's second law in one line.
- **Kinetic energy bookkeeping:** with L fixed, KE = L^2/2I, so halving I doubles the energy — the examiner checks whether you know where that energy came from.
- **Bohr connection:** in the hydrogen atom the electron's angular momentum is quantised as mvr = nh/2π — the same quantity in a different chapter.
- **Units:** kg m^2 s^-1 for L, N m for τ; both are axial vectors.

## A platform problem from start to finish

A man stands on a freely rotating platform with his arms extended: I1 = 10 kg m^2 and ω1 = 6 rad s^-1. He pulls his arms in, dropping the moment of inertia to I2 = 4 kg m^2. No external torque acts about the vertical axis (the axle is frictionless in the idealisation), so L = I1ω1 = 60 kg m^2 s^-1 survives unchanged and ω2 = 60/4 = 15 rad s^-1 — two and a half times faster. The energy audit is the part NEET loves: kinetic energy goes from (1/2)(10)(36) = 180 J to (1/2)(4)(225) = 450 J, an increase of 270 J supplied by the work his muscles do pulling the masses inward. Nothing is violated; the same conservation that holds ω in check while I changes permits the energy to rise, because conservation applies to L, never to rotational kinetic energy on its own.

## Where the questions twist

The recurring trap is conserving the wrong quantity. When a droplet or a child lands on a merry-go-round, or an insect crawls on a rotating disc, angular momentum about the axle is conserved but the event is fully inelastic — rotational kinetic energy drops, and a question asking for the "energy lost" expects you to compute both states from (1/2) Iω^2. The second twist is the direction of change: a person walking outward on the platform increases I and therefore slows ω — candidates memorise "pull in, spin faster" and then misapply it to the opposite case. Third, the conservation axis matters: gravity exerts zero torque about the centre of a planet's orbit but not about an arbitrary point, so L is conserved about the sun specifically. Finally, the parallel-axis trap: if the platform plus man is given as I about the centre but the man walks along a tangent, compute the new I from Σmr^2 afresh rather than patching the old value.

## Frequently asked questions

### Under what condition is angular momentum conserved?

When the net external torque about the chosen axis is zero; internal torques, like internal forces, cancel in pairs.

### Why does a skater spin faster after pulling in the arms?

Pulling mass toward the axis lowers I, and constancy of L = Iω forces ω to rise in inverse proportion.

### Where does the extra rotational kinetic energy of the skater come from?

From muscular work done against the outward centrifugal tendency while pulling the arms in — L is conserved, energy is not.

### How does angular momentum conservation explain Kepler's second law?

The sun's gravitational force is central, so τ = 0 and mvr is constant; the areal velocity dA/dt = L/2m is therefore constant, giving equal areas in equal times.

### What happens to the angular speed when a mass is dropped onto a rotating disc?

The moment of inertia increases by mr^2, so ω falls by the factor I_old/(I_old + mr^2) while angular momentum stays fixed and kinetic energy decreases.