# Rolling Motion

> Rolling motion for JEE Physics; the constraint v = R omega, kinetic energy split, acceleration down inclines and friction direction.

- Canonical URL: https://prepelephant.com/topics/jee/physics/rolling-motion
- Exam / course: JEE · 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: "Rolling Motion", PrepElephant, https://prepelephant.com/topics/jee/physics/rolling-motion

## Direct answer

A body rolls without slipping when the contact point is instantaneously at rest, imposing v_cm = R omega (and a_cm = R alpha). The friction involved is static and does no work in ideal rolling; the kinetic energy is KE = (1/2) M v^2 (1 + k^2/R^2), with k the radius of gyration, and the motion may be analysed about the centre of mass or about the instantaneous axis through the contact point.

## What you must remember

- Constraint of pure rolling: v_cm = R omega, a_cm = R alpha; the topmost point moves at 2 v_cm while the contact point is at rest.
- Rolling kinetic energy: KE = (1/2) M v^2 (1 + k^2/R^2); k^2 = 2R^2/5 for a solid sphere, R^2/2 for a disc, R^2 for a ring.
- Acceleration down an incline: a = g sin(theta)/(1 + k^2/R^2) — solid sphere (5/7) g sin(theta), disc (2/3) g sin(theta), hollow sphere (3/5) g sin(theta), ring (1/2) g sin(theta).
- The incline race depends only on k^2/R^2, never on mass or radius: sphere first, then disc, then hollow sphere, ring last.
- Friction acts up the incline on a body rolling down (it alone supplies the spin torque) and, being static, does no work — mechanical energy is conserved.
- Minimum mu for pure rolling down an incline: tan(theta)/(1 + R^2/k^2); below it, the body slips and rolls together.
- On a frictionless incline the body slides without rotating, with a = g sin(theta) — larger than any pure-rolling value.

## Common confusion

Friction's direction is the perpetual trap. Rolling down, gravity acts at the centre and cannot spin the body, so friction acting up the slope must supply the torque — students mark it downhill because friction "opposes motion". Equally unsettling, this friction does no work: the contact point is instantaneously at rest, so the force acts on a point with zero velocity. Only slipping friction dissipates energy.

## Exam-focused takeaway

JEE Main asks the energy split, the incline accelerations and the body ranking as numerical-value questions — dependable marks once v = R omega and the energy formula are automatic. JEE Advanced prefers the reasoning side: friction direction under applied forces, a spool or yo-yo pulled by a string at different angles, a slipping body settling into pure rolling, and the minimum-coefficient condition. The method is uniform: translation and rotation equations about the centre of mass plus the rolling constraint, solved together.

## Frequently asked questions

### Which body reaches the bottom of an incline first?

The one with the smallest k^2/R^2: a solid sphere beats a disc, which beats a hollow sphere, which beats a ring — mass and radius never enter.

### Why does friction point up the incline for a rolling body?

Gravity acting at the centre gives no torque about it; only up-slope friction can produce the torque that increases the angular speed.

### Is energy conserved while rolling down an incline?

Yes — static friction does no work in pure rolling, so the loss in potential energy equals the gain in translational plus rotational kinetic energy.

### What is the topmost point's speed on a rolling wheel?

Twice the centre's speed, since translation v_cm and rotation R omega = v_cm add in the same horizontal direction there.

### What happens on a perfectly frictionless incline?

The body slides with a = g sin(theta) without rotating, because no force can exert a torque about its centre of mass.

### What is the instantaneous axis of rotation?

The line through the contact point, about which the body is momentarily in pure rotation with omega = v_cm/R — often the quickest route to its kinetic energy.