Rotational motion (Class 11): torque, moment of inertia and rolling
Rotation follows the same pattern as straight-line motion, with torque in place of force and moment of inertia in place of mass. Roll a ring, a disc and a sphere down the same incline and see which one wins.
NCERT Class 11 Physics, Chapter 6 (Systems of Particles and Rotational Motion). Every lab below runs in 3D in your browser or on a phone, in English and Hindi.
Try the labs
Moment of inertia of different bodies, parallel/perpendicular axis · plan · Chapter 6
To find the moment of inertia of different bodies, verify the parallel-axis theorem and relate torque, moment of inertia and angular acceleration.
Rolling without slipping: ring, disc and sphere race · plan · Chapter 6
To study rolling motion down an incline and understand the role of moment of inertia and friction in rolling without slipping.
Conservation of angular momentum: rotating stool · plan · Chapter 6
To verify conservation of angular momentum and see how a change in moment of inertia changes angular velocity.
Torque and equilibrium of a rigid body (see-saw) · plan · Chapter 6
To study the equilibrium of a rigid body under coplanar forces: the conditions ΣF = 0 and Στ = 0, using a beam balanced on a pivot.
Centre of mass of a system and of a rigid body · plan · Chapter 6
To find the centre of mass of a system of particles and to show that it moves as if all the mass and all external forces were concentrated there.
Gyroscope precession · plan · Chapter 6
To understand the precession of a gyroscope: why a spinning wheel supported at one end does not fall but turns slowly about the vertical.
Ladder against a wall: friction and equilibrium · plan · Chapter 6
To apply the conditions of equilibrium (ΣF = 0, Στ = 0) to a ladder resting on a rough floor against a smooth wall.
Atwood machine with a massive pulley · plan · Chapter 6
To solve the Atwood machine with a pulley that has rotational inertia (not massless) using translational and rotational dynamics.
Torque and moment of inertia
Torque is force times the perpendicular distance from the axis, τ = r F sin θ. It produces angular acceleration, τ = Iα, where I = Σmr² depends on how the mass is spread about the axis.
The parallel axis theorem shifts the axis, I = Icm + Md², and the perpendicular axis theorem applies to flat bodies.
τ = r × Fτ = I αI = ΣmR²I = Icm + M d²
Angular momentum
L = Iω, and when no external torque acts it is conserved. A skater who pulls in her arms reduces I and spins faster. A gyroscope under gravity does not fall but precesses.
L = I ωτ = dL/dtωp = m g r / (I ω)
Rolling
A rolling body has both translational and rotational kinetic energy. On an incline the acceleration is a = g sin θ / (1 + I/mR²), so the body with the smaller I/mR² gets down first: sphere, then disc, then ring.
K = ½mv² + ½Iω²a = g sin θ / (1 + I/mR²)v = Rω (pure rolling)
Frequently asked questions
Which reaches the bottom first, a ring or a solid disc?
The disc. Its moment of inertia is ½MR² against MR² for the ring, so more of its energy goes into translation.
What are the conditions for equilibrium?
The net force and the net torque about any point must both be zero.
Where is the centre of mass of a uniform rod?
At its middle. For several bodies it is the mass-weighted average of their positions.
Why does a spinning top not fall?
Its angular momentum is large, so gravity's torque only changes the direction of L, which makes the top precess.