Torque, rotational inertia and angular momentum

Key ideas

• Torque $\tau$ is the turning effect of a force — the rotational version of force. A net torque produces angular acceleration:

$\tau = I\alpha$

• Rotational inertia $I$ (also called the moment of inertia, unit kg m²) is the rotational version of mass — how hard an object is to spin up. It depends on how the mass is spread out from the axis: mass far from the axis gives a large $I$.
• Angular momentum $L$ is the rotational version of momentum:
  • for a rigid rotating body: $L = I\omega$
  • for a point mass moving in a circle: $L = mvr$
• Conservation of angular momentum — with no external torque, $L$ stays constant:

$I_i\omega_i = I_f\omega_f$

• Rotational kinetic energy:

$E_{K(ROT)} = \tfrac{1}{2}I\omega^2$

• A rolling object has both translational and rotational kinetic energy: $E_K = \tfrac{1}{2}mv^2 + \tfrac{1}{2}I\omega^2$.

The spinning skater

• When a skater pulls their arms in, they move mass closer to the axis, so $I$ decreases.
• With no external torque, $L = I\omega$ is conserved, so $\omega$ increases — they spin faster.
• Their rotational kinetic energy rises — the extra energy comes from the muscular work done pulling the arms in against the outward pull.

Tips

• Choose the right angular-momentum formula. Use $L = I\omega$ for an extended spinning body and $L = mvr$ for a single small mass moving in a circle — mixing them is a common error.
• For a rolling object, remember it stores energy two ways; using only $\tfrac{1}{2}mv^2$ leaves out the rotational share and gives the wrong speed at the bottom of a ramp.