Newton's Second Law for Rotation

A net force changes an object's velocity; a net torque changes its angular velocity. This page puts a number on that. It is the rotational version of Newton's second law, and it is what lets you predict how a pulley, a drum or a door speeds up when something pushes it.

α = τ_net / I

For translation, a = F_net/m: more force, more acceleration; more mass, less. For rotation, swap in the partners: torque for force, rotational inertia for mass, angular acceleration for acceleration.

α = τ_net / I   or   τ_net = Iα

The angular acceleration has the direction of the net torque: a counterclockwise net torque speeds up a wheel turning counterclockwise, or slows one turning clockwise. With zero net torque, α = 0 and ω stays constant, which is the first law again.

Rotational inertia now shows why it matters. Apply the same torque to a hoop and a disk of the same mass and radius. The hoop's I = MR² is twice the disk's ½MR², so the hoop gets half the angular acceleration.

Two analyses for one system

Many problems have a part that moves and a part that turns. A block hangs from a string wound on a pulley with mass. The block falls; the pulley spins. To describe the whole thing you need one equation for each.

• The block (translation), down positive: mg − F_T = ma.
• The pulley (rotation): of the forces on it, only the string's pull has a lever arm, R. So F_TR = Iα.
• The link: the string does not slip, so a = Rα.

Two features of that answer are worth remembering. The tension (1.47 N) is less than the block's weight (1.96 N): the block speeds up downward, so the net force on it must be down. And the acceleration is less than g, because part of gravity's pull goes into spinning up the pulley.

Strings over a pulley with mass

With an ideal (massless) pulley, the tension is the same on both sides of the string. That stops being true once the pulley has mass. To give the pulley an angular acceleration, the two sides must pull with different tensions: (F_T1 − F_T2)R = Iα. The heavier block's side has the larger tension, but still less than that block's weight.

Reading data

If you apply several known torques to a wheel and measure α, then α = (1/I)τ says a graph of α against τ is a line through the origin with slope 1/I. If friction at the axle exerts a constant torque, the line keeps its slope but shifts: the wheel only starts to speed up once the applied torque exceeds the friction torque.