Torque Makes Things Spin Up: τ = Iα

Net force decides linear acceleration. We now show that net torque decides angular acceleration, and meet the quantity that plays the part of mass: the moment of inertia.

Following one particle#

Take a rigid body turning about a fixed axis with angular velocity ω and angular acceleration α. A particle of mass m at distance r from the axis moves on a circle, so the net force on it has two parts:

• a radial part $m\omega^2 r$ toward the axis. Its line passes through the axis, so its torque about the axis is zero;
• a tangential part $m\,\dfrac{dv}{dt} = m r\alpha$, perpendicular to the radius. Its lever arm is r, so its torque is $mr^2\alpha$.

The torque of the net force on that particle is therefore $m r^2 \alpha$. Add this up over all particles (they all share α):

Internal torques cancel#

The total includes the forces particles exert on each other. By the third law these come in equal, opposite pairs, and each pair acts along the line joining the two particles. The two torques $\vec r_i\times\vec F_{ij} + \vec r_j\times\vec F_{ji} = (\vec r_i - \vec r_j)\times\vec F_{ij}$ vanish because $\vec r_i - \vec r_j$ is parallel to $\vec F_{ij}$. Only external torques survive.