What Is Newton's Second Law for Rotation (τ = Iα)?

Newton's second law for rotation states that the resultant external torque on a rigid body equals its moment of inertia multiplied by its angular acceleration: $\tau_{\text{net}}=I\alpha$. This is an HL-only concept in A.4 Rigid body mechanics.

The Reasoning and Mechanism

A torque produces a turning effect about a chosen axis. For a force $F$ applied at distance $r$ from that axis, $\tau=Fr\sin\theta$, where $\theta$ is the angle between the force and position vector. Only the perpendicular force component contributes; a force through the axis produces zero torque.

The sign of each torque depends on the selected convention, such as anticlockwise positive. The resultant torque is the algebraic sum of all external torques about that axis.

The moment of inertia $I$ measures resistance to angular acceleration. It depends on both mass and its distribution relative to the chosen axis, so moving mass farther out increases $I$. Its unit is $\text{kg m}^2$; torque is measured in $\text{N m}$.

Rearranging gives $\alpha=\frac{\tau_{\text{net}}}{I}$, so the same resultant torque creates less angular acceleration when $I$ is larger.

For example, a perpendicular force of $20\,\text{N}$ applied $0.30\,\text{m}$ from an axis gives $\tau=Fr=20(0.30)=6.0\,\text{N m}$. If , then .

If an opposing torque of $2.0\,\text{N m}$ also acts, however, the resultant torque is $4.0\,\text{N m}$ and $\alpha$ falls to $2.0\,\text{rad s}^{-2}$. This demonstrates why net torque matters.

A common misconception is that torque and force are interchangeable. They are not: torque also depends on lever arm, direction, and chosen axis.

Exam Technique

In an IB Calculate question, identify the axis, assign torque signs consistently, calculate $\tau_{\text{net}}$, and then apply $\tau_{\text{net}}=I\alpha$. Show equation, substitution, and answer with units. Never substitute one individual torque when several act, and remember that belongs to the stated axis.