What Is Angular Impulse?

Angular impulse is the change in an object's angular momentum produced when a resultant torque acts over a time interval. For a constant or average torque, $J_{\mathrm{angular}}=\tau\Delta t=\Delta L$. It is the rotational analogue of linear impulse.

The Reasoning and Mechanism

A torque changes rotational motion, just as a resultant force changes linear momentum. Newton's second law for rotation is

$$\tau=\frac{\Delta L}{\Delta t}.$$

Multiplying by the time interval gives $\tau\Delta t=\Delta L$. Therefore, angular impulse equals the final angular momentum minus the initial angular momentum:

$$J_{\mathrm{angular}}=L_f-L_i=\Delta(I\omega).$$

This is an HL-only concept in A.4 Rigid body mechanics. The general expression is $\Delta(I\omega)$, not automatically $I\Delta\omega$. You may use $I\Delta\omega$ only when the moment of inertia $I$ remains constant. If no resultant external torque acts, angular momentum is conserved and the angular impulse is zero.

For example, an average torque of $6.0\,\text{N m}$ acts for $0.40\,\text{s}$. Its angular impulse is

$$J_{\mathrm{angular}}=(6.0)(0.40)=2.4\,\text{N m s}.$$

The object's angular momentum therefore changes by $2.4\,\text{kg m}^2\text{s}^{-1}$. If it initially has $L_i=1.1\,\text{kg m}^2\text{s}^{-1}$ in the chosen positive direction, then . A torque in the opposite direction would give a negative change.

For a varying torque, divide the interval into narrow sections and add each $\tau\Delta t$. In the continuous limit,

$$J_{\mathrm{angular}}=\int_{t_i}^{t_f}\tau\,dt,$$

so angular impulse is the signed area under a torque versus time graph. Areas below the time axis reduce angular momentum according to the chosen sign convention.

Exam Technique

In a Calculate question, show the equation, substitution, and final value with a consistent sign. In a graph question, determine the signed area, not the gradient. For multistage problems, calculate $\Delta L$, then use $L_f=L_i+\Delta L$; if is constant, obtain .

Common misconception: angular impulse is an extra torque. It is not; it measures torque's cumulative effect on angular momentum.