Torque changes angular momentum according to

$$\boldsymbol\tau_{\rm ext}=\frac{d\mathbf L}{dt}.$$

Integrating over a time interval from $t_i$ to $t_f$ gives

$$\int_{t_i}^{t_f}\boldsymbol\tau_{\rm ext},dt =\mathbf L_f-\mathbf L_i.$$

The integral

$$\boxed{\mathbf J_{\rm ang}=\int_{t_i}^{t_f}\boldsymbol\tau_{\rm ext},dt}$$

is the angular impulse, so

$$\boxed{\mathbf J_{\rm ang}=\Delta\mathbf L}.$$

Constant and average torque

If the torque is constant over the interval,

$$\mathbf J_{\rm ang}=\boldsymbol\tau,\Delta t.$$

More generally, define an average torque by

$$\boldsymbol\tau_{\rm avg} =\frac{1}{\Delta t}\int_{t_i}^{t_f}\boldsymbol\tau(t),dt.$$

Then

$$\Delta\mathbf L=\boldsymbol\tau_{\rm avg}\Delta t.$$

The angular impulse depends on the accumulated torque over time, not just on the largest instantaneous torque.

Fixed-axis form

For rotation about a fixed axis,

$$\Delta L_z=\int_{t_i}^{t_f}\tau_z,dt.$$

If the rotating system has constant moment of inertia $I$ about that axis,

$$L_z=I\omega,$$

so

$$I(\omega_f-\omega_i)=\int_{t_i}^{t_f}\tau_z,dt.$$

This can determine the change in angular speed without first solving the full time-dependent rotational motion.

Worked example

A flywheel has moment of inertia

$$I=2.0,\mathrm{kg,m^2}$$

and initially rotates at

$$\omega_i=1.0,\mathrm{rad/s}.$$

A constant torque of

$$\tau=3.0,\mathrm{N,m}$$

acts in the direction of rotation for

$$\Delta t=2.0,\mathrm s.$$

The angular impulse is

$$J_{\rm ang}=\tau\Delta t=(3.0)(2.0)=6.0,\mathrm{N,m,s}.$$

The initial angular momentum is

$$L_i=I\omega_i=(2.0)(1.0)=2.0,\mathrm{kg,m^2/s}.$$

Therefore

$$L_f=L_i+J_{\rm ang}=8.0,\mathrm{kg,m^2/s}.$$

Since $I$ is unchanged,

$$\omega_f=\frac{L_f}{I}=\frac{8.0}{2.0}=\boxed{4.0,\mathrm{rad/s}}.$$

Short, strong torques

Angular impulse is especially useful when torque is large and rapidly varying, as in impacts or brief actuator pulses. The detailed torque history may be complicated, but if its time integral is known, the total angular-momentum change follows directly.

Angular impulse is the time-integrated form of the torque-angular-momentum law: torque accumulated over time determines the total change in angular momentum.