# Angular impulse

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The angular impulse in physics is the product of the average of a briefly applied force F times the radius of a rigid body R times the brief time period delta t:

$\mathbf{F}\times{R}\times\delta{t}= I \times(\mathbf{\omega_f} - {\omega_i})$

where the final angular velocity is just after the impulse, and the initial angular velocity is just before the impulse.

## Example

Suppose a rotating cylinder of radius R2 is joined with a stationary cylinder of radius R1. Both feel an impulsive force until the continue rotating without slippage at different angular velocities. The above formula permits calculation of the respective angular velocities as follows:

$\mathbf{F_1}\times\delta{t_1}= -\mathbf{F_2}\times\delta{t_2}$

Thus:

$\frac{I_1\times(\omega_{f1} - \omega_{i1})}{R_1} = - \frac{I_2\times(\omega_{f2} - \omega_{i2})}{R_2}$

But:

ωi2 = 0

and:

$\omega_{f1} = \frac{\omega_{f2}\times{R_2}}{R_1}$

and thus ωf2 can be solved in terms of the initial angular velocity of the rotating cylinder ωi1.