19.5: Angular Impulse and Change in Angular Momentum
- Page ID
- 24547
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If there is a total applied torque \(\vec{\tau}_{S}\) about a point \(S\) over an interval of time \(\Delta t=t_{f}-t_{i}\), then the torque applies an angular impulse about a point \(S\), given by
\[\overrightarrow{\mathbf{J}}_{S}=\int_{t_{i}}^{t_{f}} \vec{\tau}_{S} d t \nonumber \]
Because \(\vec{\tau}_{S}=d \overrightarrow{\mathbf{L}}_{S}^{\text {total }} / d t\) the angular impulse about \(S\) is equal to the change in angular momentum about \(S\),
\[\overrightarrow{\mathbf{J}}_{S}=\int_{t_{i}}^{t_{f}} \vec{\tau}_{S} d t=\int_{t_{i}}^{t_{f}} \frac{d \overrightarrow{\mathbf{L}}_{S}}{d t} d t=\Delta \overrightarrow{\mathbf{L}}_{S}=\overrightarrow{\mathbf{L}}_{S, f}-\overrightarrow{\mathbf{L}}_{S, i} \nonumber \]
This result is the rotational analog to linear impulse, which is equal to the change in momentum,
\[\overrightarrow{\mathbf{I}}=\int_{t_{i}}^{t_{f}} \overrightarrow{\mathbf{F}} d t=\int_{t_{i}}^{t_{f}} \frac{d \overrightarrow{\mathbf{p}}}{d t} d t=\Delta \overrightarrow{\mathbf{p}}=\overrightarrow{\mathbf{p}}_{f}-\overrightarrow{\mathbf{p}}_{i} \nonumber \]