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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,

    \[\overrightarrow{\mathbf{I}}=\int_{t_{i}}^{t_{f}} \overrightarrow{\mathbf{F}} d t=\int_{t_{i}}^{t_{f}} \frac{d \overrightarrow{\mathbf{p}}}{d t} d t=\Delta \overrightarrow{\mathbf{p}}=\overrightarrow{\mathbf{p}}_{f}-\overrightarrow{\mathbf{p}}_{i} \nonumber \]


    This page titled 19.5: Angular Impulse and Change in Angular Momentum is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Peter Dourmashkin (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.