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3.7: Angular Momentum

  • Page ID
    8379
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    Notation:

    • \(\textbf{L}_{C}\)= angular momentum of system with respect to centre of mass C.
    • \(\textbf{L}\) = angular momentum of system relative to some other origin O.
    • \(\overline{\textbf{r}}\) = position vector of C with respect to O.
    • \(\textbf{P}\) = linear momentum of system with respect to O.
    • (The linear momentum with respect to C is, of course, zero.)
    Theorem:

    \[ \textbf{L} = \textbf{L}_{C} + \overline{\textbf{r}} \times \textbf{P} \tag{3.7.1}\label{eq:3.7.1} \]

    Thus:

    \[ \begin{align*} \textbf{L} &= \sum \textbf{r}_{i}\times \textbf{p}_{i} = \sum m_{i}(\textbf{r}_{i}\times \textbf{v}_{i}) = \sum m_{i}(\overline{\textbf{r}} + \textbf{r}_{i}^{\prime})\times(\overline{\textbf{v}} + \textbf{v}_{i}^{\prime}) \\[5pt] &=(\overline{\textbf{r}}\times \overline{\textbf{v}})\sum m_{i} + \overline{\textbf{r}}\times \sum m_{i}\textbf{v}_{i}^{\prime} + (\sum m_{i}\textbf{r}_{i}^{\prime}) \times \overline{\textbf{v}} + \sum \textbf{r}_{i}^{\prime} \times \textbf{p}_{i}^{\prime} \\[5pt] &=M(\overline{\textbf{r}}\times \overline{\textbf{v}}) +\overline{\textbf{r}}\times 0 + 0 \times \overline{\textbf{v}} + \textbf{L}_{C} \end{align*} \nonumber \]

    therefore

    \[\qquad \textbf{L} =\textbf{L}_{C} + \overline{\textbf{r}} \times \textbf{P} \nonumber \]

    Example \(\PageIndex{1}\)

    A hoop of radius a rolling along the ground (Figure III.6):


    alt

    The angular momentum with respect to C is LC = \(I_{C \omega}\) where \( I_{C}\) is the rotational inertia about C. The angular momentum about O is therefore

    \[ I = I_{C}\omega+M\overline{v}a=I_{C}\omega+Ma^{2}\omega=(I_{C}+Ma^{2})=I\omega \nonumber \]

    where

    \[ I = I_{C}+Ma^{2} \nonumber \]

    is the rotational inertia about O.


    This page titled 3.7: Angular Momentum is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.