12.5: Summary
In this section we have extended the principles of momentum and impulse to rigid bodies. This involved introducing the angular impulse and angular momentum, and the principle relating them. In addition we discussed the law of conservation of angular momentum, which holds when the total external angular impulse on a system is zero.
- Angular impulse
\[\overrightarrow{\boldsymbol{H}}_{\mathrm{ang}, P, 12} \equiv \int_{t_{1}}^{t_{2}} \sum_{i} \overrightarrow{\boldsymbol{M}}_{i / P, \text { ext }} \mathrm{d} t \tag{12.14} \label{12.14}\]
- Principle of angular impulse and angular momentum
\[\sum_{i} \overrightarrow{\boldsymbol{L}}_{i / P}\left(t_{1}\right)+\overrightarrow{\boldsymbol{H}}_{\mathrm{ang}, P, 12}=\sum_{i} \overrightarrow{\boldsymbol{L}}_{i / P}\left(t_{2}\right) \tag{12.15} \label{12.15}\]
In \(2 \mathrm{D}: \quad \overrightarrow{\boldsymbol{r}}_{G / P, 1} \times\left(m_{\mathrm{tot}} \overrightarrow{\boldsymbol{v}}_{G, 1}\right)+I_{G} \omega_{1} \hat{\boldsymbol{k}}+\int_{t_{1}}^{t_{2}} M_{C / P} \hat{\boldsymbol{k}} \mathrm{d} t=\)
\[\begin{array}{r} \overrightarrow{\boldsymbol{r}}_{G / P, 2} \times\left(m_{\mathrm{tot}} \overrightarrow{\boldsymbol{v}}_{G, 2}\right)+I_{G} \omega_{2} \hat{\boldsymbol{k}} \\[4pt] I_{G} \omega_{1} \hat{\boldsymbol{k}}+\int_{t_{1}}^{t_{2}} M_{C / G} \hat{\boldsymbol{k}} \mathrm{d} t=I_{G} \omega_{2} \hat{\boldsymbol{k}} \tag{12.17} \label{12.17} \end{array}\]
- Conservation of angular momentum if the angular impulse is zero:
\[\sum_{i} \overrightarrow{\boldsymbol{L}}_{i / P}\left(t_{1}\right)=\sum_{i} \overrightarrow{\boldsymbol{L}}_{i / P}\left(t_{2}\right) \tag{12.18} \label{12.18}\]