12.1: Principle of angular impulse and angular momentum
To derive the principle of angular momentum and angular impulse, we start from Euler’s second law that was found in Ch. 10, Equation 10.89:
\[\sum_{i} \overrightarrow{\boldsymbol{M}}_{i / P, \mathrm{ext}}=\frac{\mathrm{d}}{\mathrm{d} t} \sum_{i} \overrightarrow{\boldsymbol{L}}_{i / P} \tag{12.1} \label{12.1}\]
This equation holds for the angular momentum of a system of point masses with respect to a reference point \(P\) that is fixed in space in an inertial reference frame. The equation can be rewritten by multiplying both sides by \(\mathrm{d} t\) and integrating over a time interval from \(t_{1}\) to \(t_{2}\) :
\[\int_{t_{1}}^{t_{2}} \sum_{i} \overrightarrow{\boldsymbol{M}}_{i / P, \mathrm{ext}} \mathrm{d} t=\int_{t_{1}}^{t_{2}} \mathrm{~d} \sum_{i} \overrightarrow{\boldsymbol{L}}_{i / P}=\sum_{i} \overrightarrow{\boldsymbol{L}}_{i / P}\left(t_{2}\right)-\sum_{i} \overrightarrow{\boldsymbol{L}}_{i / P}\left(t_{1}\right) \tag{12.2} \label{12.2}\]
The integral on the left of Equation 12.2 is defined as the total angular impulse acting on the system \(\overrightarrow{\boldsymbol{H}}_{\text {ang }, P, 12}\) :
\[\overrightarrow{\boldsymbol{H}}_{\mathrm{ang}, P, 12} \equiv \int_{t_{1}}^{t_{2}} \sum_{i} \overrightarrow{\boldsymbol{M}}_{i / P, \text { ext }} \mathrm{d} t \tag{12.3} \label{12.3}\]
Note that only external moments contribute to the angular impulse, since internal moments sum to zero just like in Euler’s second law.
By rewriting Equation 12.2 we obtain the principle of angular impulse and angular momentum, which states that the change in the angular momentum of the system is equal to the impulse generated by external moments on the system:
The change in angular momentum is equal to the angular impulse by external moments acting on the system.
\[\sum_{i} \overrightarrow{\boldsymbol{L}}_{i / P}\left(t_{1}\right)+\overrightarrow{\boldsymbol{H}}_{\mathrm{ang}, P, 12}=\sum_{i} \overrightarrow{\boldsymbol{L}}_{i / P}\left(t_{2}\right) \tag{12.4} \label{12.4}\]
Since we are mainly concerned with the analysis of kinetics of rigid bodies, we use Equation 10.91 to derive the principle of angular impulse and angular momentum of a rigid body for the case of 2D planar kinetics:
\[\begin{array}{r} \overrightarrow{\boldsymbol{r}}_{G / P, 1} \times\left(m_{\mathrm{tot}} \overrightarrow{\boldsymbol{v}}_{G, 1}\right)+I_{G} \omega_{1} \hat{\boldsymbol{k}}+\int_{t_{1}}^{t_{2}} M_{C / P} \hat{\boldsymbol{k}} \mathrm{d} t \\[4pt] =\overrightarrow{\boldsymbol{r}}_{G / P, 2} \times\left(m_{\mathrm{tot}} \overrightarrow{\boldsymbol{v}}_{G, 2}\right)+I_{G} \omega_{2} \hat{\boldsymbol{k}} \tag{12.5} \label{12.5} \end{array}\]
If we choose \(P=G\) this principle simplifies to:
\[I_{G} \omega_{1} \hat{\boldsymbol{k}}+\int_{t_{1}}^{t_{2}} M_{C / G} \hat{\boldsymbol{k}} \mathrm{d} t=I_{G} \omega_{2} \hat{\boldsymbol{k}} \tag{12.6} \label{12.6}\]