12.2: Choosing a convenient reference point
Although the principle of angular impulse and angular momentum equation Equation 12.5 can be used for any reference point \(P\) it can often be significantly simplified by a smart choice of the reference point \(P\). Let us discuss some convenient choices:
- The centre of mass \(G\) of the system of point masses. This causes all the cross products in Equation 12.5 to disappear because \(\overrightarrow{\boldsymbol{r}}_{G / G}=\overrightarrow{\mathbf{0}}\) and results in Equation 12.6. Moreover, this is the only option for choosing an accelerating reference point.
- Since the moment of an external force is \(\overrightarrow{\boldsymbol{M}}_{i / P}=\overrightarrow{\boldsymbol{r}}_{i / P} \times \overrightarrow{\boldsymbol{F}}_{i}\), choosing the point of action of that force as a reference point \(P=i\) will result in the angular impulse of that point becoming zero because \(\overrightarrow{\boldsymbol{r}}_{i / i}=\overrightarrow{\mathbf{0}}\). This is especially useful if the actual force \(\overrightarrow{\boldsymbol{F}}_{i}\) is unknown.
- In some cases the reference point \(P\) can be chosen such that either \(\overrightarrow{\boldsymbol{v}}_{G}\) is parallel to \(\overrightarrow{\boldsymbol{r}}_{G / P}\), or that one or more of the external forces \(\overrightarrow{\boldsymbol{F}}_{i}\) is parallel to \(\overrightarrow{\boldsymbol{r}}_{i / P}\), which can simplify the situation because either \(\overrightarrow{\boldsymbol{r}}_{G / P} \times \overrightarrow{\boldsymbol{v}}_{G}=\overrightarrow{\mathbf{0}}\) or \(\overrightarrow{\boldsymbol{r}}_{i / P} \times \overrightarrow{\boldsymbol{F}}_{i}=\overrightarrow{\mathbf{0}}\)