10.9: Angular momentum of a rigid body
10.9.1 Angular momentum of a point mass
Similar to the momentum \(\overrightarrow{\boldsymbol{p}}_{i}=m_{i} \overrightarrow{\boldsymbol{v}}_{i}\) (which is also called the linear momentum), we defined in Equation 10.17 the angular momentum of a point mass as:
\[\overrightarrow{\boldsymbol{L}}_{i / P} \equiv \overrightarrow{\boldsymbol{r}}_{i / P} \times \overrightarrow{\boldsymbol{p}}_{i}=\overrightarrow{\boldsymbol{r}}_{i / P} \times\left(m_{i} \overrightarrow{\boldsymbol{v}}_{i}\right) \tag{10.65} \label{10.65}\]
10.9.2 Angular momentum of a system
Using Equation 10.65, we can now determine the angular momentum of a system of point masses, by summing the angular momentum of all point masses, for the same reference point \(P\) :
\[\overrightarrow{\boldsymbol{L}}_{C, P}=\sum_{i} \overrightarrow{\boldsymbol{L}}_{i / P}=\sum_{i} \overrightarrow{\boldsymbol{r}}_{i / P} \times\left(m_{i} \overrightarrow{\boldsymbol{v}}_{i}\right) \tag{10.66} \label{10.66}\]
Equation 10.66 can be simplified, as we will derive below, by utilising the properties of the CoM and these kinematic equations:
\[\begin{align} \overrightarrow{\boldsymbol{r}}_{i / P} & =\overrightarrow{\boldsymbol{r}}_{G / P}+\overrightarrow{\boldsymbol{r}}_{i / G} \tag{10.67} \label{10.67}\\[4pt] \overrightarrow{\boldsymbol{v}}_{i} & =\overrightarrow{\boldsymbol{v}}_{G}+\overrightarrow{\boldsymbol{v}}_{i / G} \tag{10.68} \label{10.68}\end{align}\]
We substitute these two equations into Equation 10.66, and use that the terms that do not contain the index \(i\) can be taken outside of the sums:
\[\begin{align} \overrightarrow{\boldsymbol{L}}_{C, P} & =\sum_{i} \overrightarrow{\boldsymbol{L}}_{i / P}=\sum_{i}\left(\overrightarrow{\boldsymbol{r}}_{G / P}+\overrightarrow{\boldsymbol{r}}_{i / G}\right) \times\left[m_{i}\left(\overrightarrow{\boldsymbol{v}}_{G}+\overrightarrow{\boldsymbol{v}}_{i / G}\right)\right] \tag{10.69} \label{10.69}\\[4pt] & =\overrightarrow{\boldsymbol{r}}_{G / P} \times \overrightarrow{\boldsymbol{v}}_{G}\left(\sum_{i} m_{i}\right)+\overrightarrow{\boldsymbol{r}}_{G / P} \times\left(\sum_{i} m_{i} \overrightarrow{\boldsymbol{v}}_{i / G}\right) \\[4pt] & +\left(\sum_{i} m_{i} \overrightarrow{\boldsymbol{r}}_{i / G}\right) \times \overrightarrow{\boldsymbol{v}}_{G}+\sum_{i} m_{i} \overrightarrow{\boldsymbol{r}}_{i / G} \times \overrightarrow{\boldsymbol{v}}_{i / G} \tag{10.70} \label{10.70}\\[4pt] & =\overrightarrow{\boldsymbol{r}}_{G / P} \times\left(m_{\mathrm{tot}} \overrightarrow{\boldsymbol{v}}_{G}\right)+\sum_{i} m_{i} \overrightarrow{\boldsymbol{r}}_{i / G} \times \overrightarrow{\boldsymbol{v}}_{i / G} \tag{10.71} \label{10.71}\\[4pt] & =\overrightarrow{\boldsymbol{L}}_{G / P}+\sum_{i} \overrightarrow{\boldsymbol{L}}_{i / G} \tag{10.72} \label{10.72}\end{align}\]
We obtained four terms in Equation 10.70, of which the middle two are zero, because it follows from the definition of the CoM Equation 7.37 that if \(P=G\) we have \(\sum_{i} m_{i} \overrightarrow{\boldsymbol{r}}_{i / G}=\left(\sum_{i} m_{i}\right) \overrightarrow{\boldsymbol{r}}_{G / G}=\overrightarrow{\mathbf{0}}\), and the same holds for its time derivative \(\sum_{i} m_{i} \overrightarrow{\boldsymbol{v}}_{i / G}=\overrightarrow{\mathbf{0}}\)
We have thus derived with Equation 10.72 that the angular momentum of a system of point masses with respect to an arbitrary point \(P\) is identical to the sum of the angular momentum of a point mass equal to the total mass \(m_{\text {tot }}\) of the system at the CoM with respect to point \(P\), and the angular momentum of the system with respect to point \(G\), its CoM.
10.9.3 Angular momentum of a rigid body
The term \(\sum_{i} \overrightarrow{\boldsymbol{L}}_{i / G}\) in the angular momentum expression Equation 10.72 is still a bit difficult to evaluate, since it is a sum over many point masses. It can be simplified for rigid bodies by making use of the property that we derived in the previous chapter that all point masses in a rigid body \(C\) have the same angular velocity \(\omega_{C}\).
From Equation 9.26 we have the kinematic equation that holds for all point masses \(i\) in the rigid body:
\[\overrightarrow{\boldsymbol{v}}_{i / G}=\overrightarrow{\boldsymbol{\omega}} \times \overrightarrow{\boldsymbol{r}}_{i / G} \tag{10.73} \label{10.73}\]
We substitute this equation in Equation 10.71 for the angular momentum \(\overrightarrow{\boldsymbol{L}}_{C, G}=\) \(\sum_{i} \overrightarrow{\boldsymbol{L}}_{i / G}\) for a rigid body with respect to its CoM:
\[\begin{align} \sum_{i} \overrightarrow{\boldsymbol{L}}_{i / G} & =\sum_{i} m_{i} \overrightarrow{\boldsymbol{r}}_{i / G} \times \overrightarrow{\boldsymbol{v}}_{i / G} \tag{10.74} \label{10.74}\\[4pt] & =\sum_{i} m_{i} \overrightarrow{\boldsymbol{r}}_{i / G} \times\left(\overrightarrow{\boldsymbol{\omega}} \times \overrightarrow{\boldsymbol{r}}_{i / G}\right) \tag{10.75} \label{10.75}\\[4pt] & =\sum_{i} m_{i}\left[\left(\overrightarrow{\boldsymbol{r}}_{i / G} \cdot \overrightarrow{\boldsymbol{r}}_{i / G}\right) \overrightarrow{\boldsymbol{\omega}}-\left(\overrightarrow{\boldsymbol{r}}_{i / G} \cdot \overrightarrow{\boldsymbol{\omega}}\right) \overrightarrow{\boldsymbol{r}}_{i / G}\right] \tag{10.76} \label{10.76}\\[4pt] \sum_{i} \overrightarrow{\boldsymbol{L}}_{i / G, 2 D} & =\sum_{i} m_{i}\left[\left(x_{i / G}^{2}+y_{i / G}^{2}+z_{i / G}^{2}\right) \omega \hat{\boldsymbol{k}}\right. \\[4pt] & \left.-\left(z_{i / G} \omega\right)\left(x_{i / G} \hat{\boldsymbol{\imath}}+y_{i / G} \hat{\boldsymbol{\jmath}}+z_{i / G} \hat{\boldsymbol{k}}\right)\right] \tag{10.77} \label{10.77}\\[4pt] & =\sum_{i} m_{i}\left[\rho_{i / G}^{2} \omega \hat{\boldsymbol{k}}-z_{i / G} \omega\left(x_{i / G} \hat{\boldsymbol{\imath}}+y_{i / G} \hat{\boldsymbol{\jmath}}\right)\right] \tag{10.78} \label{10.78}\\[4pt] & =\left(I_{z z} \hat{\boldsymbol{k}}+I_{x z} \hat{\boldsymbol{\imath}}+I_{y z} \hat{\boldsymbol{\jmath}}\right) \omega \tag{10.79} \label{10.79}\end{align}\]
In this derivation we used this vector identity for a triple product to obtain Equation 10.76: \(\overrightarrow{\boldsymbol{a}} \times(\overrightarrow{\boldsymbol{b}} \times \overrightarrow{\boldsymbol{c}})=(\overrightarrow{\boldsymbol{a}} \cdot \overrightarrow{\boldsymbol{c}}) \overrightarrow{\boldsymbol{b}}-(\overrightarrow{\boldsymbol{a}} \cdot \overrightarrow{\boldsymbol{b}}) \overrightarrow{\boldsymbol{c}}\). In Equation 10.77 we introduced a Cartesian coordinate system with origin \(G\) and \(z\)-axis in the direction \(\overrightarrow{\boldsymbol{\omega}}=\omega \hat{\boldsymbol{k}}\), such that \(\overrightarrow{\boldsymbol{r}}_{i / G}=x_{i / G} \hat{\boldsymbol{\imath}}+y_{i / G} \hat{\boldsymbol{\jmath}}+z_{i / G} \hat{\boldsymbol{k}}\). In Equation 10.78 we write the expression in terms of the distance \(\rho_{i / G}\) to the \(z\)-axis with \(\rho_{i / G}^{2}=x_{i / G}^{2}+y_{i / G}^{2}\). We also find that in 3D the angular momentum vector \(\sum_{i} \overrightarrow{\boldsymbol{L}}_{i / G}\) is not necessarily parallel to the angular velocity vector \(\overrightarrow{\boldsymbol{\omega}}\) since it also contains a term along the \(\hat{\boldsymbol{\imath}}\) and \(\hat{\boldsymbol{\jmath}}\) directions. In the last equation Equation 10.79 we simplify the expression by introducing the components \(I_{z z}, I_{x z}\) and \(I_{y z}\) of the inertia tensor \(\mathbf{I}_{C}\) of the rigid body \(C\), that follow from the equation \(\overrightarrow{\boldsymbol{L}}=\mathbf{I} \overrightarrow{\boldsymbol{\omega}}\). The two subscripts, e.g. in \(I_{z z}\), indicate the relation between the \(L_{z}\) and \(\omega_{z}\) vector component.
10.9.4 Derivation of the inertia tensor
Note that a similar equation to the parallel axis theorem can be derived for the products of inertia by substituting \(\overrightarrow{\boldsymbol{r}}_{i / P}=\overrightarrow{\boldsymbol{r}}_{i / G}+\overrightarrow{\boldsymbol{r}}_{G / P}\) in Equation 10.76 and Equation 10.79, while using the properties of the CoM:
\[I_{P, x z}=-m_{\mathrm{tot}} x_{G / P} z_{G / P}+I_{G, x z} \tag{10.80} \label{10.80}\]
The full inertia tensor with respect to an arbitrary point \(P\) can thus be written as the sum of an inertia tensor of the CoM \(\mathbf{I}_{G, P}\) and an inertia tensor \(\mathbf{I}_{C, G}\) of object \(C\) with respect to the CoM:
\[\mathbf{I}_{C, P}=\mathbf{I}_{G, P}+\mathbf{I}_{C, G} \tag{10.81} \label{10.81}\]
\[\begin{gathered} \mathbf{I}_{G, P}=m_{\mathrm{tot}}\left[\begin{array}{ccc} \left(y_{G / P}^{2}+z_{G / P}^{2}\right) & -x_{G / P} y_{G / P} & -x_{G / P} z_{G / P} \\[4pt] -y_{G / P} x_{G / P} & \left(x_{G / P}^{2}+z_{G / P}^{2}\right) & -y_{G / P} z_{G / P} \\[4pt] -z_{G / P} x_{G / P} & -z_{G / P} y_{G / P} & \left(x_{G / P}^{2}+y_{G / P}^{2}\right) \end{array}\right] \tag{10.82} \label{10.82}\\[4pt] \mathbf{I}_{C, G}=\left[\begin{array}{ccc} I_{G, x x} & I_{G, x y} & I_{G, x z} \\[4pt] I_{G, y x} & I_{G, y y} & I_{G, y z} \\[4pt] I_{G, z x} & I_{G, z y} & I_{G, z z} \end{array}\right] \tag{10.83} \label{10.83}\end{gathered}\]