PC1672 Advanced dynamics

4.2 Rigid bodies

A rigid body is an object all of whose atoms are (at least on average) in fixed positions relative to each other. It can be described using six coordinates, three for the position of its centre of mass and three angles for its orientation. Its state of motion can be specified using two vector velocities: the linear velocity of its centre-of-mass and its angular velocity.

We shall find it convenient to use a body-fixed coordinate system, whose axes point along fixed directions in the body. We shall chose the origin to be either some point that is held fixed (if the body is pivoted at that point) or at the centre of mass (if the body is free to move in space).

If the body is rotating with some angular velocity, the angular momentum of each atom has the form we met above. The total angular momentum of the body is

$\begin{displaymath}{\bf L}=\sum_\alpha m_\alpha\left[{\bf x}_\alpha^2 \hbox{\bol... ..._\alpha \cdot\hbox{\boldmath {$\omega$}}) {\bf x}_\alpha\right]\end{displaymath}$

where $\alpha$ runs over all the atoms. The components of ${\bf L}$ are

$\begin{displaymath}L_i=\sum_\alpha m_\alpha\left[{\bf x}_\alpha^2\omega_i -x_{\alpha i}\sum_{j=1}^3 x_{\alpha j}\omega_j\right]\end{displaymath}$

This shows that the angular momentum of the body is linearly related to its angular velocity, but not simply proportional to it.

This equation can be written in matrix form

$\begin{displaymath}L_i=\sum_j I_{ij}\omega_j\end{displaymath}$

but we should remember that ${\bf L}$ and $\hbox{\boldmath {$\omega$ }}$ are vectors with geometrical meanings that do not depend on our choice of axes. Hence we can also write this relation in the form

$\begin{displaymath}{\bf L}=\underline{\underline{\bf I}}\cdot\hbox{\boldmath {$\omega$}}\end{displaymath}$

where $\underline{\underline{\bf I}}$ is the moment-of-inertia tensor.

Textbook references

K&K 7.6
F&C 9.1
B&O 6.7
M 12.1, 12.3
M&T 11.1, 11.3

Mike Birse
17th May 2000