Mechanics/Rigid Bodies

If the set of particles in the previous chapter form a rigid body, rotating with angular velocity ω about its centre of mass, then the results concerning the moment of inertia from the penultimate chapter can be extended.

We get

$I_{ij}=\sum _n{m}_n(r_n^2{\delta }_{ij}-{r_n}_i{r_n}_j)$

where (r_n1, r_n2, r_n3) is the position of the n^th mass.

In the limit of a continuous body this becomes

$I_{ij}=\underset{V}{\int }\rho (𝐫)(r^2{\delta }_{ij}-r_i{r}_j)dV$

where ρ is the density.

Either way we get, splitting L into orbital and internal angular momentum,

$L_i=M{ϵ}_{ijk}{R}_j{V}_k+I_{ij}{\omega }_j$

and, splitting T into rotational and translational kinetic energy,

$T=\frac{1}{2}M{V}_i{V}_i+\frac{1}{2}{\omega }_i{I}_{ij}{\omega }_j$

It is always possible to make I a diagonal matrix, by a suitable choice of axis.

The moments of inertia of simple shapes of uniform density are well known.

• Spherical shell, mass M, radius a

$I_{xx}=I_{yy}=I_{zz}=\frac{2}{3}M{a}^2$

• Solid ball, mass M, radius a

$I_{xx}=I_{yy}=I_{zz}=\frac{2}{5}M{a}^2$

• Thin rod, mass M, length a, orientated along z-axis

$I_{xx}=I_{yy}=\frac{1}{12}M{a}^2{I}_{zz}=0$

• Disc, mass M, radius a, in x-y plane

$I_{xx}=I_{yy}=\frac{1}{4}M{a}^2{I}_{zz}=\frac{1}{2}M{a}^2$

• Cylinder, mass M, radius a, length h orientated along z-axis

$I_{xx}=I_{yy}=M(\frac{a^2}{4}+\frac{h^2}{12})I_{zz}=\frac{1}{2}M{a}^2$

• Thin rectangular plate, mass M, side length a parallel to x-axis, side length b parallel to y-axis

$I_{xx}=M\frac{b^2}{12}{I}_{yy}=M\frac{a^2}{12}{I}_{zz}=M(\frac{a^2}{12}+\frac{b^2}{12})$