# General 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 (rn1, rn2, rn3) is the position of the nth mass.

In the limit of a continuous body this becomes

$I_{ij}=\int_V \rho(\mathbf{r})(r^2\delta_{ij}-r_i r_j) \, dV$

where ρ is the density.

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

$L_i=M\epsilon_{ijk}R_j V_k+ I_{ij}\omega_j$

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.

# Mass Moments Of Inertia Of Common Geometric ShapesEdit

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

### Spherical shellEdit

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

### Solid ballEdit

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

### Thin rodEdit

mass M, length a, orientated along z-axis

$I_{xx} = I_{yy} = \frac{1}{12}Ma^2 \quad I_{zz}=0$

### DiscEdit

mass M, radius a, in x-y plane

$I_{xx} = I_{yy} = \frac{1}{4}Ma^2 \quad I_{zz}=\frac{1}{2}Ma^2$

### CylinderEdit

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

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

### Thin rectangular plateEdit

mass M, side length a parallel to x-axis, side length b parallel to y-axis

$I_{xx}=M\frac{b^2}{12} \quad I_{yy}=M\frac{a^2}{12} \quad I_{zz}=M \left( \frac{a^2}{12}+\frac{b^2}{12} \right)$