General Mechanics/Rigid Bodies

If the set of particles in the previous chapter form a rigid body, rotating with angular velocity &omega; 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 &rho; 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 Shapes=

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

Spherical shell
mass M, radius a
 * $$I_{xx} = I_{yy} = I_{zz}=\frac{2}{3}Ma^2$$

Solid ball
mass M, radius a
 * $$I_{xx} = I_{yy} = I_{zz}=\frac{2}{5}Ma^2$$

Thin rod
mass M, length a, orientated along z-axis
 * $$I_{xx} = I_{yy} = \frac{1}{12}Ma^2 \quad I_{zz}=0$$

Disc
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$$

Cylinder
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 plate
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)$$