# 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

where (*r*_{n1}, *r*_{n2}, *r*_{n3}) is the position of the n^{th} mass.

In the limit of a continuous body this becomes

where ρ is the density.

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

and, splitting *T* into rotational and translational kinetic energy,

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

# Mass Moments Of Inertia Of Common Geometric Shapes

[edit | edit source]The moments of inertia of simple shapes of uniform density are well known.

### Spherical shell

[edit | edit source]mass *M*, radius *a*

### Solid ball

[edit | edit source]mass *M*, radius *a*

### Thin rod

[edit | edit source]mass *M*, length *a*, orientated along *z*-axis

### Disc

[edit | edit source]mass *M*, radius *a*, in *x-y* plane

### Cylinder

[edit | edit source]mass *M*, radius *a*, length *h* orientated along *z*-axis

### Thin rectangular plate

[edit | edit source]mass *M*, side length *a* parallel to *x*-axis, side length *b* parallel to *y*-axis