General Mechanics/Rigid Bodies

< General Mechanics

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 (rn1, rn2, rn3) is the position of the nth 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 ShapesEdit

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

Spherical shellEdit

mass M, radius a

 

Solid ballEdit

mass M, radius a

 

Thin rodEdit

mass M, length a, orientated along z-axis

 

DiscEdit

mass M, radius a, in x-y plane

 

CylinderEdit

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

 

Thin rectangular plateEdit

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

 

further readingEdit