General Mechanics/Rigid Bodies

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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 Shapes[edit]

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

Spherical shell[edit]

mass M, radius a

Solid ball[edit]

mass M, radius a

Thin rod[edit]

mass M, length a, orientated along z-axis

Disc[edit]

mass M, radius a, in x-y plane

Cylinder[edit]

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

Thin rectangular plate[edit]

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

further reading[edit]