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 (r_n1, r_n2, r_n3) is the position of the n^th 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,

$T={\frac {1}{2}}MV_{i}V_{i}+{\frac {1}{2}}\omega _{i}I_{ij}\omega _{j}$

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

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

further reading

• Statics/Moment of Inertia (contents)
• Statics/Geometric Properties of Solids