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

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 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} 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[edit]

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

Spherical shell[edit]

mass M, radius a

I_{xx} = I_{yy} = I_{zz}=\frac{2}{3}Ma^2

Solid ball[edit]

mass M, radius a

I_{xx} = I_{yy} = I_{zz}=\frac{2}{5}Ma^2

Thin rod[edit]

mass M, length a, orientated along z-axis

I_{xx} = I_{yy} = \frac{1}{12}Ma^2 \quad I_{zz}=0


mass M, radius a, in x-y plane

I_{xx} = I_{yy} = \frac{1}{4}Ma^2 \quad 


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 

Thin rectangular plate[edit]

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[edit]