Question

Calculate the moment of inertia of a wheel about its axis which having rim of mass $24M$ and twenty four spokes each of mass $M$ and length $l$.

Answer 1

Please note that although the diagram shows only 8 spokes, please assume that there are 24 ones.

We know that for a wheel, the moment of inertia

Here, the moment of inertia $I_R$ of the rim of the wheel and those of the spokes $I_S$ will add up.

The axis of the wheel is an axis passing through the inner end and perpendicular to the spokes.

$I_R=24M{l}^2$ because here the radius is the length of each spoke.

$I_S=I_C+M{l}^2$ by theorem of parallel axis.

$\therefore I_S=\frac{M{l}^2}{12}+M{l}^2=\frac{13M{l}^2}{12}$

$total$ $I_S=\frac{13M{l}^2}{12}\times 24=26M{l}^2$

$\therefore total$ $inertia=I_T=50M{l}^2$

Answer $50M{l}^2$