Question

Density of a sphere of mass $M$ , radius $R$ varies with distance $r$ from its centre as $P=P.(1+\frac{r}{R})P_0$ is a $+ve$ constant find its MOI about its diameter.

Answer 1

The moment of inertia of the sphere of mass$M$and radius$R$is given as

Let us consider a sphere of radius$r$at a distance$x$

$dV=\pi r^{2}dx$

$dV=\pi \left(R^2-x^2\right)dx$

$dm=\pi \rho \left(R^2-x^2\right)dx$

The moment of inertia

$I=\int r^{2}dm$

By substituting the values we get

$I=\pi \rho \underset{0}{\overset{R}{\int }}{\left(R^2-x^2\right)}^{2}dx$

$I=\frac{8\pi \rho }{15}{R}^5$

By substituting the the density given we get the moment of inertia of the sphere as,

$I=\frac{8\pi R^5}{15}\left(P.\left(1+\frac{r}{R}\right)P_0\right)$