Question

Moments of inertia of a hollow sphere of mass M and inner radius R and outer radius 2R having uniform mass distribution about diameter axis is:

• A. $\frac{31M{R}^2}{70}$
• B. $\frac{62M{R}^2}{35}$
• C. $\frac{15M{R}^2}{80}$
• D. None of these

Answer 1

The correct option is B $\frac{62M{R}^2}{35}$

Density of sphere $\left(d\right)=\frac{M}{V}$

$V=\frac{4x}{3}({\left(2r\right)}^3-r^3)=\frac{4\pi \times 7r^3}{3}$

Now consider a hollow sphere of thickness $dx$ at a distance $x$ from center ..($x$ is in between $r$ to $2r$)

Mass of this sphere $=dm=d(4\pi x^2.dx)dm=\frac{3m{x}^{2}dx}{7r^3}$

now $MI$ of this elemental hollowsphere is $dI$

$I=\frac{2dm{x}^2}{3}(hollow=\frac{2m{r}^2}{3})dI=\frac{2}{7}\frac{m}{r^3}{x}^4.dxI={[\frac{2}{35}\frac{m}{r^3}\times 5]}_0^{2r}I=\frac{62}{35}m{r}^2$