Question

A rod of length $L$ rotates about an axis passing through one of its ends and perpendicular to its plane. If the linear mass density of the rod varies as $\rho =(A{r}^3+B)kg/m$, then the moment of inertia of the rod about the given axis of rotation is

• A. $\frac{L^3}{3}[\frac{A{L}^3}{2}+B]$
• B. $\frac{L}{3}[\frac{A{L}^2}{2}+B]$
• C. $\frac{L^3}{3}[\frac{AL}{2}+B]$
• D. None of these

Answer 1

The correct option is A $\frac{L^3}{3}[\frac{A{L}^3}{2}+B]$

Consider a small element of the rod, having length $dr$, situated at a distance $r$ from the axis.
Mass of element, $dm=\rho dr$
$=(A{r}^3+B)dr$
Moment of inertia of the rod about the axis
$I={\int }_0^{L}dm.r^2={\int }_0^L(A{r}^3+B)r^{2}dr$
$={\int }_0^{L}A{r}^{5}dr+{\int }_0^{L}B{r}^{2}dr=A{\left[\frac{r^6}{6}\right]}_0^L+B{\left[\frac{r^3}{3}\right]}_0^L$
$=\frac{A{L}^6}{6}+\frac{B{L}^3}{3}=\frac{L^3}{3}[\frac{A{L}^3}{2}+B]$.