Question

Linear mass density (mass/ length) of a rod depends on the distance from one end (say A) as ${\lambda }_x=(\alpha x+\beta )$ Here,$\alpha$and $\beta$ are constants. Find the moment of inertia of this rod about an axis passing through $A$ and perpendicular to the rod. Length of the rod is $l$.

Answer 1

$I={\int }_0^{1}dI={\int }_0^1\left(dm\right)X^2$
$={\int }_0^1\left({\lambda }_{x}dX\right)X^2$
$={\int }_0^1(\alpha x+\beta )X^{2}dX$
$=\frac{\alpha l^4}{4}+\frac{\beta l^3}{3}$