Question

A rod of length ${}^{\prime }{L}^{\prime }$ has variable mass density, $\lambda (x)={\left(\lambda \right)}_0\left(1+\frac{x}{L}\right).$ There exists an axis parallel to $AB$ about which the moment of inertia of the rod is minimum. If the distance of such an axis from $AB$ is of the form $\frac{5L}{N},$ then find $N.$

Answer 1

Answer: 9

Moment of inertia is minimum about an axis passing through $COM.$
$x_{cm}=\frac{\int xdm}{\int dm}=\frac{\underset{0}{\overset{L}{\int }}{\lambda }_0\left(1+\frac{x}{L}\right)xdx}{\underset{0}{\overset{L}{\int }}{\lambda }_0\left(1+\frac{x}{L}\right)dx}=\frac{5L}{9}$