Question

A nonuniform rod $OM$ of length $L$ has linear mass density that varies with the distance $x$ from left end of the rod according to $λ={λ}_0\left(\frac{x^3}{L^3}\right)$ ; Where ${λ}_0$ is constant, is kept along $x$ -axis and is rotating about an axis $AB$ , which is perpendicular to the rod as shown in the figure. What is the value of $x$ so that moment of inertia of rod about axis $AB\left(I_{AB}\right)$ is minimum?

• A. $\frac{7}{15}L$
• B. $\frac{2}{5}L$
• C. $\frac{8}{15}L$
• D. $\frac{4}{5}L$

Answer 1

Answer: (D)

The moment of inertia is minimum about an axis passing through the centre of mass of the body.
Therefore, $x$ should be the centre of mass of the rod
$⇒x_{cm}=\frac{\frac{{λ}_0}{L^3}\underset{0}{\overset{L}{∫}}{x}^3â‹\dots xdx}{\frac{{λ}_0}{L^3}\underset{0}{\overset{L}{∫}}{x}^{3}dx}=\frac{4}{5}L$