Question

A nonuniform rod $O M$ of length $L$ has linear mass density that varies with the distance $x$ from left end of the rod according to $\lambda = \lambda _{0} \left(\frac{x^{3}}{L^{3}}\right)$ ; Where $\lambda _{0}$ is constant, is kept along $x$ -axis and is rotating about an axis $A B$ , 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 $A B \left(I_{A B}\right)$ 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
$\Rightarrow x_{c m}=\frac{\frac{\lambda _{0}}{L^{3}} \int\limits_{0}^{L} x^{3} \cdot x d x}{\frac{\lambda _{0}}{L^{3}} \int\limits _{0}^{L} x^{3} d x}=\frac{4}{5}L$