Question

A thin rod of length 4l mass 4 m is bent at the points as shown in the fig. What is the moment of inertia of the rod about the axis passing point O& perpendiular to the plane of the paper

• A. $\frac{{m\ell }^2}{3}$
• B. $\frac{10{m\ell }^2}{3}$
• C. $\frac{{m\ell }^2}{12}$
• D. $\frac{{m\ell }^2}{24}$

Answer 1

Answer: (B)

$\frac{10{m\ell }^2}{3}$

$\begin{array}{c}\text{we know moment of Inertia of a Rod about}\\ \text{an axis passing perpendicular through its end-}\\ \Rightarrow I_{0B}=I_{0C}=\frac{m{L}^2}{3}\end{array}$

$\begin{array}{c}\text{Now moment of Inertia of Part}CD\\ \begin{array}{cc}{I}_{CD}=I_C+m{d}^2& I_C=I\text{about untreof Rod}\\ d=& \text{Distance of centre}\\ \text{from the axis.}& \end{array}\\ \Rightarrow I_{CD}=\frac{m{L}^2}{12}+m{\left(\frac{\sqrt{5}L}{2}\right)}^2=\frac{4}{3}m{L}^2\\ \Rightarrow I_{CD}=I_{AB}=\frac{4}{3}m{L}^2\text{(Both are similar)}\\ \Rightarrow I_{\text{Total}}=I_{AB}+I_{OB}+I_{0C}+I_{CD}=\frac{10}{3}m{L}^2\end{array}$