Question

A square is made by joining four rods each of mass M and length L. Its moment of inertia about an axis PQ , in its plane and passing through one its corner is

• A. 6 $6{ML}^2$
• B. $\frac{4}{3}{ML}^2$
• C. $\frac{8}{3}{ML}^2$
• D. $\frac{10}{3}{ML}^2$

Answer 1

Answer: (C)

$\frac{8}{3}{ML}^2$

$\begin{array}{c}\text{We know that -}\\ \text{moment of Ineutia of a Rodabout an axis passing}\\ \text{through its end at an angle}\theta \text{is.}\\ I=I_0\cdot {\sin }^2\theta =\frac{m{L}^2}{3}\cdot {\sin }^2\left({45}^{\circ }\right)=\frac{m{L}^2}{6}\\ \Rightarrow I_{AB}=I_{AD}=\frac{M{L}^2}{6}\text{(Similar, cases)}\end{array}$

$\begin{array}{cc}{I}_{BC}=I_0{\sin }^2\theta +M{d}^2& \text{Because it is inclined at}\\ & \text{an angle}(\theta ={45}^{\circ })\text{and}\\ \text{at a distanu}(\text{d) from ax is.}& \\ \Rightarrow I_{BC}=\frac{M{L}^2}{12}\times {\sin }^2{45}^{\circ }+M{\left(\frac{3L}{2\sqrt{2}}\right)}^2& \\ I_{BC}=I_{CD}=\frac{7M{L}^2}{3}& \\ \Rightarrow I_{\text{square}}=I_{AB}+I_{BC}+I_{CD}+I_{AD}=\frac{8}{3}M{L}^2& \end{array}$