Question

A rectangular loop has mass $M$ and sides $a$ and $b$. An axis $O O$ ' passes through the centre $C$ of the loop and is parallel to side $a$ (lie in the plane of the loop). Then the radius of gyration of the loop, for the axis $O O^{\prime}$ is

• A. $\frac{b}{2} \sqrt{\frac{b+3 a}{3(b+a)}}$
• B. $\sqrt{\frac{\left(a^{2}+b^{2}\right)}{12}}$
• C. $\sqrt{\frac{b^{2}+3 a^{2}}{12}}$
• D. none of these

Answer 1

Answer: (A)

Let $\lambda$ is linear mass density.
Mass of side $a: m_{1}=\lambda a$,
Mass of side $b: m_{2}=\lambda b$
$I_{OO'} =2\left[m_{1}\left(\frac{b}{2}\right)^{2}+\frac{m_{2} b^{2}}{12}\right] $
Now $K_{O O^{\prime}}=\sqrt{\frac{I_{o o^{\prime}}}{2\left(m_{1}+m_{2}\right)}}$