Question

Radius of gyration of a uniform disc about a line perpendicular to the plane of disc is equal to its radius $R$. If the distance of the line from the center is $\frac{R}{\sqrt{x}},$ find the value of $x$.

• A. $2$
• B. $4$
• C. $1$
• D. $8$

Answer 1

The correct option is A $2$

Let the mass of disc is $M$
Let the distance of the line from the center is $d.$
Moment of inertia about the line $\left(I\right)$
Radius of gyration about the line $\left(K\right)$
Applying parallel axes theorem,
$I=I_{com}+M{d}^2$
We know,
$K=\sqrt{\frac{I}{M}}$
So, $K=\sqrt{\frac{I_{com}+M{d}^2}{M}}$
Given: $K=R$
$\Rightarrow R=\sqrt{\frac{\frac{M{R}^2}{2}+M{d}^2}{M}}$

$\Rightarrow R=\sqrt{\frac{R^2}{2}+d^2}$
Squaring on both sides,
$R^2=\frac{R^2}{2}+d^2$
$\Rightarrow d=\frac{R}{\sqrt{2}}$
On comparing with $\frac{R}{\sqrt{x}}$, $x=2$