Question

The ratio of the radii if gyration of a circular disc and a circular ring of same masses and radii about a tangential axis parallel to their plane is

• A. $\surd 6:\surd 5$
• B. $1:\surd 2$
• C. $\surd 5:\surd 6$
• D. $Noneofthese$

Answer 1

The correct option is C $\surd 5:\surd 6$

We know $I_2\left(disk\right)=\frac{1}{2}MR$

By $\perp$ axis theorem,

$I_x+I_y=\frac{1}{2}M{R}^2$

From Symmetry, we can say $I_x=I_y$

$\Rightarrow 2I_x=\frac{1}{2}M{R}^2$

$\Rightarrow I_x=\frac{1}{4}M{R}^2$

By $\parallel$ axis theorem,

$I_t=I_x+M{R}^2=\frac{1}{4}M{R}^2+M{R}^2=\frac{5}{4}M{R}^2$

Similarly, $I_t\left(ring\right)=\frac{3}{2}M{R}^2$

$\therefore$ Radius of gyration of disc $=k_d=R\sqrt{\frac{5}{2}}$

Radius of gyration of ring $=k_r=R\sqrt{3}$

Ratio$=\frac{k_d}{k_r}=\frac{\sqrt{5}}{\sqrt{6}}$