Question

Thickness of a wooden circular plate is same as the thickness of a metal circular plate but density of metal plate is $8$ times density of wooden plate. If moment of inertia of wooden plate is twice the moment of inertia of metal plate about their natural axes, then the ratio of radii of wooden plate to metal plate is:

• A. $1:2$
• B. $1:4$
• C. $4:1$
• D. $2:1$

Answer 1

The correct option is D $2:1$

Let thickness be $t$. Let the radii be $r_w$ and $r_m$ and density be ${\rho }_w$ and ${\rho }_m$ for the wooden and metal plate respectively.

So $I_w=\frac{\pi r_w^{2}t{\rho }_w{r}_w^2}{2}=\frac{\pi r_w^{4}t{\rho }_w}{2}$

and $I_m=\frac{\pi r_m^{2}t{\rho }_m{r}_m^2}{2}=\frac{\pi r_m^{4}t{\rho }_m}{2}$

Given that, $I_w=2I_m,{\rho }_m=8{\rho }_w$

we get $r_w^4=16r_m^4\Rightarrow r_w=2r_m$

so ratio of radii of wooden plate to metal plate is $2:1$