Question

Three identical square plates rotate about the axes shown in the figure in such a way that their kinetic energies are equal. Each of the rotation axes passes through the centre of the square. Then the ratio of angular speed ${\omega }_1:{\omega }_2:{\omega }_3$ is:

• A. $1:1:1$
• B. $\sqrt{2}:\sqrt{2}:1$
• C. $1:\sqrt{2}:1$
• D. $1:2:\sqrt{2}$

Answer 1

Answer: (B)

$\sqrt{2}:\sqrt{2}:1$

Let the mass and the side length of the square plates be $M$ and $L$ respectively.

For (1) : $I_1=\frac{M{L}^2}{12}$

For (2) : Using perpendicular axis theorem, $I_z=I_x+I_y$ where $I_x=I_y$

$\therefore$ $\frac{M{L}^2}{6}=2I_y$ $⟹I_2=I_y=\frac{M{L}^2}{12}=I_1$

For (3) : $I_3=\frac{M{L}^2}{6}=2I_1$

Rotational kinetic energy $E=\frac{1}{2}I{\omega }^2$

$\therefore$ $\omega =\frac{2E}{\sqrt{I}}$ $⟹\omega \propto \frac{1}{\sqrt{I}}$ $(\because E=constant)$

$\therefore$ ${\omega }_1:{\omega }_2:{\omega }_3=\frac{1}{\sqrt{I_1}}:\frac{1}{\sqrt{I_2}}:\frac{1}{\sqrt{I_3}}$

${\omega }_1:{\omega }_2:{\omega }_3=\frac{1}{\sqrt{I_1}}:\frac{1}{\sqrt{I_1}}:\frac{1}{\sqrt{2I_1}}$

$⟹$ ${\omega }_1:{\omega }_2:{\omega }_3=\sqrt{2}:\sqrt{2}:1$