Question

A disc and a ring of same mass are rolling and if their kinetic energies are equal, then the ratio of their velocities will be

• A. $\sqrt{4}:\sqrt{3}$
• B. $\sqrt{3}:\sqrt{4}$
• C. $\sqrt{3}:\sqrt{2}$
• D. $\sqrt{2}:\sqrt{3}$

Answer 1

Answer: (A)

$\sqrt{4}:\sqrt{3}$

Answer 2

$K_{disc} = \frac{1}{2}mv_{d}^{2}\left( 1 + \frac{k^{2}}{R^{2}} \right)$

$= \frac{3}{4}mv_{d}^{2}$ $\left\lbrack As\frac{k^{2}}{R^{2}} = \frac{1}{2}\text{for disc} \right\rbrack$

$K_{ring} = \frac{1}{2}mv_{r}\left( 1 + \frac{k^{2}}{R^{2}} \right) = mv_{r}^{2}$ $\left\lbrack As\frac{k^{2}}{R^{2}} = 1\text{for ring} \right\rbrack$

According to problem $K_{disc} = K_{ring}$ ⇒ $\frac{3}{4}mv_{d}^{2} = mv_{r}^{2}$

⇒ $\frac{v_{d}}{v_{r}} = \sqrt{\frac{4}{3}}$.