Question

If a ring rolls down from top to bottom of an inclined plane, it takes time $t_1$. If it slides, it takes time $t_2$. Then the ratio $\frac{t_2^2}{t_1^2}$ is

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $\frac{1}{4}$
• D. $\frac{1}{2}$
• E. $\frac{2}{5}$

Answer 1

Answer: (D)

$t_1\to$ Rolling
$t_2\to$ sliding
$a_{\text{roll }}=\frac{g\sin \theta }{1+\frac{k^2}{R^2}}$
$a_{\text{slide }}=g\cdot \sin \theta$
$t_1=\sqrt{\frac{2s}{a}}$
$=\sqrt{\frac{2s\left(1+\frac{k^2}{R^2}\right)}{g\sin \theta }}$
$t_2=\sqrt{\frac{2s}{g\sin \theta }}$
$t_{{\frac{2}{2}}^2}=\frac{1}{t_1^2}$
$=\frac{1}{1+\frac{k^2}{R^2}}=\frac{1}{2}$