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)

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