$A$ circular disc reaches from top to bottom of an inclined plane of length $L$. When it slips down the plane,it takes time $t_{1}$. When it rolls down the plane,it takes time $t_{2}$. The value of $\frac{t_{2}}{t_{1}}$ is $\sqrt{\frac{3}{x}}$. The value of $x$ will be .... .

$A$ small ring is rolling without slipping on the circumference of a large bowl as shown in the figure. The ring is moving down at $P_{1}$,comes down to the lowest point $P_{2}$,and is climbing up at $P_{3}$. Let $v_{CM}$ denote the velocity of the centre of mass of the ring. Choose the correct statement regarding the frictional force on the ring.

$A$ solid cylinder and a solid sphere having the same mass and radius roll down on the same smooth inclined plane. The ratio of the acceleration of the cylinder $(a_c)$ to that of the sphere $(a_s)$ is

$A$ cylinder of mass $M_c$ and a sphere of mass $M_s$ are placed at points $A$ and $B$ of two inclines,respectively (See Figure). If they roll on the incline without slipping such that their accelerations are the same,then the ratio $\frac{\sin \theta_c}{\sin \theta_s}$ is

$A$ solid sphere is rolling without slipping on a horizontal rough surface and starts rising on an inclined rough surface as shown in the figure. Assume pure rolling throughout the motion. Choose the $INCORRECT$ statement.

The ratio of the accelerations for a solid sphere (mass $m$ and radius $R$) rolling down an incline of angle $\theta$ without slipping and slipping down the incline without rolling is