The figure shows a smooth inclined plane of inclination $\mathrm{\theta }$ fixed in a car. A sphere is set in pure rolling on the incline. For what value of the acceleration of the car in the horizontal direction the sphere will continue pure rolling?

A uniform disc of mass $M$ and radius $R$ is released from the shown position. $PQ$ is a string, $OP$ is a horizontal line, $O$ is the centre of the disc and distance$OP$ is $\frac{R}{2}$. Then, the tension in the string just after the disc is released will be

The string of a step rolling wheel is pulled by applying force $F$ with different lines of action in two situations as shown. The wheel starts rolling without slipping due to application of the force.

Consider a ball of radius $R$, rolling without slipping on a horizontal plane. If the radius of gyration of the ball about an axis passing through its centre of mass is $K$, then what is the ratio of rotational kinetic energy to that of the total energy associated with the ball?