A wheel of mass M and radius a and M.I. $I _G$ (about centre of mass) is set rolling with angular velocity $\omega$ up a rough inclined plane of inclination $\theta$. The distance travelled by it up the plane is :

A disc of mass $m$ of radius $r$ is placed on a rough horizontal surface. A cue of mass $m$ hits the disc at a height $h$ from the axis passing through centre and parallel to the surface. The cue stop and falls down after impact. The disc starts pure rolling for

A uniform rigid rod has length $L$ and mass $m$. It lies on a horizontal smooth surface, and is rotated at a uniform angular velocity $\omega$ about a vertical axle passing through one of its ends. The force exerted by the axle on the rod will be

A disc and a sphere of same radius but different masses roll off on two inclined planes of the same altitude and length. Which one of the two objects gets to the bottom of the plane first?

A body is given translational velocity and kept on a surface that has sufficient friction. Then:

(a) A child stands at the center of a turntable with his two arms outstretched. The turntable is set rotating with an angular speed of $40$ rev/min. How much is the angular speed of the child if he folds his hands back and thereby reduces his moment of inertia to $2/5$ times the initial value? Assume that the turntable rotates without friction. (b) Show that the child’s new kinetic energy of rotation is more than the initial kinetic energy of rotation. How do you account for this increase in kinetic energy?

Two small kids weighing 10 kg and 15 kg are trying to balance a seesaw of total length 5m, with the fulcrum at the centre. If one of the kids is sitting at an end, where should the other sit?

 If principle of moments for any object holds, then object is in state of

A uniform dice of mass $10kg$ radius $1m$ is placed on a rought horizontal surface. The coefficient of friction between the disc and the surface is $0.2$. A horizontal time varying force is applied on the centre of the disc whose variation with time is shown in graph.
List-I                                                         List-IIDisc rolls without slipping                   at $t=7s$Disc rolls with slipping                       at $t=3s$  Disc starts slipping at                         at $t=4s$Friction force is $10N$ at              None

When slightly different weights are placed on the two pans of a beam balance, the beam comes to rest at an angle with the horizontal. The beam is supported at a single point P by a pivot. Then which of the following statement(s) is/are true ?