**Subject :**Elements of Mechanical Engineering

## Rolling Motion: Kinetics of body

**Rolling Motion:**** **Now, we will discuss the other type of plane motion: motion of a disk or wheel rolling on a plane surface. If the disk is constrained to roll without sliding, the acceleration of its mass center and its angular acceleration or related. For a balanced disk i.e. for a disk whose mass center and geometrical center coincide, the acceleration of mass center is angular acceleration times the radius. Because the body is in plane motion, the kinetic diagram of the body consists of a horizontal force applied at the center and a couple.

When a disk rolls without slippling, there is no relative motion between the point of the disk in contact with the ground and the ground itself. The friction force *F *will be self adjusting with the limiting value of .

When the disk rotates and slides at the same time, a relative motion exists between the point of the disk which is in contact with the ground and the ground itself and the force of friction has the magnitude , where is the coefficient of kinetic friction. In this case, however, the motion of the mass center G of the disk and the rotation of the disk about G are independent, and the acceleration of the center is not equal to the product of angular acceleration and radius.

The three different cases are summarized as follows:

Rolling, no sliding

Rolling, sliding impending

Rolling, and sliding ** **a and independent.

When it is not known whether or not a disk slides, it should first be assumed that the disk rolls without sliding. If *F *is found smaller than, or equal to, , the assumption is proved correct. If F is found larger than , the assumption is incorrect and the problem should be started again, assuming rolling and sliding.

Now, let us solve a problem on rolling.

A metal hook with a radius *r* is released from the rest on the incline. If the coefficient of static and kinetic friction are , determine the angular acceleration of the hook and time *t *for the hook to move a distance of *S* down the inclined. [ Figure A ]

The counterclockwise angular acceleration requires a counterclockwise moment about *G*, so *F* must be upword.