16.·1. Rolling and Sliding. When sliding takes place the friction F bears a constant ratio μ to the normal reaction R but when rolling takes place the friction has generally a much smaller value.

When two bodies in contact at a point A have a relative motion the process of determining whether this motion involves rolling or sliding at A is as follows:

Write down the equations of motion which involve the friction F and the normal reaction R, and assuming

(i) that rolling takes place also write down the kinematical condition which expresses the fact that there is no relative tangential velocity at A. If from the solution of these equations we find that F/R is less than the coefficient of friction μ, the assumption that rolling takes place is justified and rolling will continue until F/R becomes greater than μ.

(ii) Assuming that sliding takes place, write μR instead of F in the equations of motion and solve the equations without the kinematical condition above referred to. If the solution shews that there is a relative tangential velocity at A and the direction of motion is opposed to what has been assumed as the direction of the friction then we have found the true motion and it will continue until relative velocity at A vanishes.

16·2. A wheel spinning about a horizontal axis is projected along a rough horizontal plane, to determine the subsequent motion.

Let m be the mass and a the radius of the wheel, mκ2 its moment of inertia about its centre G, and A the point of contact with the plane.