Question

An inextensible rope tied to the axle of a wheel of mass m and radius r is pulled in the horizontal direction in the plane of the wheel. The wheel rolls without jumping over a grid consisting of parallel horizontal rods arranged at a distance l from one another (l << r).

Determine the average tension T of the rope at which the wheel moves at a constant velocity v, assuming the mass of the wheel to be concentrated at its axle.

Answer 1

Let us suppose that at a certain instant, the wheel is in one of the positions such that its centre of mass is above a rod, and its velocity is v. At the moment of the impact against the next rod (Fig. 171), the centre of mass of the wheel has a

certain velocity v' perpendicular to the line connecting it to the previous rod. This velocity can be obtained from the energy conservation law:

By hypothesis (the motion is without jumps), the impact of the wheel against the rod is perfectly inelastic. This means that during the impact, the projection of the momentum of the wheel on the straight line connecting the centre of the wheel to the rod vanishes. Thus, during each collision, the energy

where sin α ≈ l/r, is lost (converted into heat). For the velocity v to remain constant, the work done by the tension T of the rope over the path l must compensate for this energy loss. Therefore,