Chapter 13, Problem 72PQ

A solid sphere and a hollow cylinder of the same mass and radius have a rolling race down an incline as in Example 13.9 (page 372). They start at rest on an incline at a height h above a horizontal plane. The race then continues along the horizontal plane. The coefficient of rolling friction between each rolling object and the surface is the same. Which object rolls the farthest? (Justify your answer with an algebraic expression.)

72. Conservation of energy provides a very simple approach to this problem. Each object starts at rest on the incline, and each object stops on the horizontal surface. Along the way there is an increase in thermal energy between the surface and the object. Let’s include the Earth, the rolling object, and the surface in the system. We set the reference configuration to the horizontal surface. We can create an energy bar chart as we’ve done

Chapter 13 - Rotation II: A Conservation Approach                    13-44

previously to see that the initial gravitational potential energy is eventually dissipated as thermal energy as the object rolls a total distance S.

$U_{gi}=\Delta E_{th}$

$mgh={\mu }_r{F}_{N}S$

$S=\frac{mgh}{{\mu }_r{F}_N}$

Each object has the same mass m, is released from the same height h, and has the same coefficient of rolling friction ${\mu }_r$ with the surface. The normal force exerted by the surfaces on the each object is also the same since the objects have the same mass. So, S is the same for both objects. In other_words, they travel the same distance from the starting point. This result may be surprising, but it is not a race in the traditional sense. We didn’t ask which object arrived at the finish line first. Instead, we asked where the finish line is. The sphere gets there sooner because it has a small rotational inertia, so it rolls down the incline at a higher speed.

Figure P13.72ANS