Question
Consider two cylinders that start down identical inclines from rest except that one is frictionless. Thus one cylinder rolls without slipping, while the other slides frictionlessly without rolling. They both travel a short distance at the bottom and then start up another incline. (a) Show that they both reach the same height on the other incline, and that this height is equal to their original height. (b) Find the ratio of the time the rolling cylinder takes to reach the height on the second incline to the time the sliding cylinder takes to reach the height on the second incline. (c) Explain why the time for the rolling motion is greater than that for the sliding motion.

a) see solution video

b) $\sqrt{\dfrac{3}{2}}$

c) Translational velocity for the rolling case is smaller since some energy is allocated to rotational kinetic energy. A smaller translational velocity means a greater time taken.

Solution Video