Question

The objects described below all have the same mass. The cube has edge lengths R and is allowed to slide from rest down an inclined ramp. Friction between the cube and the ramp is sufficiently small that you may ignore it. The other objects have the same radius R and are allowed to roll from rest the same distance down the same ramp. Some friction is necessary for each object to roll on the ramp. Assume that there is adequate friction to allow rolling. With some thought you can see that this friction does no work as the objects roll and therefore won't complicate your analysis. Order the various objects from first to last by the time each requires to reach the bottom of the ramp.

Answer 1

In rolling motion down an inclined plane, a cube sliding without friction reaches the bottom first, followed by rounded objects in order of increasing moment of inertia. A solid sphere, with its lower moment of inertia, will descend quicker than a solid cylinder, and a hoop will be the slowest. When comparing two cylinders, one rolling without slipping and one sliding, both reach the same height on another incline, regardless of their different motions, due to the conservation of energy.

Step-by-step explanation:

The rolling motion of objects down an inclined plane without slipping means that the objects undergo both translational and rotational motion. When comparing the times it takes various objects to reach the bottom of an inclined ramp, the moment of inertia plays a significant role because it determines the distribution of mass and affects the object's angular acceleration. A cube sliding down a frictionless surface will only experience translational motion, reaching the bottom first. Rounded objects such as cylinders, spheres, and hoops have different moments of inertia, influencing their acceleration and, hence, how quickly they reach the bottom.

A solid sphere will reach the bottom before a solid cylinder due to its lower moment of inertia, whereas a hoop will take the longest time because it has the highest moment of inertia. This is because the moment of inertia affects how much of the gravitational potential energy is converted into rotational kinetic energy, with a lower value allowing for faster translational speeds. By contrast, for the case of the cylinders starting on identical inclines where one rolls without slipping and the other slides without friction, they reach the same height on another incline due to the conservation of energy.