The objects described below all have the same mass. The cube has edge lengths R

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.

Cube                        [ Select ] ["4", "3", "1", "2"]

Thin Hoop                        [ Select ] ["3", "2", "1", "4"]

Cylinder                        [ Select ] ["2", "3", "1", "4"]

Sphere                        [ Select ] ["4", "1", "3", "2"]

Explanation / Answer

Order:

1)Cube

2)Sphere

3) Cylinder

4)Hoop

Cube reaches first since it's motion is frictionless, while for others there is friction.

Among others, solid sphere has the least radius of gyration, so has the maximum acceleration and reaches the bottom first (among the rolling objects). Again, velocity of cylinder while rolling down (=4/3 gh) is greater than that of hoop (=gh), so it reaches earlier than the hoop.

[Note: The order is from the object reaching the bottom 1st to the one reaching at last]

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