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Question

A hoop, a uniform solid cylinder, a spherical shell, and a uniform solid sphere are released from rest at the top of an incline. What is the order in which they arrive at the bottom of the incline? Does it matter whether or not the masses and radii of the objects are all the same?
Explain.

Answer 1

If they slide down the incline, it doesn't matter. But if they roll down the incline, it does.

When a object rolls, the distance it travels is proportional to its angular velocity. If it starts at rest, it has to undergo angular acceleration in order to change its angular velocity.

For linear acceleration, the formula is F = MA
For angular acceleration, the formula is T = IA where:
T is the torque
I is the moment of inertia
A is the angular acceleration

All these objects have different moments of inertia:
http://en.wikipedia.org/wiki/List_of_moments_of_inertia

so even with the same torques, their angular accelerations will be different.

In this case, the torque is produced by the frictional force. In an ideal rolling situation, the point of the rolling object touching the surface has a net velocity of 0 - it is the center around which the rest of the object rotates.

The torque depends on the mass of the object, its radius, and the slope of the incline:
http://en.wikipedia.org/wiki/Torque

If this still isn't enough, then look at it:
http://www.schoolforchampions.com/science/friction_rolling_start.htm

and then:
http://cnx.org/content/m14384/latest/