W2.5 Hollow versus Solid
A ring and a disc of equal mass are released from rest at the top of an inclined plane, as shown in the photograph below.
A race will ensue, with both objects rolling down the incline.
When the race is started by releasing both rolling objects from rest simultaneously at the same point on the incline, what is the result?
- the ring will win.
- the disc will win.
- the race will end in a tie.
Solution
Reveal

Solution
The answer is (b): the disc will win the race, as can be seen in this MPEG movie clip.
The moment of inertia of the disc about its axis is ,
but the moment of inertia of the ring about its axis is mr2.
Employing the parallel axis theorem, when the masses of the ring
and the disc are the same, an additional moment of inertia of
mr2 must be added to each of the above moments to obtain the
moment of inertia about the point of contact of the roller with
the inclined plane, giving:

The torque due to gravity acting on either roller is mgrsina, where a is the angle of the incline. The angular acceleration of the rollers is then:

The linear acceleration of each down the incline is the equal to the radius of the object multiplied by the angular acceleration:

Related Problems
Suppose that two uniform discs with different masses were raced down the incline. What would happen?
Suppose that a ring and a disc with different mass were raced down the incline. What would happen?
Suppose that two uniform discs with different radii were raced down the incline. What would happen?
By looking up the moment of inertia of various rolling objects in a table, place the following in order of how fast they would roll down an incline if released from rest at the same point at the top of the incline:
- a light uniform disc
- a heavy ring
- a large, light ball
- a heavy uniform disc
- a light ring
- a small, heavy ball