Rolling Down the Ramp: Which Object Reaches the Bottom First?

Question:

If a uniform disk, a uniform hoop, and a uniform sphere are released at the same time at the top of an inclined ramp and they all roll without slipping, in what order do they reach the bottom of the ramp?

a. disk, hoop, sphere b. sphere, hoop, disk c. hoop, sphere, disk d. sphere, disk, hoop e. hoop, disk, sphere

Answer:

D. The sphere, the disk, and the hoop

When the uniform disk, uniform hoop, and uniform sphere are released at the top of an inclined ramp and they all roll without slipping, the sphere will reach the bottom of the ramp first, followed by the disk, and finally the hoop.

The reason for this order lies in the moment of inertia of each object. The sphere has a moment of inertia of 2/5mR², which is the smallest among the three objects. The disk has a moment of inertia of 1/2mR², while the hoop has a moment of inertia of mR². Moment of inertia is the resistance or opposition to rotation, so the object with the smaller moment of inertia will have less resistance and move faster.

Therefore, due to their different moments of inertia, the sphere will move the fastest and reach the ground first, followed by the disk and then the hoop.