Question

A uniform sphere and a uniform disk, each with mass M and radius R, are both initially at rest on identical inclined planes. Both objects roll the same distance down the inclined

planes without slipping. Which object has the greater total kinetic energy when it reaches the bottom of the ramp?

A) The disk has greater kinetic energy when it reaches the bottom.

B) The sphere has greater kinetic energy when it reaches the bottom.

C) Both objects have the same kinetic energy when they reach the bottom.

D) There is not enough information to answer this question.

Answer 1

When a uniform sphere and a uniform disk roll down identical inclined planes without slipping, both objects will have the same total kinetic energy when they reach the bottom.

Step-by-step explanation:

When a uniform sphere and a uniform disk roll down identical inclined planes without slipping, both objects will have the same total kinetic energy when they reach the bottom.

The kinetic energy of a rotating object can be divided into two components: translational kinetic energy (KEtrans) and rotational kinetic energy (KErot). The total kinetic energy (KEtotal) is the sum of these two components. In this case, both the sphere and the disk have the same mass and radius, so they will have the same moment of inertia (I) and hence the same rotational kinetic energy. Therefore, the only difference in their total kinetic energies will come from their translational kinetic energies. However, the translational kinetic energy depends on the linear velocity and since both objects are rolling without slipping and travel the same distance, their linear velocities will be the same and therefore their translational kinetic energies will also be the same. Thus, both objects will have the same total kinetic energy when they reach the bottom of the ramp.