Suppose a solid uniform sphere of mass M and radius R rolls without slipping down an inclined plane starting from rest.The angular velocity of the sphere at the bottom of the incline depends on
A)the mass of the sphere.
B)the radius of the sphere.
C)both the mass and the radius of the sphere.
D)neither the mass nor the radius of the sphere.

Question

Quizplus · Accepted Answer

The angular velocity of the sphere at the bottom of the incline only depends on the radius of the sphere. This can be derived from the conservation of energy principle, where the potential energy at the top of the incline is converted into both kinetic energy of translation and rotational kinetic energy at the bottom of the incline. Since the sphere is rolling without slipping, the kinetic energy of translation is related to the rotational kinetic energy through a factor of (2/5)MR^2, where M is the mass and R is the radius of the sphere. Therefore, the final angular velocity is proportional to the square root of the ratio of the rotational kinetic energy to (2/5)MR^2, which only depends on the radius of the sphere.

Suppose a Solid Uniform Sphere of Mass M and Radius