A sphere and a cylinder having the same mass and radii start from rest and roll down the same inclined plane. Which body gets to the bottom first? A. Sphere with greater rotational energy at bottom than cylinder B. Sphere with lesser rotational energy at bottom than cylinder C. Cylinder with greater rotational energy at bottom than sphere D. Both reach the bottom simultaneously with equal rotational energy at bottom

The correct answer is: Both reach the bottom simultaneously with equal rotational energy at bottom.

The sphere and the cylinder have the same mass and radii, so they have the same potential energy at the top of the inclined plane. When they roll down the plane, they both convert some of their potential energy into kinetic energy. The sphere will have a greater linear velocity than the cylinder, but the cylinder will have a greater rotational velocity. The total kinetic energy of the sphere and the cylinder will be the same.

The sphere and the cylinder will reach the bottom of the plane at the same time. The sphere will have a greater linear velocity, but the cylinder will have a greater rotational velocity. The total kinetic energy of the sphere and the cylinder will be the same.

The following is a brief explanation of each option:

• Option A: The sphere with greater rotational energy at bottom than cylinder. This is not correct because the sphere and the cylinder have the same mass and radii, so they have the same potential energy at the top of the inclined plane. When they roll down the plane, they both convert some of their potential energy into kinetic energy. The sphere will have a greater linear velocity than the cylinder, but the cylinder will have a greater rotational velocity. The total kinetic energy of the sphere and the cylinder will be the same.
• Option B: The sphere with lesser rotational energy at bottom than cylinder. This is not correct because the sphere and the cylinder have the same mass and radii, so they have the same potential energy at the top of the inclined plane. When they roll down the plane, they both convert some of their potential energy into kinetic energy. The sphere will have a greater linear velocity than the cylinder, but the cylinder will have a greater rotational velocity. The total kinetic energy of the sphere and the cylinder will be the same.
• Option C: Cylinder with greater rotational energy at bottom than sphere. This is not correct because the sphere and the cylinder have the same mass and radii, so they have the same potential energy at the top of the inclined plane. When they roll down the plane, they both convert some of their potential energy into kinetic energy. The sphere will have a greater linear velocity than the cylinder, but the cylinder will have a greater rotational velocity. The total kinetic energy of the sphere and the cylinder will be the same.
• Option D: Both reach the bottom simultaneously with equal rotational energy at bottom. This is the correct answer because the sphere and the cylinder have the same mass and radii, so they have the same potential energy at the top of the inclined plane. When they roll down the plane, they both convert some of their potential energy into kinetic energy. The sphere will have a greater linear velocity than the cylinder, but the cylinder will have a greater rotational velocity. The total kinetic energy of the sphere and the cylinder will be the same.