# Explain why a uniform solid sphere and a uniform solid cylinder of the same mass and radius reach the bottom of an inclined plane at different times when released from rest simultaneously and allowed to roll without slipping.

> IB DP Physics Higher Level (2023 syllabus) — A.4 Rigid body mechanics (HL only) · Explain · 4 marks

> Two uniform solid objects — a sphere and a cylinder — have identical masses and radii. They are released from rest at the top of the same inclined plane and roll without slipping to the bottom.

## Mark scheme (4 marks)

1. For rolling without slipping, the total kinetic energy is split between translational and rotational kinetic energy.
2. The sphere has a smaller moment of inertia (2/5 mr²) compared to the cylinder (1/2 mr²) for the same mass and radius.
3. A smaller moment of inertia means less energy is stored in rotation, so more energy goes into translational kinetic energy for the sphere.
4. Therefore the sphere reaches the bottom first because it attains a greater translational speed (linear acceleration) down the slope.

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