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.
Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).
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.
Model answer (4 marks)
For a body rolling without slipping the gravitational potential energy lost is divided between translational and rotational kinetic energy:
(mgh = frac12 mv^2 + frac12 Iomega^2) with (omega = v/r).
For a solid sphere (I = frac25 mr^2) and for a solid cylinder (I = frac12 mr^2). The cylinder therefore has a larger moment of inertia. A larger (I) means a larger fraction of the energy is stored in rotation, leaving less for translation. Consequently the sphere attains a higher linear speed (and greater linear acceleration) down the slope. Because the sphere reaches a higher speed, it reaches the bottom earlier than the cylinder.
(mgh = frac12 mv^2 + frac12 Iomega^2) with (omega = v/r).
For a solid sphere (I = frac25 mr^2) and for a solid cylinder (I = frac12 mr^2). The cylinder therefore has a larger moment of inertia. A larger (I) means a larger fraction of the energy is stored in rotation, leaving less for translation. Consequently the sphere attains a higher linear speed (and greater linear acceleration) down the slope. Because the sphere reaches a higher speed, it reaches the bottom earlier than the cylinder.
Examiner tips
- Show the energy split equation and the relation (omega=v/r).
- State the two moments of inertia and compare them.
- Explain how a smaller (I) gives more translational energy.
- Conclude that the sphere reaches the bottom first.
Common mistakes
- Using the wrong moment of inertia for the sphere or cylinder.
- Ignoring the rotational term and treating both as sliding blocks.
Mark scheme (4 marks)
- For rolling without slipping, the total kinetic energy is split between translational and rotational kinetic energy.
- The sphere has a smaller moment of inertia (2/5 mr²) compared to the cylinder (1/2 mr²) for the same mass and radius.
- A smaller moment of inertia means less energy is stored in rotation, so more energy goes into translational kinetic energy for the sphere.
- Therefore the sphere reaches the bottom first because it attains a greater translational speed (linear acceleration) down the slope.
Related
- All IB DP Physics Higher Level (2023 syllabus) revision notes →
- How to answer a "Explain" question →
- Decode the mark scheme abbreviations →
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