Explain why a torque applied to a rigid body for a given time interval produces a greater change in angular velocity when the mass of the body is distributed close to the axis of rotation compared with when it is distributed far from the axis of rotation.
Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).
Model answer (4 marks)
Torque equals the rate of change of angular momentum (τ = ΔL/Δt), so a given torque applied for a given time produces a fixed change in angular momentum ΔL.
Angular momentum equals the product of moment of inertia and angular velocity (L = Iω), so ΔL = IΔω.
When mass is distributed close to the axis, the moment of inertia I is smaller, because I depends on mr² and r is smaller for each mass element.
Since ΔL is fixed and I is smaller, Δω = ΔL/I must be larger, so the body gains a greater change in angular velocity.
Angular momentum equals the product of moment of inertia and angular velocity (L = Iω), so ΔL = IΔω.
When mass is distributed close to the axis, the moment of inertia I is smaller, because I depends on mr² and r is smaller for each mass element.
Since ΔL is fixed and I is smaller, Δω = ΔL/I must be larger, so the body gains a greater change in angular velocity.
Examiner tips
- Use the equation τ = ΔL/Δt to show ΔL is fixed for a given torque and time. Use L = Iω to relate ΔL to I and Δω. Explain that I is smaller when mass is nearer the axis. Show that Δω = ΔL/I, so a smaller I gives a larger Δω.
Common mistakes
- Confusing torque with angular acceleration instead of angular momentum. Forgetting to state that ΔL is fixed for the same torque and time. Using the wrong relationship between I and r (e.g., saying I is larger when mass is close to the axis).
Mark scheme (4 marks)
- Torque equals the rate of change of angular momentum (τ = ΔL/Δt), so a given torque applied for a given time produces a fixed change in angular momentum ΔL.
- Angular momentum equals the product of moment of inertia and angular velocity (L = Iω), so ΔL = IΔω.
- When mass is distributed close to the axis, the moment of inertia I is smaller, because moment of inertia depends on mr² and r is smaller for each mass element.
- Since ΔL is fixed and I is smaller, Δω = ΔL/I must be larger, so the body gains a greater change in angular velocity.
Key terms in this question
torque · angular velocity · axis of rotation
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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