Rolling Motion

Rolling motion combines translation of the centre of mass with rotation about the centre of mass. For rolling without slipping, the key constraint is: v_cm = ω R When a contact-force limit is relevant, the 9814 syllabus uses: F ≤ μ N

1. Definitions (Must Know)

• Pure rolling (no slip): the point of contact with the ground has zero velocity relative to the ground at the instant of contact.
• Rolling with slipping: the contact point slides; friction does kinetic-energy dissipation and v_cm ≠ ω R in general.
• No-slip condition (pure rolling): in magnitude, v_cm = ω R and, for the tangential acceleration components, a_cm = α R. Signed equations depend on the chosen angular convention.
• Kinetic energy split (about CM): Eₖ = 1/2 Mv_cm² + 1/2 I_cmω²
• Syllabus contact-force test: F ≤ μ N. The official 2027 H3 Physics syllabus does not distinguish static and kinetic coefficients in this model.
• Symbols used in this lesson: v_cm (m s⁻¹), a_cm (m s⁻²), ω (rad s⁻¹), α (rad s⁻²), R (m), M (kg), I_cm (kg m²), F friction (N), N normal reaction (N), μ coefficient of friction (dimensionless).

2. Key Ideas (What Earns Marks)

• Model rolling as: translation of CM + rotation about CM (H3 syllabus statement).
• For pure rolling:
  • Use the constraint v_cm = ω R early to reduce unknowns.
  • Use a_cm = α R for dynamics problems with acceleration.
• A contact force may be needed to enforce no slip, but it can be zero for uniform rolling on a horizontal surface. When the problem supplies μ, check whether the required force exceeds μ N.

Quick comparison:

3. Detailed Explanations

A. Why “translation + rotation about CM” is the right picture

Any rigid body motion can be decomposed into:

1. motion of the centre of mass, plus
2. rotation about an axis through the centre of mass.

For rolling problems, this decomposition helps you:

• write linear equations for the CM (e.g. ∑ F = Ma_cm), and
• write rotational equations about the CM (e.g. ∑ τ_cm = I_cmα).

B. Deriving the no-slip constraint

For a wheel of radius R rolling without slipping:

• the contact point’s speed relative to the ground is zero.

The contact point’s velocity is the vector sum of:

• translational CM velocity v_cm (to the right, say), and
• rotational velocity about the CM of magnitude ω R (to the left at the bottom point).

So they cancel: v_cm - ω R = 0 ⇒ v_cm = ω R

Differentiating gives: a_cm = α R

C. Contact force and the condition F ≤ μ N

In a more detailed contact model, no relative sliding corresponds to static friction. For this H3 syllabus, however, no distinction is made between static and kinetic coefficients: calculate the required contact-force magnitude F and test F ≤ μ N

If the required force exceeds μ N, the assumed no-slip motion is not possible. If the required force is zero, pure rolling can continue without a frictional force.

4. Common Mistakes

• Assuming friction must be non-zero or must remove energy in every rolling problem.
• Applying v_cm = ω R even when the question implies slipping.
• Using I about the wrong axis (use I_cm when taking moments about the centre of mass).
• Forgetting to check the no-slip condition with F ≤ μ N when μ is given.

5. Exam Tips

• If the phrase “rolls without slipping” appears, write v_cm = ω R immediately.
• If μ is given (or “rough surface” is emphasised), expect a slip check:
  1. solve assuming no slip,
  2. compute required F,
  3. compare with μ N.
• Use separate equations for translation and rotation:
  • ∑ F = Ma_cm
  • ∑ τ_cm = I_cmα plus a_cm = α R for no slip.

6. Worked Examples

7. Mind Stretchers

8. Optional/Enrichment: Rolling as Instantaneous Rotation

In pure rolling, the wheel’s motion can be seen as instantaneous rotation about the contact point. This viewpoint can be helpful, but the CM-decomposition method (translation + rotation about CM) is usually the most robust for exam problems.

Next step

Return to the Rotational Motion hub, or continue to Electric and Magnetic Fields, the next maintained H3 topic.