Rotational Dynamics & Rolling (IPhO Mechanics)

Rigid body questions become straightforward once you separate (1) translation of the centre of mass from (2) rotation about a chosen axis, and treat rolling as a constraint rather than “a friction force you guess”.

Learning objectives

By the end, you should be able to:

• IPHO-M-ROT-01: combine ∑ F = ma_cm, ∑τ_cm = I_cmα and the correct contact constraint;
• IPHO-M-ROT-02: infer static-friction direction from incipient slip and verify |fₛ| ≤ μₛN; and
• check the chosen moment of inertia, sign convention and limiting behaviour.

Scroll diagram horizontally to read all labels.

1. Definitions

• Torque about point O: vector τ_O = vector r × vector F, and in general ∑ vector τ_O = d vector L_O/dt.
• Moment of inertia about an axis: I = ∫ r_⊥² dm (distance perpendicular to the axis).
• Parallel axis theorem: I_O = I_cm + md².
• Angular momentum about the centre (fixed axis through centre): L_cm = I_cmω.
• Rolling without slipping (radius R on a stationary surface):
  v_cm = ω R, a_cm = α R
  along the direction of motion.
• Static friction fₛ: adjusts to enforce no slip, with |fₛ| ≤ μₛ N.
• Instantaneous contact inertia: for a rolling body, I_P = I_cm + mR² about the contact point P.

2. Key ideas

• Default template:
  ∑ Fₓ = m a_cm, ∑τ_cm = I_cmα, a_cm = α R.
• On a stationary surface with pure rolling, static friction does no work at the contact (instantaneous contact speed is zero). Energy methods are often the fastest.
• The rolling acceleration down an incline has a memorisable form:
  a = (g sin α)/(1 + I_cm/(mR²)).
• Friction direction is set by the tendency to slip at the contact, not by the direction of the centre-of-mass velocity.

3. Detailed explanation

3.1 Translation plus rotation (why it is reliable)

For planar rolling with one translational coordinate x and one rotation angle θ, you can nearly always write:

• translation of the centre of mass,
• rotation about the centre of mass,
• the no-slip constraint relating x and θ.

This avoids guessing friction and gives you a closed system of equations.

3.2 Why torque about contact is not the default

The contact point is an instantaneous centre of zero velocity, but it is generally not a fixed inertial pivot and may have nonzero acceleration. Therefore the fixed-axis equation ∑τ_P = I_Pα cannot be invoked solely because the contact point is instantaneously at rest.

Use centre-of-mass translation plus centre-of-mass torque as the reliable dynamics route. An angular-momentum calculation about a moving point needs the full transport relation and an explicit justification; it is rarely the shortest safe competition solution.

3.3 Static friction: direction and size

A clean approach:

1. Decide the likely direction of slip if the surface were frictionless.
2. Set friction opposite that slip tendency.
3. Solve for fₛ from dynamics.
4. Verify |fₛ| ≤ μₛ N. If not, rolling without slipping is impossible and the motion must involve slipping.

4. Common mistakes

• Using v = ω R even though the body is slipping (constraint broken).
• Using the wrong axis for I or forgetting the parallel axis theorem.
• Assuming friction always opposes motion of the centre of mass.
• Writing ∑τ = Iα about a point that is accelerating without justification.

5. Problem-solving tips

1. State your sign convention early (clockwise positive or anticlockwise positive).
2. If you compute friction, always check it against μₛ N.
3. Use the limit I_cm → 0 as a quick checksum: you should recover particle-like motion.
4. If a surface is moving (belt), do not assume static friction does no work in the lab frame.

6. Worked examples

7. Extension

8. Practice and evidence