Kinematics of Angular Motion

Kinematics of angular motion describes how a rigid body rotates about a fixed axis. The payoff is that the rotational equations under uniform angular acceleration are direct analogues of SUVAT.

1. Definitions (Must Know)

• Angular displacement, θ (rad): angle turned about a fixed axis.
• Angular velocity, ω (rad s⁻¹): rate of change of angular displacement, ω = dθ/dt.
• Angular acceleration, α (rad s⁻²): rate of change of angular velocity, α = dω/dt = d²θ/dt².
• Fixed axis rotation: every point on the rigid body moves in a circle centred on the axis, with the same θ, ω, α at any instant.
• Radian: θ = s/r where s is arc length and r is radius.
• Symbols used in this lesson: θ,θ₀ angle (rad), ω,ω₀ angular speed (rad s⁻¹), α angular acceleration (rad s⁻²), t time (s), r radius (m), s arc length (m), v tangential speed (m s⁻¹), aₜ tangential acceleration (m s⁻²).

2. Key Ideas (What Earns Marks)

• Rotational kinematics mirrors linear kinematics:
  • x ↔ θ, v ↔ ω, a ↔ α.
• For a point at distance r from the axis: s = rθ, v = rω, aₜ = rα
• Under uniform angular acceleration (α constant), use the rotational “SUVAT” set:
  ω = ω₀ + α t; θ = θ₀ + ω₀ t + (1/2)α t²; ω² = ω₀² + 2α(θ-θ₀)

Quick map (use a table when asked to compare):

3. Detailed Explanations

A. Linear–rotational links for a point on a rotating rigid body

If a point is at radius r from the axis:

• The arc length travelled is s = rθ.
• Differentiating gives tangential speed: v = ds/dt = rdθ/dt = rω
• Differentiating again gives tangential acceleration: aₜ = dv/dt = rdω/dt = rα

These are especially useful when a question gives a linear speed/acceleration and asks for ω or α (or vice versa).

B. Deriving the constant-α equations (one clean route)

If α is constant: α = dω/dt ⇒ ω = ω₀ + α t

Also ω = dθ/dt, so (use a dummy variable τ):

To eliminate t, use α = dω/dt = (dω/dθ)dθ/dt = ωdω/dθ:

4. Common Mistakes

• Using degrees in calculations without converting to radians (units of ω and α require radians).
• Confusing tangential acceleration aₜ = rα with centripetal acceleration aᵣ = rω² (different directions, different causes).
• Forgetting that r can differ for different points on the same rigid body (so different points have different v and aₜ, even though ω and α are shared).
• Substituting into the constant-α equations when α is not constant (the question must justify “uniform angular acceleration”).

5. Exam Tips

• Write a one-line map before you start: “use θ–ω–α as the rotational analogues of s–v–a.”
• Keep a clear sign convention (clockwise vs anticlockwise) and stick to it.
• Use radians by default; only convert to degrees at the end if the question asks for it.
• If a question mixes linear and rotational quantities, immediately write v = rω and aₜ = rα for the point of interest.

6. Worked Examples

7. Mind Stretchers

8. Optional/Enrichment: Variable Angular Acceleration (Calculus Form)

If α is not constant, the constant-α equations do not apply. The safe starting points are: ω = dθ/dt, α = dω/dt and you integrate/differentiate according to what the question gives.

Next step

Return to the Rotational Motion hub, or continue in sequence to Moment of Inertia.