The big idea: Rotational motion is straight-line motion gone round in a circle. Every quantity has a spinning twin:
- distance → angle turned θ (the angular displacement, in radians) - velocity → angular velocity ω (the angle turned per second, rad s⁻¹) - acceleration → angular acceleration α (how fast ω changes, rad s⁻²)
What is a radian?: One radian is the angle whose arc length equals the radius. A full turn = 2π rad ≈ 6.28 rad = 360°. Always work in radians for these formulas.
On an angular-velocity–time (ω–t) graph you read two things, exactly like a v–t graph: the slope is the angular acceleration α, and the area under the line is the angle turned θ. For constant α the motion obeys the rotational suvat equations.
- final / initial angular velocity (rad s⁻¹)
- angular acceleration (rad s⁻²)
- angle turned (rad)
- time (s)
Worked example — find the angular acceleration
On the graph above the angular velocity rises from ω₀ = 2.0 rad s⁻¹ to ω = 18 rad s⁻¹ over t = 8.0 s. Find the angular acceleration.
Solution
- Start from the given equation:
- Put in the values read off the graph:
- Solve for α — keep the unit:
Final answer
α = 2.0 rad s⁻².
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What makes something spin?: A torque (τ) is the turning effect of a force — the rotational version of force. It depends on the force F, the distance r from the pivot, and the angle θ between them. Only the part of the force perpendicular to the arm turns the body.
- torque (N m)
- applied force (N)
- distance from pivot to where the force acts (m)
- angle between the force and the arm r
Worked example — tightening a bolt
A spanner is 0.18 m long. You push on its end with 25 N at right angles to the spanner. Find the torque on the bolt.
Solution
- Write the given formula first:
- At right angles θ = 90°, sin 90° = 1:
- Work it out — keep the unit:
Final answer
τ = 4.5 N m. (Push at 40° instead and only 25 × 0.18 × sin 40° = 2.9 N m — the angle matters.)
When is a body balanced?: A rigid body is in rotational equilibrium when the total torque about any point is zero:
clockwise torques = anticlockwise torques.
(For full equilibrium the forces must also balance, ΣF = 0.)
Worked example — balancing a beam
A light beam rests on a central pivot. A 40 N weight hangs 0.60 m to the left. A downward force F is applied 0.80 m to the right. Find F so the beam balances.
Solution
- Balanced ⇒ anticlockwise torque = clockwise torque (torque = force × distance):
- Make F the subject:
- Work it out:
Final answer
F = 30 N.
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Where it shows up: Rotational motion and torque are HL only (A.4):
- Paper 1A — a one-step 'how many revolutions?', 'what is α?', or 'which has the largest torque?'. - Paper 2 — determine a force or tension by taking torques about a clever pivot, or find a final ω/angle with the rotational suvat.
Three easy marks: (1) Convert any revolutions to radians (× 2π) first. (2) For torque use the perpendicular distance (the sin θ). (3) Balanced ⇒ set clockwise = anticlockwise.
IB-style question — a turntable speeding up
A turntable starts from rest and speeds up uniformly to 12 rad s⁻¹ in 3.0 s. Determine the angle it turns through, and hence how many revolutions it makes.
Solution
- Angle = area under the ω–t line. Use the given average-velocity formula:
- Substitute (from rest, ω₀ = 0):
- Revolutions = angle ÷ one full turn (2π rad):
Final answer
θ = 18 rad ≈ 2.9 revolutions (2 complete turns plus a bit more).