The big idea: Watch a merry-go-round spin up, or a wheel roll faster and faster: that is rotational motion, straight-line motion gone round in a circle. Every quantity has a spinning twin.
| Straight-line quantity | Spinning twin | Unit |
|---|---|---|
| distance | angle turned θ (the angular displacement) | 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.
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On an angular-velocity–time (ω–t) graph you read two things, exactly like a v–t graph.
What the ω–t graph tells you
- The slope is the angular acceleration α.
- The area under the line is the angle turned θ.
- For constant α the motion obeys the rotational suvat equations.
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- final / initial angular velocity (rad s⁻¹)
- angular acceleration (rad s⁻²)
- angle turned (rad)
- time (s)
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.
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Rolling without slipping
A wheel or rod that rolls without slipping links its turning to its moving along:
Turns into distance: A wheel of radius 0.35 m that rolls 6.0 m has turned θ = s ÷ r = 6.0 ÷ 0.35 = 17 rad, about 2.7 revolutions.
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What makes something spin?: A torque (τ) is the turning effect of a force: the rotational version of force.
Torque depends on
- the force F
- the distance r from the pivot
- the angle θ between them
Only the perpendicular part turns: 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
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.
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An unbalanced torque spins it up: If the torques do not cancel, the resultant torque gives the body an angular acceleration, just as a resultant force gives a linear one.
A rod rolling under a moving board: friction from the board pushes the top one way, friction from the ground pushes the bottom the other way. The two make a torque in the same sense, so the rod spins up.
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.)
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.
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Where the weight acts: Treat the whole weight of a rigid body as acting at its centre of mass: the middle of a uniform beam or box.
The same balance tells you when a box tips over:
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How this is tested — 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.
The classic trap: Working in degrees, or forgetting to convert revolutions to radians (× 2π) before using the formulas — always work in radians.
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.
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.
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