Angular velocity and rotational kinematics
Rotation about a fixed axis
A ceiling fan, a wheel on its axle, a door on its hinges. Each is a rigid body turning about a fixed axis.
In pure rotation, every particle of the body moves in a circle. The circles lie in planes perpendicular to the axis, with their centres on the axis.
Mark three dots on a spinning disc: one near the centre, one halfway out, one at the rim. In the same time, all three turn through the same angle.
So the whole body has one angular velocity ω, the rate at which its angle changes: ω = dθ/dt, measured in rad/s.
But the dots do not have the same speed. The rim dot goes round a bigger circle, so it moves faster. Its speed is v = ωr, where r is its distance from the axis.
Angular velocity is a vector along the axis. Curl your right-hand fingers the way the body turns, and your thumb points along ω.
With vectors, the velocity of a particle is v = ω × r. For a fixed axis, ω keeps the same direction; only its size can change.
Equations of rotational motion
Switch on a ceiling fan. It starts slowly and speeds up. Its angular velocity is changing.
The rate of change of angular velocity is the angular acceleration: α = dω/dt, measured in rad/s².
When α is constant, rotation follows equations just like straight-line motion. θ plays the part of x, ω plays v, and α plays a.
First: ω = ω₀ + αt. The angular velocity changes steadily, so the ω–t graph is a straight line.
Second: θ = ω₀t + ½αt². The angle turned is the area under the ω–t graph.
Third: ω² = ω₀² + 2αθ. Use it when the time is not given.
Example: a wheel starts at 4 rad/s and speeds up at 2 rad/s² for 3 s. Then ω = 4 + 2 × 3 = 10 rad/s, and θ = 4 × 3 + ½ × 2 × 9 = 21 rad.
Check with the third equation: 4² + 2 × 2 × 21 = 16 + 84 = 100, and 10² is 100. It works.
Try it: spin up a wheel
- Press Play. The green arrow is the velocity of the amber dot.
- Move the dot towards the axle: ω stays the same, but v = ωr gets smaller.
- Make α negative and watch the wheel slow down, stop and turn back.
- The last two readouts check the equation ω² = ω₀² + 2αθ.
Quick check
A rigid disc spins about its axis. Which quantity is the same for a point near the rim and a point near the centre?
- Linear speed v
- Angular velocity ω
- Distance travelled in one second
- Centripetal acceleration
Answer
B. Every point of a rigid body turns through the same angle in the same time, so ω is the same everywhere. The speed v = ωr, distance travelled and centripetal acceleration ω²r all grow with r.
Work it out
The angular speed of a motor wheel rises from 1200 rpm to 3120 rpm in 16 s. What is its angular acceleration, assuming it is uniform?
- 2π rad/s²
- 4π rad/s²
- 8π rad/s²
- 120 rad/s²
Answer
B. Convert to rad/s: ω₀ = 2π × 1200/60 = 40π rad/s and ω = 2π × 3120/60 = 104π rad/s. Then α = (ω − ω₀)/t = 64π/16 = 4π rad/s², about 12.6 rad/s².
Summary
- Speed of a particle:
v = ωr (vector form v = ω × r) - Angular acceleration:
α = dω/dt - Constant α:
ω = ω₀ + αt - Angle turned:
θ = ω₀t + ½αt² - Without time:
ω² = ω₀² + 2αθ
- In rotation about a fixed axis, every particle moves in a circle and the whole body has one angular velocity ω.
- Particles farther from the axis move faster: v = ωr.
- ω points along the axis, given by the right-hand rule.
- With constant α, the rotation equations match the straight-line ones with x → θ, v → ω, a → α.