Angular velocity and tangential speed
Key idea: H2 Physics lessons on angular displacement, angular velocity, tangential speed, centripetal acceleration and radial force models.
Continue where you stopped
The core idea
Build the idea
Learn the idea
Big question: How does one angular speed produce different linear speeds?
Angular velocity is the rate of change of angular displacement, ω = Δθ/Δt. Every point on a rigid rotating body shares ω, but tangential speed v = rω grows with radius. Velocity is tangent to the path and continually changes direction even when its magnitude is constant.
Describe rotation with angular velocity
Angular velocity is the rate of change of angular displacement: ω = Δθ/Δt. For uniform rotation, one turn of 2π rad in period T gives ω = 2π/T = 2πf.
All points fixed to one rigid rotating body sweep the same angle in the same time, so they share ω, frequency and period. This remains true even though they travel different arc lengths.
Check your understanding: A disc rotates at 3.0 revolutions per second. What is its angular velocity?
ω = 2πf = 6π = 18.8 rad s⁻¹.
Turn swept angle into tangential speed
Since arc length s = rθ, differentiating the relationship in uniform motion gives v = rω. A point twice as far from the axis travels twice the distance in each turn and therefore has twice the tangential speed.
Tangential velocity points along the tangent, perpendicular to the radius. Angular velocity is instead represented along the rotation axis using the right-hand rule; the two quantities should not be drawn as parallel arrows in the plane.
Check your understanding: Two points lie 0.10 m and 0.30 m from the same axis. Compare their tangential speeds.
They share angular velocity, so the outer point's tangential speed is three times the inner point's.
Key ideas to keep
- Do not confuse revolutions per second with radians per second: ω = 2πf.
- Points at different radii have the same period but different tangential speeds.
- Tangential velocity is perpendicular to the radius.
See the reasoning
Worked example
Connect rotation rate to belt speed
Question: A motor pulley of radius 0.080 m rotates at 1500 revolutions per minute without slipping against a belt. Find angular velocity and belt speed.
Step 1: Convert the frequency
Why: Angular equations require turns per second, not per minute.
Working: f = 1500/60 = 25.0 Hz.
Step 2: Find angular velocity
Why: Each revolution sweeps 2π rad.
Working: ω = 2πf = 157 rad s⁻¹.
Step 3: Use the no-slip boundary
Why: The belt and pulley rim share tangential speed at contact.
Working: v = rω = 0.080(157) = 12.6 m s⁻¹.
Answer: Angular velocity is 157 rad s⁻¹ and belt speed is 12.6 m s⁻¹.
Check: The rim travels circumference 0.503 m on each of 25 turns per second, also giving about 12.6 m s⁻¹.
Another worked model
Question
A centrifuge rotates at 1800 rpm. Find its angular velocity and the speed of a sample 0.12 m from the axis.
Check the worked solution
1800 rpm = 30.0 Hz, so ω = 2πf = 60π = 188 rad s⁻¹. The tangential speed is v = rω = 0.12(60π) = 22.6 m s⁻¹.
Use a hint if needed
Practise with support
Try this
A fan rotates at 3.0 Hz. Find its angular velocity and the speed of a blade tip 0.20 m from the axis.
Hint: Use ω = 2πf before v = rω.
Check your answer
ω = 2π(3.0) = 18.8 rad s⁻¹. Hence v = 0.20(18.8) = 3.77 m s⁻¹.
Now work without the hint
Practise independently
Your turn
Two points are 0.10 m and 0.40 m from the axis of the same rigid disc rotating at 15 rad s⁻¹. Compare their angular velocities and tangential speeds.
Check your answer
Both points have angular velocity 15 rad s⁻¹. Their speeds are v₁ = 0.10(15) = 1.5 m s⁻¹ and v₂ = 0.40(15) = 6.0 m s⁻¹.
Avoid these traps
Common mistakes
Common mistake
Every point on a rigid rotating body has the same tangential speed.
What is wrong with this reasoning?
Show better thinking
All points share angular velocity, but v = rω means a point farther from the axis has a greater tangential speed.
Write for the examiner
Exam guidance
Convert rotations or degrees to radians, then decide whether the question asks for angular or tangential speed.
Exam-style practice [6 marks]
A disc rotates uniformly at 480 revolutions per minute. Points A and B are 0.12 m and 0.30 m from its axis. Find the angular velocity, both tangential speeds and the angular displacement in 2.5 s. Compare the periods of A and B.
Plan before you answer
- Convert revolutions per minute to hertz.
- Use one shared angular velocity in v = rω.
- Use θ = ωt for uniform rotation.
Mark your answer and compare the model
Marking points
Tick each point only if your answer states it clearly.
Model answer
f = 480/60 = 8.0 Hz, so ω = 2πf = 16π = 50.3 rad s⁻¹. Therefore vA = 0.12(50.3) = 6.03 m s⁻¹ and vB = 0.30(50.3) = 15.1 m s⁻¹. In 2.5 s, θ = ωt = 16π(2.5) = 40π = 126 rad. Both points complete each turn together, so they have the same period.
Come back in three days
Check what stayed with you
Recall question 1
How are angular frequency and ordinary frequency related?
Check the answer
ω = 2πf.
Recall question 2
Which rotational quantities are shared by all points on a rigid disc?
Check the answer
Angular velocity, frequency and period.
Recall question 3
Where does tangential velocity point?
Check the answer
Along the tangent, perpendicular to the radius at that point.
Syllabus and review details
This lesson covers the listed H2 Physics 9478 outcomes. The official topic states no explicit exclusions. Radial equations in this chain are always built from the real forces acting on the body; centripetal force is not added as a separate force.
- GCE A-Level H2 PhysicsTopic 7(b) / Topic 7(c) · 2027Checked against the syllabus · partial topic coverageOfficial 9478 syllabus
Course and syllabus information
- Course
- GCE A-Level H2 Physics
- Edition
- GCE A-Level H2 Physics 2027