Use angular speed, period and tangential speed
Relate angular speed to period and connect angular speed to the linear speed around a circle.
A Level; Cambridge International Physics 9702, 2025–2027 syllabus, Topic 12. These original calculations are study aids; full timed assessment and human subject-teacher review remain outstanding.
Before you begin
- Use radians, angular speed, velocity, acceleration, mass and resultant force with consistent SI units.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does angular speed measure?
What you will learn
- Understand angular speed as angle turned per unit time
- Use ω = 2π/T
- Use v = rω for tangential speed
Angular speed is radians turned each second
Angular speed ω is the rate of change of angular displacement. For uniform circular motion, one full turn of 2π rad takes one period T, so ω = 2π/T.
Its unit is rad s⁻¹. Frequency f = 1/T, so ω = 2πf. A shorter period means a larger angular speed.
Tangential speed depends on radius
In time t, a point turns through angle θ and travels arc length s = rθ. Dividing by t gives v = rω for tangential speed v.
For a rigid rotating object all points share the same angular speed, but points at greater radius have greater tangential speed. State r in metres and ω in rad s⁻¹ to obtain v in m s⁻¹.
Pause and explain
A point has angular speed 4.0 rad s⁻¹ at radius 0.50 m. Find tangential speed.
Worked example
A wheel rotates with period 0.40 s. Find angular speed and tangential speed at radius 0.15 m.
Show the worked solution
- ω = 2π/T = 2π/0.40 = 5π rad s⁻¹.
- v = rω = 0.15 × 5π.
- Evaluate and give units.
Answer ω ≈ 15.7 rad s⁻¹; v ≈ 2.36 m s⁻¹.
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A rotor completes one revolution every 0.25 s. Find angular speed.
Give me a hint
Use 2π radians for one revolution and divide by period.
Compare my reasoning
- ω = 2π/T.
- ω = 2π/0.25.
- Give rad s⁻¹.
8π rad s⁻¹, approximately 25.1 rad s⁻¹.
Look for these in your work
- Full-turn angle used
- Period substituted
- Angular-speed unit stated
A point rotates at 6.0 rad s⁻¹ on a 0.20 m radius. Find its tangential speed.
Give me a hint
Multiply angular speed by radius in metres.
Compare my reasoning
- v = rω.
- v = 0.20 × 6.0.
- State m s⁻¹.
1.2 m s⁻¹.
Look for these in your work
- Relation selected
- Product calculated
- Speed unit stated
Two points on a rotating disc have radii 0.10 m and 0.30 m. The disc's angular speed is fixed. Compare their tangential speeds.
Give me a hint
Use v = rω and keep the common ω fixed.
Compare my reasoning
- v is directly proportional to r for fixed angular speed.
- The second radius is three times the first.
- Its tangential speed is therefore three times as large.
The point at 0.30 m has three times the tangential speed of the point at 0.10 m.
Look for these in your work
- Common angular speed identified
- Radius ratio calculated
- Tangential-speed comparison stated
Common mistakes
- Using frequency where period is required without taking the reciprocal
- Using degrees per second in v = rω
- Assuming points on a wheel all have the same tangential speed
What can you explain now?
What does angular speed measure?
Compare with the explanation
Angle turned per unit time
It is the rate of angular displacement.
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Make it stick
Why can two points on a rigid disc share angular speed but have different linear speeds?
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