Measure angular displacement in radians
Define a radian from arc length and express angular displacement without confusing angle and distance.
Section 12 · Motion in 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 arc length subtends one radian?
What you will learn
- Define one radian using arc length and radius
- Express angular displacement in radians
- Relate a full revolution to 2π radians
A radian compares arc length with radius
One radian is the angle at the centre of a circle that subtends an arc equal in length to the radius. For arc length s and radius r, angular displacement θ = s/r when θ is in radians.
A full revolution has arc length 2πr, so θ = 2πr/r = 2π rad. Degrees and radians measure the same angle using different scales; keep the unit consistent in rotational equations.
Use radians for circular-motion equations
If a point moves through 0.50 m of arc on a circle of radius 0.20 m, its angular displacement is 0.50/0.20 = 2.5 rad.
The ratio s/r is dimensionless, though the angle is conventionally labelled in radians. Do not divide a straight-line chord by radius when the problem supplies arc length.
Pause and explain
What is the angular displacement in one full revolution?
Worked example
A point travels 0.75 m along an arc of radius 0.25 m. Find angular displacement.
Show the worked solution
- Use θ = s/r.
- θ = 0.75/0.25.
- State radians.
Answer θ = 3.0 rad.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
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.
An arc of length 0.60 m lies on a circle of radius 0.30 m. Find the angular displacement.
Give me a hint
Divide arc length by radius.
Compare my reasoning
- θ = s/r.
- θ = 0.60/0.30.
- Give radians.
2.0 rad.
Look for these in your work
- Arc used
- Radius used
- Radian value stated
A point completes three quarters of a revolution. Find its angular displacement in radians.
Give me a hint
A complete revolution is 2π rad.
Compare my reasoning
- Fraction of revolution = 3/4.
- θ = (3/4)(2π).
- Simplify and state rad.
3π/2 rad.
Look for these in your work
- Revolution fraction applied
- Radians converted
- Exact result stated
A student uses a 0.30 m chord instead of the 0.30 m arc to calculate an angle on a circle of radius 0.50 m. Explain why the ratio is not the required angular displacement.
Give me a hint
Distinguish the straight-line chord from the curved distance around the circle.
Compare my reasoning
- θ = s/r uses arc length s.
- A chord cuts across the circle and is shorter than the corresponding arc for a non-zero angle.
- Using chord/r therefore underestimates the angular displacement.
The given straight chord is not arc length; angular displacement requires the curved distance s divided by r.
Look for these in your work
- Correct length identified
- Chord and arc distinguished
- Effect explained
Common mistakes
- Using chord length instead of arc length
- Leaving angle in degrees in a radian-only equation
- Calling one revolution one radian
What can you explain now?
What arc length subtends one radian?
Compare with the explanation
An arc equal to the radius
That is the definition of one radian.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Why is arc length, rather than chord length, used in θ = s/r?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.