Angular displacement in radians
Key idea: H2 Physics lessons on angular displacement, angular velocity, tangential speed, centripetal acceleration and radial force models.
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The core idea
Build the idea
Learn the idea
Big question: Why are radians the natural unit for circular motion?
Angular displacement in radians is arc length divided by radius, θ = s/r. One complete turn is 2π rad. Because the definition is a direct ratio, the familiar circular-motion equations work without extra degree-conversion factors when angles are in radians.
Build the radian from a circle
An angular displacement of one radian subtends an arc whose length equals the radius. In general, θ = s/r, provided s and r use the same length unit. The ratio has no physical dimension, although writing rad keeps an angular answer clear.
A complete turn has arc length 2πr, so it contains 2π radians. This gives 180° = π rad and makes degree–radian conversion a proportion rather than a rule to memorise blindly.
Check your understanding: A quarter-circle has radius 0.60 m. What arc length does it contain?
A quarter-turn is π/2 rad, so s = rθ = 0.60(π/2) = 0.942 m.
Keep displacement, distance and sign distinct
Angular displacement records the signed change in orientation. Anticlockwise may be chosen positive, making clockwise displacement negative. Angular distance instead counts the total angle travelled and cannot be negative.
The small-angle-looking equation s = rθ is exact for a circular arc when θ is in radians. Substituting degrees directly is the common error that introduces a hidden factor of π/180.
Check your understanding: A wheel turns 270° clockwise and then 90° anticlockwise. What is its angular displacement if anticlockwise is positive?
The displacement is −270° + 90° = −180° = −π rad. Its angular distance travelled is 360° = 2π rad.
Key ideas to keep
- Convert degrees to radians before using s = rθ.
- Angular displacement can be signed to show rotation direction.
- A radian is dimensionless but should usually be written as rad for clarity.
See the reasoning
Worked example
Find signed angular displacement and arc distance
Question: A point on a wheel of radius 0.25 m rotates 3.5 turns anticlockwise, then 0.75 turn clockwise. Find its angular displacement and total distance travelled along the rim.
Step 1: Choose the sign convention
Why: The two rotations have opposite directions.
Working: Take anticlockwise as positive: net turns = 3.5 − 0.75 = 2.75.
Step 2: Convert the net rotation
Why: One turn corresponds to 2π rad.
Working: θ = 2.75(2π) = 5.5π = 17.3 rad anticlockwise.
Step 3: Use total travel for distance
Why: Distance uses 3.5 + 0.75 turns rather than the net displacement.
Working: Distance = 4.25(2πr) = 4.25(2π)(0.25) = 6.68 m.
Answer: Angular displacement is 17.3 rad anticlockwise; the point travels 6.68 m.
Check: The travelled distance is larger than r|θ| because the clockwise section partly cancels displacement but still adds distance.
Another worked model
Question
A wheel of radius 0.35 m turns anticlockwise through 135°. Find the angular displacement and distance travelled by a point on its rim.
Check the worked solution
θ = 135π/180 = 3π/4 = 2.36 rad. The rim distance is s = rθ = 0.35(3π/4) = 0.825 m.
Use a hint if needed
Practise with support
Try this
A point on a wheel of radius 0.40 m makes five-eighths of a revolution clockwise. Taking anticlockwise as positive, find its angular displacement and distance travelled.
Hint: One revolution is 2π rad; distance travelled is a positive path length.
Check your answer
θ = −(5/8)(2π) = −5π/4 = −3.93 rad. The distance travelled is r|θ| = 0.40(5π/4) = 1.57 m.
Now work without the hint
Practise independently
Your turn
A radius turns anticlockwise from 30° to 250° on a circle of radius 0.45 m. Find its angular displacement and the arc length swept out.
Check your answer
The displacement is 220° = 11π/9 = 3.84 rad. The arc length is s = rθ = 0.45(11π/9) = 1.73 m.
Avoid these traps
Common mistakes
Common mistake
An angle in degrees can be substituted directly into s = rθ.
What is wrong with this reasoning?
Show better thinking
The direct arc-length relation requires θ in radians. Convert with θ(rad) = θ(°)π/180 before using s = rθ.
Common mistake
Angular displacement has no direction because radians are dimensionless.
What is wrong with this reasoning?
Show better thinking
Declare a positive sense of rotation. Angular displacement is signed; use rad to communicate the angle measure even though it is dimensionless in SI base units.
Write for the examiner
Exam guidance
Check that an angular answer for a fraction of a turn is a sensible fraction of 2π.
Exam-style practice [5 marks]
A circular track has radius 1.20 m. A trolley travels clockwise through an arc of 2.50 m, then anticlockwise through 0.70 m. Taking anticlockwise as positive, find its angular displacement and angular distance travelled.
Plan before you answer
- Convert each arc using θ = s/r.
- Apply signs only to displacement.
- Add magnitudes for angular distance.
Mark your answer and compare the model
Marking points
Tick each point only if your answer states it clearly.
Model answer
The clockwise angle is 2.50/1.20 = 2.083 rad and the anticlockwise angle is 0.70/1.20 = 0.583 rad. Hence angular displacement = −2.083 + 0.583 = −1.50 rad, or 1.50 rad clockwise. Angular distance = 2.083 + 0.583 = 2.67 rad.
Come back in three days
Check what stayed with you
Recall question 1
Define one radian.
Check the answer
The angle subtended at a circle's centre by an arc equal in length to the radius.
Recall question 2
Convert 60° to radians.
Check the answer
60π/180 = π/3 rad.
Recall question 3
When is s = rθ valid without a conversion factor?
Check the answer
When θ is measured in radians.
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(a) · 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