Radians & Angular Displacement

1. Definitions (Must Know)

A. Radian (rad)

One radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius.

So for an arc length s on a circle of radius r:

θ = s/r

Angles in circular motion formulas must be in radians.

B. Angular displacement, θ

Angular displacement, θ (rad), is the angle swept out at the centre of the circle.

Scroll diagram horizontally to read all labels.

2. Key Ideas (What Earns Marks)

• Convert degrees to radians before using s = rθ.
• One full circle is 2π rad, so:
  • 360^° = 2π rad
  • 180^° = π rad
• The radian definition gives both θ = s/r and s = rθ.
• State a rotational sign convention if direction matters; for example, anticlockwise positive.

3. Detailed Explanations

A. Why radians are “natural” for circles

Because θ = s/r, radians are a ratio of two lengths, so they are dimensionless.

That is why formulas like s = rθ only work cleanly when θ is in radians (not degrees).

B. Converting degrees to radians

Use:

θ (rad) = θ (^°) × π/180

Mini-example:

30^° = 30 × π/180 = π/6 rad

C. Signed angular displacement

Angular displacement describes the change in angular position. If a sign convention is needed, state it before calculating. For example, with anticlockwise positive, a quarter-turn clockwise is

Δθ = -π/2 rad.

The distance travelled along the arc is a scalar and remains positive:

s = rlvertΔθrvert.

D. From arc length to angular speed

If an object moves an arc length s in time t:

• θ = s/r
• average angular speed ω = θ/t

For uniform motion, dividing s = rθ by the same time interval gives:

v = s/t = rθ/t = rω

The next lesson develops angular speed, period and frequency fully.

Data table

s = rθ (r = 0.50 m)
Angle, θ (rad)	Arc length, s (m)
0	0
2	1
3	2
5	2
6	3

4. Common Mistakes

• Using degrees inside s = rθ.
• Using diameter where the formula requires radius.
• Confusing angular displacement with the distance travelled around the arc.

5. Exam Tips

• Keep exact multiples of π until the final step when possible.
• State the positive rotational sense in questions where direction matters.
• Check that θ = s/r is dimensionless; metres divide by metres.

6. Worked Examples

Example 1: Convert degrees to radiansCore

Convert 135^° to radians.

Show Answer

135^° = 135 × π/180 = 3π/4 rad

Example 2: Arc length from an angleCore

A point moves through 120^° on a circle of radius 0.75 m. Find the distance travelled along the arc.

Show Answer

Convert the angle first:

120^° = 2π/3 rad

Then use s = rθ:

s = (0.75)(2π/3) = 1.57 m.

Example 3: Angular displacement with directionCore

Anticlockwise is positive. A wheel turns 1.75 revolutions clockwise. Find its angular displacement.

Show Answer

One revolution is 2π rad, and clockwise is negative:

Δθ = -1.75(2π) = -3.5π rad.

7. Mind Stretchers

Mind stretcher 1: What goes wrong if you use degrees in s = rθ?Extension

Suppose an angle is 60^°. A student substitutes θ = 60 into s = rθ instead of converting to radians.

By what factor is their arc length wrong?

Show Answer

The correct angle in radians is: θ = 60^° × π/180 = π/3

Correct arc length is sₜᵣᵤₑ = r(π/3).

The student’s arc length is swᵣₒₙg = r(60).

So: swᵣₒₙg/sₜᵣᵤₑ = 60/π/3 = 180/π ≈ 57.3

They overestimate the arc length by a factor of about 57.3.

8. Practice, Quiz and Next Step

Close your notes and use Radians & Angular Displacement in the supplied context below. This requires a constructed explanation or working, not recognition of an option.

Fresh context: An unfamiliar data set or physical system requires you to apply Radians & Angular Displacement while stating the model, regime and assumptions.

1. Retrieve: define radians & angular displacement in your own words, including units, sign or conditions where relevant.
2. Represent: Choose and label an appropriate diagram, graph, table or symbolic model; derive or justify the relationship used.
3. Apply: Reach a conclusion, then evaluate it using units, uncertainty, a limiting case and one practical or modelling limitation.

Check the response before looking back

• The model, regime, coordinates and assumptions are explicit.
• The derivation or multi-step reasoning is visible rather than implied.
• The conclusion is tested against units, data quality and a limiting case.
• A practical control, uncertainty or model limitation is evaluated where applicable.

If one check fails, name that exact gap, revisit the matching explanation or worked example, and redo the task with different values or a different situation. Then use theA-Level Physics course hub orpractice browser for an independent re-test.