12.1.1Physical quantities and kinematics

Radians and Angular Displacement

5 min read 10 parts Lesson 19 of 22 1 quiz

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Lesson summary

12.1.1 Radians and Angular Displacement — AS PHYSICS 9702 notes: Define the radian, convert between degrees and radians and use the arc-length relation θ = s ÷ r.Radian (rad) = the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle.In …

In physics radians connect an angle directly to lengths along a circle
A radian is a ratio of two lengths
For an arc of length s on a circle of radius r
Both distances must be in the same units
For a full circle the arc is the circumference 2πr
Degrees to radians

Section 1 of 10The radian

1Radian (rad)

Radian (rad) the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle.

2In physics radians connect an angle directly to lengths along a circle

In physics radians connect an angle directly to lengths along a circle formulae for circular motion such as v = ωr are only valid if the angle is in radians.

3A radian is a ratio of two lengths

A radian is a ratio of two lengths so it is dimensionless; "rad" is written to show the unit in use.

One radian is about 57.3°.

Section 2 of 10Arc length and angle

1For an arc of length s on a circle of radius r

θ = s ÷ r, so s = rθ.

2Both distances must be in the same units

Both distances must be in the same units for example metres.

3For a full circle the arc is the circumference 2πr

For a full circle the arc is the circumference 2πr so the angle is 2πr ÷ r = 2π radians.

Hence 360° = 2π rad, 180° = π rad and 90° = π ÷ 2 rad.

Section 3 of 10Converting between degrees and radians

Converting between degrees and radians

Degrees to radians

Degrees to radians multiply by π ÷ 180.

Radians to degrees

Radians to degrees multiply by 180 ÷ π.

Set the calculator to radian mode (R)

Set the calculator to radian mode (R) when a formula uses an angle in radians,
- and back to degrees (D) when required; wrong mode is a common cause of wrong answers.

Section 4 of 10Common angles

Degrees	30°	45°	60°	90°	180°	270°	360°
Radians	π ÷ 6	π ÷ 4	π ÷ 3	π ÷ 2	π	3π ÷ 2	2π

Section 5 of 10Angular displacement

1Angular displacement

Angular displacement the change in angle, in radians, of a body as it rotates around a circle.

θ = distance travelled around the circle ÷ radius of the circle.

3A complete revolution is an angular displacement of 2π rad

A complete revolution is an angular displacement of 2π rad and it can exceed 2π for several turns, for example 5 revolutions give 10π rad.

4The distance along the circle is the arc length s = rθ

The distance along the circle is the arc length s = rθ the straight-line displacement is the chord length.

Section 6 of 10Small angles and sectors

Small angles and sectors

For small angles sin θ ≈ tan θ ≈ θ

For small angles sin θ ≈ tan θ ≈ θ with θ in radians.

Area of a sector of angle θ

½r²θ, from the fraction θ ÷ 2π of the area πr².

A graph of θ against time for steady rotation

A graph of θ against time for steady rotation is a straight line through the origin with gradient equal to the angular speed.

Section 7 of 10Where radians are used

Where radians are used

Circular motion

Circular motion s = rθ and v = ωr.

Oscillations and waves

Oscillations and waves phase differences are quoted in radians, so π means antiphase and π ÷ 2 means a quarter of a cycle.

The rate at which angular displacement changes

The rate at which angular displacement changes is the angular speed ω (see 12.1.2).

Section 8 of 10More from the specification

Velocity

Rate of change of displacement.

Linear

A graph that is a straight line through the origin, so resistance is constant.

Frequency

Revolutions per second, f = 1/T.
What the symbols mean: T = period, the time for one complete wave (s) · f = frequency (Hz)

Uniform

Same density throughout, so the weight acts at the centre.

Linear speed

The distance travelled along the circle per unit time, v = ωr.
What the symbols mean: v = speed (m s⁻¹) · ω = angular speed (rad s⁻¹) · r = radius (m) · T = period (s)

Angular velocity

Δθ/Δt in rad/s.

Section 9 of 10Key terms

Term	Meaning
Radian	The angle subtended at the centre of a circle by an arc equal in length to the radius.
Angular displacement	The angle, in radians, through which a body turns about an axis.
Arc length	The distance along the circumference, s = rθ.
Revolution	One full turn, 2π radians.
Dimensionless	Having no units, as a ratio of two lengths.
Sector	A part of a circle bounded by two radii and an arc.
Chord	The straight line joining two points on a circle.
Small-angle approximation	sin θ ≈ tan θ ≈ θ for small θ in radians.

Section 10 of 10Exam technique

Using a calculator in degree mode for formulae that need radians.

Writing that one radian equals 1° — it is about 57.3°.

Forgetting that angular displacement can exceed 2π for several turns.

Treating the radian as a unit with dimensions — it is a ratio of lengths.

Using s = rθ with θ in degrees.