5.1.2Torque and Rotational Dynamics

Angular Displacement — AP PHYSICS 1 Revision Notes & Topical Questions

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5.1.2 Angular Displacement — AP PHYSICS 1 notes: Define angular displacement, convert between degrees, radians and revolutions and relate it to arc length.Angular position = the angle of a point on a rigid system relative to a reference line, measured about the axis of rotation.A…

Not total rotation
Directions
Convention
Dimensionless
Calculator
Degrees to radians

Section 1 of 10Angular position and displacement

Angular position

Angular position the angle of a point on a rigid system relative to a reference line, measured about the axis of rotation.

Angular displacement

Δθ = θ_f − θ₀, the change in angular position, in rad.

Reference line

Reference line a coordinate axis or another position vector.
- (a point at 30° from the x-axis; a point rotated 75° clockwise from its start)

Not total rotation

Not total rotation a wheel turning two revolutions forward and one backward has net displacement one revolution.

Section 2 of 10Direction and sign

Directions

Directions clockwise or counterclockwise about the axis; both are vector-like.

Convention

Convention either direction may be positive only if the other is negative;
- counterclockwise is usually positive.

Example

Δθ = −3π rad is one and a half revolutions clockwise.

Size

Size angular displacement can exceed 2π because the object may turn many times.

Section 3 of 10Radians

Radian

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

θ = s/r

θ = s/r arc length s = rθ with θ in radians.

Dimensionless

Dimensionless a ratio of lengths, so the radian is not strictly a unit.

Calculator

Calculator must be in radian mode for formulas with ω, α and arc length.

Section 4 of 10Conversions

Angle	Degrees	Radians
Full revolution	360°	2π (about 6.28)
Three quarters	270°	3π/2
Half	180°	π
Quarter	90°	π/2 (about 1.57)

Degrees to radians

θ(rad) = θ(°) × π/180°; 1 rad is about 57.3°.

Revolutions

N = Δθ/2π; 25 rad is about 4.0 revolutions.

rev/s to rad/s

rev/s to rad/s multiply by 2π.

Section 5 of 10Arc length

Arc length

Full revolution

Full revolution s = 2πr, the circumference.

Part of a revolution

Part of a revolution s = rθ.

Rolling without slipping

Rolling without slipping the centre moves Δx = RΔθ when the wheel turns through Δθ.

Rigid body

Rigid body all points share Δθ but points at larger r move through greater arc length.

Section 6 of 10Angular position–time graphs

Angular position–time graphs

Gradient

Gradient is the angular velocity; a steeper line means faster rotation.

Straight line

Straight line constant ω.

Curve concave upward

ω increasing, so there is angular acceleration.

Flat line

Flat line no rotation.

Displacement over an interval

Displacement over an interval difference of the two vertical-axis values.

Section 7 of 10Benchmarks

Second hand

2π rad every 60 s, so 0.105 rad/s.

Minute hand

60 times slower, 1.7 × 10⁻³ rad/s.

Section 8 of 10More from the specification

Section 9 of 10Key terms

Term	Meaning
Angular displacement	Angle turned by an object, Δθ = θ_f − θ_i.
Radian	Angle for which arc length equals radius; 2π rad = 1 revolution.
Arc length	Distance along a circle, s = rθ.
Revolution	One complete turn, 2π rad.

Section 10 of 10Exam technique

Using degrees in s = rθ.

Ignoring signs when a rotation reverses direction.

Treating a revolution as 1 radian rather than 2π.

Using the diameter instead of the radius for arc length.

Believing angular displacement is limited to 2π.

Last updated: 15 October 2026