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Radian
The angle subtended at the centre of a circle by an arc equal in length to the radius of the circle.
Angle in Radians for a Complete Circle
The angle in radians for a complete circle (360°) is equal to 2π.
Conversion from Degrees to Radians
Use the equation θ° × (π/180) = θ rad to convert from degrees to radians.
Angular Displacement
The change in angle, in radians, of a body as it rotates around a circle.
Angular Displacement Equation
∆θ = distance travelled around the circle / radius of the circle.
Δθ
Angular displacement, or angle of rotation (radians).
S
Length of the arc, or the distance travelled around the circle (m).
r
Radius of the circle (m).
Degrees to Radians Table
Common conversions include: 360° = 2π, 270° = 3π/2, 180° = π, 90° = π/2.
Visual Definition of Radian
When the angle is equal to one radian, the length of the arc (S) is equal to the radius (r) of the circle.
Worked Example for Angular Displacement
Convert 3 rad into degrees: 3 rad × (180/π) = 60°.
Units for Angular Displacement
Angular displacement is more conveniently measured in radians rather than degrees.
Arc Length in Radians
An angle in radians, subtended at the centre of a circle, is the arc length divided by the radius of the circle.
Common Degrees to Radians Conversion
270° = 3π/2, 180° = π, 90° = π/2.
Equation for Angular Displacement
The change in angle in radians can be expressed as ∆θ = S/r.
Measurement Units
Both distances in the angular displacement equation must be measured in the same units, e.g., metres.
Visual Representation of Angular Displacement
A diagram showing the relationship between arc length, radius, and angle in radians.
Complete Circle in Radians
The complete angle of a circle is represented as 2π radians.
Definition of Angular Speed
Angular speed is the rate of change of angular displacement with respect to time.
Unit of Angular Speed
Angular speed is typically measured in radians per second (rad/s).
Relationship Between Linear and Angular Speed
Linear speed (v) is related to angular speed (ω) by the equation v = rω.
Degree (Deg) mode
A calculator setting for trigonometric functions to provide answers in degrees.
Radians (Rad) mode
A calculator setting for trigonometric functions to provide answers in radians.
Angular Speed (⍵)
The rate of change in angular displacement with respect to time.
Angular Speed Unit
Measured in rad s⁻¹.
Uniform Circular Motion
Motion where an object travels at a constant speed in a circular path, but its velocity changes due to changing direction.
Angular Displacement (Δθ)
The change in angular position of an object, measured in radians.
Time Interval (Δt)
The duration over which angular displacement occurs, measured in seconds.
Time Period (T)
The time taken to complete one full cycle of motion, measured in seconds.
Frequency (f)
The number of complete cycles per second, measured in Hertz (Hz).
Angular Velocity
The vector quantity that represents the rate of rotation, equivalent to angular speed but includes direction.
Linear Speed (v)
The speed of an object moving along a circular path, measured in m s⁻¹.
Radius of Orbit (r)
The distance from the center of the circular path to the object, measured in meters.
Angular Speed Formula
ω = Δθ / Δt.
Angular Speed in terms of Time Period
ω = 2π / T.
Angular Speed in terms of Frequency
ω = 2πf.
Linear Speed Equation
v = rω.
Angular Velocity Equation
ω = v / r.
Example of Angular Speed Calculation
A bird flies in a horizontal circle with an angular speed of 5.25 rad s⁻¹ and radius 650 m.
Linear Speed Calculation Example
v = 650 × 5.25 = 3410 m s⁻¹.
Frequency Calculation Example
f = 5.25 / (2π) = 0.836 Hz.
Effect of Rotation Angle on Angular Velocity
The greater the rotation angle θ in a given amount of time, the greater the angular velocity ⍵.
Effect of Radius on Angular Velocity
An object rotating further from the center of the circle (larger r) moves with a smaller angular velocity (smaller ⍵).
