CAIE A Level Physics: Kinematics of Uniform Circular Motion

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 ⍵).