Kinematics of uniform circular motion
Section: Motion in a Circle | Syllabus: Cambridge A Level Physics (9702)
Angles in Radians
The radian (rad) is the SI unit of angular measurement. It is the ratio of arc length to radius, making it a "dimensionless" unit essential for circular motion equations.
- Definition: One radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle.
- For a complete circle, the arc length is the circumference (2π r). Therefore, the angle in a full circle is 2π radians.
Show a circle where the arc length s is equal to the radius r. Highlight the angle θ as 1 radian (approx. 57.3^).
Examiner Conversion TipAlways check your calculator mode! To convert degrees to radians: _rad = θ^ × (π/180).
| Angle (Degrees) | Angle (Radians) |
|---|---|
| 90^ | π/2 (approx. 1.57) |
| 180^ | π (approx. 3.14) |
| 270^ | 3π/2 (approx. 4.71) |
| 360^ | 2π (approx. 6.28) |
Angular Displacement (θ)
Angular displacement is the angle swept out by the radius joining the object to the centre of the circle.
s = rθWhere s is the arc length and r is the radius. Note: θ must be in radians.
Displacement vs. DistanceIn one full revolution:
- Linear distance: 2π r
- Linear displacement: 0 (returns to start)
- Angular displacement: 2π rad
Question: A point on the Earth's equator travels a circumference of 4.0 × 10^7 m in 24 hours. Calculate its angular displacement in 1 hour.
Solution In 24 hours, θ = 2π rad. In 1 hour, θ = (2π/24) = (π/12) ≈ 0.26 radAngular Speed (ω)
Angular speed is the rate of change of angular displacement.
ω = (Δθ/Δ t)Units: rad s^-1
The Period-Frequency Link
For one complete rotation (Δθ = 2π):
ω = (2π/T) or ω = 2π fRelationship: Linear speed and Angular speed
An object at radius r moving with angular speed ω has a tangential linear speed (v).
v = rω Physical Insight: Rigid Body RotationFor a rigid rotating object (like a spinning CD or the Earth):
- Every point has the same angular speed ω.
- The linear speed v increases linearly with the distance from the axis (r).
Show the Earth rotating. Compare a person at the Equator (r = 6400 km) and a person at 80^ North (r = 1100 km). Both rotate through the same angle in 1 hour, but the person at the Equator has a much higher linear speed.
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