Kinematics of uniform circular motion
Section: Motion in a Circle | Syllabus: Cambridge AS 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. Diagram: Definition of the Radian 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 Tip Always check your calculator mode! To convert degrees to radians: θ_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 = rθ Where s is the arc length and r is the radius. Note: θ must be in radians. Displacement vs. Distance In one full revolution: Linear distance: 2π r Linear displacement: 0 (returns to start) Angular displacement: 2π rad Worked Example: Rotating Earth 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 rad Angular 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 f Relationship: 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 Rotation For 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).
Diagram: Latitude vs. Speed 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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