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AS & A-Level Physics 12 — Circular Motion

AS & A-Level Physics 12 — Circular Motion

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Independent Deckloop revision aligned with the Cambridge International AS & A Level Physics (9702) syllabus, 2025–2027. Not affiliated with or endorsed by Cambridge International Education. Chapter 12 of 26: Circular motion. Concepts, worked applications and misconception checks.

Physics EN A-Level
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The Radian for Angular Displacement

An angle in radians is a ratio of lengths. Measure the arc s swept around a circle and divide by its radius r: θ=s/r\theta=s/r. Both lengths must use the same units. When the arc and radius have equal lengths, the central angle is one radian. For a complete turn, the arc is the circumference 2πr2\pi r, so the angle is 2π2\pi radians. This gives the conversion 180∘=π180^\circ=\pi rad and approximately 57.3° per radian. Although the ratio has no physical dimension, writing rad identifies the quantity as an angle. Equations such as v=rωv=r\omega assume angular displacement is measured in radians; inserting degree values without conversion gives an incorrect numerical result.

Key points

  • A radian is the angle subtended when the arc length equals the radius.
  • Angular displacement θ=s/r\theta = s/r, where ss is arc length and rr is radius.
  • A full circle is 2π2\pi radians, so 360∘=2π rad360^\circ = 2\pi \text{ rad}.
  • Angles in physics formulae are almost always in radians.

Worked example

Question

A child's toy train travels along a circular track with a radius of 0.80 m. If the train moves along an arc of 1.2 m, what is its angular displacement in radians? Convert this angle to degrees.

Solution

1. Calculate angular displacement in radians using θ=s/r\theta = s/r:
2. θ=1.2 m/0.80 m=1.5 rad\theta = 1.2\text{ m} / 0.80\text{ m} = 1.5\text{ rad}
3. 2. Convert radians to degrees using the conversion factor 1 rad=360∘/(2π)1\text{ rad} = 360^\circ / (2\pi):
4. θdegrees=1.5 rad×(360∘/(2π rad))=1.5×57.295...∘≈85.9∘\theta_{\text{degrees}} = 1.5\text{ rad} \times (360^\circ / (2\pi\text{ rad})) = 1.5 \times 57.295...^\circ \approx 85.9^\circ

The angular displacement is 1.5 rad, which is approximately 85.9°.

Common pitfalls

  • Forgetting to switch calculator mode between degrees and radians when solving problems involving trigonometric functions or angular motion equations.
  • Confusing arc length (ss) with angular displacement (θ\theta); arc length is a distance, angular displacement is an angle.

Prerequisites

  • Basic division and multiplication for unit conversion.
  • Understanding of angles and circles.
Further resources