Ch 1 · 12 Motion in a Circle · A Level Physics 9702
Topic 1 of 2 in Circular Motion and Gravitation — A Level Physics 9702 — 36 lessons.
Introduction
A car rounding a bend, a satellite orbiting the Earth and an electron curving in a magnetic field all move in circles. Even at constant speed, they are accelerating, because their direction keeps changing. This topic gives you the language and equations for circular motion, used again in gravitation and magnetism. (Syllabus topic 12: sections 12.1–12.2.)
Key ideas — 12.1 (a) The radian
One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.
🧑🏫 Teacher's explanation
Imagine you are walking along the edge of a circular track. If you walk a distance exactly equal to the radius of that track, the angle you have swept out from the center is exactly one radian. It is a way of measuring how much 'turning' has occurred rather than just how far you have traveled linearly.
📖 Definitions
🔍 Worked example — Easy
A pendulum swings through an angle of 0.50 radians. Calculate the arc length traveled by the tip of the pendulum if the length of the pendulum is 0.80 m.
🔍 Worked example — Standard
A wheel rotates at a constant angular speed of 12 rad s^-1. Calculate the time taken for the wheel to complete 5 full revolutions.
🔍 Worked example — Exam-style
A particle moves in a circle of radius 0.25 m with a constant linear speed of 4.0 m s^-1. Calculate its angular speed ω.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — 12.1 (b) Angular speed
Angular speed ω is the angle swept out per unit time (rate of change of angular displacement), in rad s⁻¹:
🧑🏫 Teacher's explanation
Imagine you are standing on a spinning carousel. If you look at a horse near the center, it seems to complete a full circle very quickly. If you look at a horse on the very edge, it takes the same amount of time to complete a circle, but it actually has to travel a much larger distance in that same time. This is the core difference between angular speed and linear speed.
📖 Definitions
🔍 Worked example — Standard
A wheel rotates at a frequency of 12 Hz. Calculate the angular speed ω in rad s^-1.
🔍 Worked example — Exam-style
A stone is whirled on a string of length 1.2 m at a constant angular speed of 5.0 rad s^-1. Calculate the linear speed of the stone.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — 12.2 (a) Centripetal acceleration and force
An object moving in a circle at constant speed has a velocity that changes direction continuously, so it is accelerating. The acceleration is directed towards the centre: the centripetal acceleration.
🧑🏫 Teacher's explanation
Imagine you are swinging a ball on a string above your head. Even if you swing it at a steady speed, the ball is constantly 'turning' to stay in that circle. Because velocity is a vector (it has a direction), and that direction is constantly changing, the ball is technically accelerating. This acceleration isn't making the ball go faster or slower; it is just pulling it into a curve.
📖 Definitions
🔍 Worked example — Easy
A stone is whirled in a horizontal circle of radius 0.50 m at a constant speed of 4.0 m s^-1. Calculate the centripetal acceleration of the stone.
🔍 Worked example — Standard
A car of mass 1200 kg travels around a flat, circular bend of radius 40 m at a constant speed of 15 m s^-1. Calculate the magnitude of the centripetal force required.
🔍 Worked example — Exam-style
A satellite orbits a planet in a circular path. The satellite has an angular speed of 0.0025 rad s^-1 and moves in a circle of radius 7.0 × 10^6 m. Calculate the centripetal acceleration of the satellite.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked example
Example 1. A wheel of radius 0.30 m makes 120 revolutions per minute. Find its angular speed and the speed of a point on its rim.
🔍 Worked example — Standard
A 1200 kg car travels at a constant speed of 15 m s^-1 around a level bend with a radius of 50 m. Calculate the magnitude of the friction force required to maintain this circular motion.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) The angle subtended at the centre of a circle by an arc equal in length to the radius. (2) π/4 = 0.785 rad; 143°. (3) ω = 2π/5400 = 1.16 × 10^-3 rad s⁻¹; v = 6.7 × 10^6 × 1.16 × 10^-3 = 7.8 × 10^3 m s⁻¹. (4) ω = 2π × 8.33 = 52.4 rad s⁻¹; a = 0.060 × 52.4^2 = 165 m s⁻². (5) The force is always perpendicular to the velocity, so it does no work and changes only the direction, not the magnitude, of the velocity.