12.1 Kinematics of Uniform Circular Motion

2026 📋 Syllabus Objectives

By the end of these notes, you should be able to:

  1. Define the radian and express angular displacement in radians
  2. Understand and use the concept of angular speed
  3. Recall and use ω = 2π/T and v = rω

📖 Main Content

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Part 1: Angular Displacement and Radians

What is Angular Displacement?

When an object moves in a circle, we need a way to describe where it is along that circle. We use something called angular displacement — this is simply the angle that the object has swept through from its starting position.

Think of it like the hand of a clock. Instead of saying "the hand moved 5 centimetres," we say "the hand rotated through an angle of 30°." That angle is the angular displacement.

Angular displacement (θ) = the angle, measured at the centre of the circle, through which an object has moved from its starting point.

The symbol used for angular displacement is the Greek letter θ (said "theta").

Why Use Radians Instead of Degrees?

You are already familiar with measuring angles in degrees (e.g. a full circle = 360°). In physics, however, we use a different unit called the radian — and here is why:

Radians connect the angle directly to the actual distances involved in circular motion, which makes calculations much cleaner and more natural. Degrees are a human invention; radians come directly from the geometry of the circle itself.

Defining the Radian

Imagine a circle with radius r. Now mark off a piece of the circumference (the curved edge) that has a length exactly equal to r. The angle at the centre of the circle that "opens up" to that arc is defined as 1 radian.

1 radian (1 rad) = the angle at the centre of a circle formed when the arc length (the curved distance) is exactly equal to the radius.

This gives us a very useful formula:

θ=sr\theta = \frac{s}{r}

  • θ = angular displacement (in radians, rad)
  • s = arc length — the distance travelled along the curved path (in metres, m)
  • r = radius of the circle (in metres, m)

Because both s and r are lengths measured in metres, they cancel out — so the radian has no unit (it is dimensionless). We still write "rad" after the number to make it clear we are talking about angles.

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