Circular Motion: Angular Velocity

When an object moves in a circle, we need new quantities to describe its motion. Linear velocity alone is not enough — we need angular velocity to describe how quickly the object sweeps through an angle.

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Radian Measure

Before we can define angular velocity, we need to be comfortable with radians. A radian is defined as the angle subtended at the centre of a circle by an arc equal in length to the radius.

For a circle of radius r, an arc of length s subtends an angle θ (in radians) given by:

$\theta = \frac{s}{r}$θ=rs​

Key radian values:

• Full circle: 2π rad (360°)
• Half circle: π rad (180°)
• Quarter circle: π/2 rad (90°)
• One radian ≈ 57.3°

To convert degrees to radians: multiply by π/180. To convert radians to degrees: multiply by 180/π.

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Angular Displacement

Angular displacement is the angle through which an object has rotated, measured in radians. If a satellite completes one full orbit, its angular displacement is 2π rad. If a wheel rotates three complete turns, its angular displacement is 6π rad.

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Angular Velocity