Circular motion fundamentals

1. What is uniform circular motion?

An object undergoing uniform circular motion travels in a circular path at constant speed. Although the magnitude of velocity is constant, the direction changes continuously — which means the velocity vector is always changing. By Newton's first law, a changing velocity requires a net force. This is the central idea underpinning all circular motion analysis in Unit 3.

It is critical to distinguish speed (scalar, constant in UCM) from velocity (vector, continuously changing in UCM). A common misconception is that "constant speed" implies "no acceleration" — this is false. Acceleration is defined as the rate of change of velocity, and because direction changes, there is always acceleration present.

Key kinematic quantities for UCM:

• Period (T): time taken for one complete revolution, measured in seconds (s).
• Frequency (f): number of revolutions per second, measured in hertz (Hz = s⁻¹). Related by f = 1/T.
• Angular speed (ω): rate of change of angular displacement, in rad s⁻¹. Given by ω = 2π/T = 2πf.
• Linear (tangential) speed (v): the instantaneous speed along the circular path, in m s⁻¹. Given by v = 2πr/T, where r is the radius of the circle in metres.

Worked example: A car travels around a roundabout of radius 12 m and completes one full loop in 8.0 s. Find its speed.
v = 2πr/T = 2π × 12 / 8.0 ≈ 9.42 m s⁻¹

This speed is constant throughout the motion (assuming UCM), but the direction of the velocity vector rotates through 360° each period.

2. Centripetal acceleration — derivation and formula

Because the velocity direction changes continuously in UCM, there must be an acceleration directed toward the centre of the circle. This is called centripetal acceleration (from Latin centrum = centre, petere = to seek). It is always directed radially inward — toward the centre of the circular path — and is therefore perpendicular to the velocity at every instant.

The two equivalent expressions for centripetal acceleration are:

• a_c = v² / r  (using linear speed and radius)
• a_c = 4π²r / T²  (using radius and period — derived by substituting v = 2πr/T)

Unit analysis: For a_c = v²/r: units are (m s⁻¹)² / m = m² s⁻² / m = m s⁻². For a_c = 4π²r/T²: units are m / s² = m s⁻². Both expressions give SI units of m s⁻², consistent with acceleration. Note that 4π² is dimensionless.

Direction convention (QCAA): Centripetal acceleration is defined as pointing toward the centre of the circular path. This direction changes continuously as the object moves around the circle. It is never tangential and never outward. Do not describe it as pointing "upward" or "downward" in general — state specifically "toward the centre" or use a diagram showing the inward radial direction.

Worked example — Brisbane Story Bridge climb: Tourists on a Story Bridge climb walk across an arc of radius approximately 30 m at a constant speed of 1.2 m s⁻¹ (extremely slow stroll). Find the centripetal acceleration.
a_c = v²/r = (1.2)²/30 = 1.44/30 = 0.048 m s⁻²
This is tiny compared to g = 9.8 m s⁻², confirming that for slow motion on gentle curves, centripetal effects are negligible.

Worked example — motor racing at Queensland Raceway: A racing car takes a bend of radius 80 m at 60 m s⁻¹. Find the centripetal acceleration and express it as a multiple of g.
a_c = v²/r = (60)²/80 = 3600/80 = 45 m s⁻²
45 / 9.8 ≈ 4.6g
Drivers experience approximately 4.6 times their body weight directed horizontally toward the centre of the turn — a significant physiological load.