Linear Speed Along a Circular Path

From Rotation to Distance

A point moving around a circle can be described in two related ways. Its angular position tells us how far it has turned around the centre; its linear motion tells us how far it has travelled along the circular path. The connection is simple when the radius stays fixed, but needs care when the radius changes.

Angles in the formulas below are measured in radians. This is essential: the arc-length formula uses radians, not degrees.

Here \(r\) is measured in units of length, \(\theta\) in radians, and time in units such as seconds. Radians are dimensionless in dimensional analysis, so \(r\omega\) has units of length per time.

Why the Arc-Length Formula Works

A complete turn has angle \(2\pi\) radians and travels one circumference, \(2\pi r\). An angle \(\theta\) therefore covers the fraction \(\theta/(2\pi)\) of the circumference when \(\theta\) is nonnegative. Multiplying that fraction by the circumference gives the arc length:

For signed angles, \(s=r\theta\) records displacement along the circle in a chosen orientation; reversing direction makes \(s\) decrease. The physical distance along the arc for an angular change \(\Delta\theta\) is \(r|\Delta\theta|\).

When the radius is constant, differentiating the relation is direct. The derivative of the constant \(r\) is zero, so the rate of change of arc position is proportional to angular velocity.

The signed quantity \(ds/dt\) gives direction along the circle. Speed is nonnegative, so it is the magnitude \(v=|ds/dt|=r|\omega|\). If the point moves in the positive angular direction, the signed rate is positive; if it moves in the negative direction, it is negative.

Converting Revolutions into Angular Velocity

Sometimes the given rotational rate is in revolutions per unit time rather than radians per unit time. One complete revolution is \(2\pi\) radians. Convert the rate to radians per unit time before applying \(v=r|\omega|\).

When the Radius Changes

The formula \(s=r\theta\) describes the arc on a circle of radius \(r\). If the radius is changing, it is important not to treat \(r\theta\) as the total distance the moving point has travelled. A changing-radius point can move both around the centre and toward or away from it. Its total speed combines these two perpendicular components.

To see this precisely, describe the point’s Cartesian coordinates in terms of its radius and angle: \(x=r\cos\theta\) and \(y=r\sin\theta\). Differentiating both coordinates with respect to time gives

The speed is the magnitude of the velocity, \(\sqrt{(dx/dt)^2+(dy/dt)^2}\). Squaring the two coordinate rates and adding them cancels the mixed terms. Using \(\sin^2\theta+\cos^2\theta=1\) leaves

Taking the nonnegative square root gives the total speed. The radial component has signed rate \(dr/dt\); the tangential component has signed rate \(r\,d\theta/dt\). Their magnitudes are perpendicular components, not quantities to add directly.

A Common Mistake: Differentiating \(r\theta\) as Total Distance

If both \(r\) and \(\theta\) vary, the product rule gives \(d(r\theta)/dt=r(d\theta/dt)+\theta(dr/dt)\). That is the derivative of the quantity \(r\theta\), but it is not generally the total speed of the moving point. The product \(r\theta\) is an arc coordinate for the circle at the current radius; when that radius changes, it does not record the complete path travelled.

For the values in the previous example, this product-rule calculation would give

This is not the point’s total speed, which is \(\sqrt{65}/2\ \text{m/s}\). Nor is \(d(r\theta)/dt\) its tangential speed, which is \(4\ \text{m/s}\). The changing-radius motion must be handled using the radial and tangential components.