Motion in a circle

Learning objectives

• Use angular speed and radians.
• Calculate centripetal acceleration.
• Identify the real forces providing radial resultant force.

The big idea

An object moving in a circle changes velocity because its direction changes. For uniform circular motion, the acceleration points towards the centre even though the speed is constant.

Centripetal force names the required inward resultant, not an extra interaction to add to a force diagram. Depending on the situation, tension, gravity, friction or several forces together supply this resultant.

1. Constant speed, changing velocity

Velocity includes direction, so an object can accelerate while maintaining constant speed. In uniform circular motion, successive velocity vectors turn towards the centre, producing inward acceleration v²/r. The instantaneous velocity remains tangent to the circle. A photograph of one position does not establish the direction of acceleration; compare velocity vectors over a short interval instead.

2. Radians connect turning with distance

An angle in radians is arc length divided by radius. One complete revolution is 2π radians, so angular speed is 2π/T for period T. Tangential speed is ωr. Two points fixed on a rotating turntable have the same angular speed but different tangential speeds if their radii differ. Revolutions per minute must be converted before substitution into equations using radians per second.

3. Find the actual inward interactions

Centripetal force describes the required inward resultant, not an additional force to add beside tension, friction or gravity. For a car turning on a level road, sideways friction can supply it. For a satellite, gravity supplies it. Resolve all real forces radially and use their resultant as mv²/r. In a vertical circle, weight changes its contribution to the inward resultant with position.

4. Check the scaling prediction

For a 0.50 kg object moving at 4 m/s in a circle of radius 2 m, the required inward force is 0.50 × 16/2 = 4 N. Doubling speed at fixed radius requires four times the force. If the available interaction cannot provide that force, the object does not continue on the same circle with unchanged conditions. Its subsequent path follows the actual remaining forces.

A method you can reuse

1. Sketch the circle, tangent velocity and centre direction.
2. Convert period or rotation rate into angular speed if needed.
3. Draw real forces on the chosen body only.
4. Resolve their inward resultant and equate it to mv²/r.
5. Check the radius definition and the squared-speed dependence.

Equations and relationships

Investigate and explain

Use a circular-motion model to compare radial acceleration at fixed radius for two speeds; predict the factor before measuring.

Use circular settings to identify tangent velocity and inward acceleration, then compare elliptical cases. Orbital models do not directly represent a string, road friction or every rotating-machine constraint.

Open the interactive investigation

Check your understanding