Circular Motion & Gravitation

You’ll be able to

Compute centripetal acceleration from speed and radius
Identify which real force supplies the centripetal force in a situation
Apply Newton’s law of universal gravitation

Turning is accelerating

An object moving in a circle at constant speed is still accelerating, because its direction — and therefore its velocity vector — keeps changing. This centripetal acceleration always points toward the center of the circle, and its size is a_c = v²/r. Faster motion or a tighter circle means a larger centripetal acceleration, and it grows with the square of the speed.

Centripetal force is a role, not a new force

By Newton’s second law, that inward acceleration needs an inward net force, F_c = mv²/r. But "centripetal force" is not a distinct kind of force — it is a job done by some ordinary force. For a car it is friction; for a ball on a string it is tension; for the Moon it is gravity. There is no outward "centrifugal force": the feeling of being flung outward is just your inertia trying to go straight while the real force pulls you inward.

Centripetal acceleration and force

a_c = v² / r · F_c = mv² / r

Both point toward the center of the circle. Doubling the speed quadruples both, because v is squared.

Newton’s law of universal gravitation

F = G·m₁·m₂ / r²

G = 6.67 × 10⁻¹¹ N·m²/kg². The force is an inverse-square law: triple the separation and the force drops to one-ninth.

Worked example

A 2 kg ball on a 0.5 m string is whirled in a horizontal circle at 4 m/s. Find its centripetal acceleration and the tension in the string.

1.Centripetal acceleration: a_c = v²/r = (4)² ÷ 0.5 = 16 ÷ 0.5 = 32 m/s².
2.The tension supplies the centripetal force, so F_c = m·a_c.
3.Compute: F_c = 2 × 32 = 64 N.
4.That inward force is the string tension: T = 64 N.

Answer: a_c = 32 m/s² and T = 64 N, both directed toward the center

Watch out

Do not add an outward "centrifugal force" to a free-body diagram. The net force in circular motion points inward. If your forces balance to zero, the object would go straight, not curve.

Checkpoint

A car rounds a curve of radius 50 m at a constant 10 m/s. What is its centripetal acceleration?

Checkpoint

A car drives around a flat, level circular track. Which force provides the centripetal force that keeps it turning?

On the exam

When a circular-motion problem asks for a force, first ask "which real force points toward the center?" Then set that force equal to mv²/r. Naming the actual force (tension, friction, gravity, normal) is what earns the reasoning point.

Answer the 2 checkpoints as you read.