2.7 Circular Motion

Centripetal Acceleration

An object moving in a circle at constant speed still accelerates because its velocity direction constantly changes. This acceleration, called centripetal acceleration, always points toward the center of the circle. Its magnitude is a_c = v²/r, where v is speed and r is the radius. No acceleration means no net force, so circular motion always requires a net inward force.

Net Force Causes the Turn

There's no separate 'centripetal force' — it's just the net force directed toward the center, supplied by real forces like tension, gravity, friction, or normal force. Using Newton's second law: F_net = ma_c = mv²/r. To solve circular motion problems, identify all real forces, sum their components toward the center, and set that sum equal to mv²/r.

Common Scenarios

In a vertical circle (like a roller coaster loop), the net inward force changes composition depending on position: at the top, gravity and normal force both point toward the center; at the bottom, normal force must overcome gravity. For a car on a flat curve, friction provides the centripetal force; on a banked curve, the normal force also contributes toward the center.

Key terms

Centripetal acceleration formula

a_c = v²/r, directed toward the center of the circular path.

Is there a special 'centripetal force'?

No — centripetal force is just the net force (from real forces like tension, gravity, friction, normal) pointing toward the center.

Newton's second law for circular motion

F_net = ma_c = mv²/r, where F_net is the sum of all real forces directed toward the center.

Top of a vertical loop

Both gravity and normal force point toward the center (downward), combining to supply centripetal force.

Flat curve driving

Friction between tires and road provides the centripetal force needed to keep the car turning.

Try an AP-style question

A car moves at 10 m/s around a curve of radius 5 m. What is its centripetal acceleration?

1. 20 m/s²
2. 50 m/s²
3. 2 m/s²
4. 0.5 m/s²