Centripetal Force
Anything moving in a circle is constantly changing direction, and that change requires a force pointed toward the center of the circle. Without this inward pull, the object would fly off in a straight line instead of curving.
The formula
a = v² / r
- a — centripetal acceleration (m/s²): how quickly the object's direction changes toward the center
- v — speed (m/s): how fast the object moves along the circular path
- r — radius (m): the distance from the object to the center of the circle
Worked example
An object moves in a circle of radius 2 meters at a speed of 4 meters per second. What is its centripetal acceleration?
- v = 4 m/s
- r = 2 m
- a = v² / r
- a = (4 m/s)² / 2 m
a = 8 m/s²
Test yourself
The same object now circles at 8 meters per second, still at a 2 meter radius. What is its centripetal acceleration?
- 16 m/s²
- 8 m/s²
- Correct answer: 32 m/s²
Right! Speed is squared, so doubling it quadruples the acceleration: 8² ÷ 2 = 32 m/s².
Keep the speed at 4 meters per second, but double the radius to 4 meters. What happens to the acceleration?
- Correct answer: 4 m/s²
- 16 m/s²
- 8 m/s²
Exactly — a = v² / r: 4² ÷ 4 = 4 m/s². Doubling the radius cuts the acceleration in half.
Where you see this
A car rounding a curve needs friction between its tires and the road pulling it toward the curve's center, or it slides off; a swung bucket of water needs the string's tension pulling inward, or the bucket flies off in a straight line the instant it lets go. Both are the same inward force, just supplied by something different.
Common mistakes
Centripetal force is often described as a real, separate force pushing outward — there is no outward "centrifugal force" acting on the object itself; what's actually happening is the object's own inertia trying to go straight while something (friction, tension, gravity) pulls it inward. It's also easy to think a faster circular speed needs proportionally more force, but a = v²/r means doubling the speed quadruples the required force, not doubles it.
How it connects
This is Newton's second law again, applied to an object whose acceleration always points toward a center rather than along its direction of travel. Universal gravitation, next, supplies the actual force — gravity — that keeps planets and satellites moving in the circular or near-circular orbits centripetal force describes.