Centripetal Force Formula
The centripetal force formula F = mv²/r calculates the inward force needed for circular motion.
Learn with step-by-step examples.
The Formula
Centripetal force is the net inward force that keeps an object moving in a circular path. Without this force, an object would fly off in a straight line due to inertia, following Newton's first law.
The word "centripetal" comes from Latin, meaning "center-seeking." This force can be provided by gravity (planets orbiting the Sun), tension (a ball on a string), friction (a car turning a corner), or any combination of forces directed toward the center of the circular path.
An alternative form uses angular velocity: F = mω²r, where ω is the angular velocity in radians per second. The centripetal acceleration alone is a = v²/r.
Variables
| Symbol | Meaning |
|---|---|
| F | Centripetal force (in newtons, N) |
| m | Mass of the object (in kilograms, kg) |
| v | Tangential velocity (in meters per second, m/s) |
| r | Radius of the circular path (in meters, m) |
Example 1
A 1,200 kg car takes a curve of radius 50 m at 15 m/s. What centripetal force is needed?
Identify the values: m = 1,200 kg, v = 15 m/s, r = 50 m
Apply the formula: F = mv²/r = 1,200 × 15² / 50
F = 1,200 × 225 / 50 = 270,000 / 50
F = 5,400 N
Example 2
A 0.5 kg ball on a 1.2 m string is swung in a circle. The tension in the string is 20 N. How fast is the ball moving?
Rearrange the formula: v² = Fr/m
v² = 20 × 1.2 / 0.5 = 24 / 0.5 = 48
v = √48
v ≈ 6.93 m/s
When to Use It
Use the centripetal force formula for any object moving in a circular or curved path.
- Determining the friction needed for a car to safely navigate a curve
- Calculating forces on a roller coaster at the top or bottom of a loop
- Designing centrifuges for separating substances
- Analyzing satellite orbits and planetary motion