Centripetal Acceleration Formula
Reference for centripetal acceleration a = v squared over r and a = omega squared times r.
Covers the link to centripetal force with orbit and turn examples.
The Formula
Centripetal acceleration is the rate of change of velocity direction for an object moving along a curved path. It always points toward the center of the circle.
Even if the speed stays constant, the direction changes continuously. That change in direction IS acceleration — centripetal acceleration.
Variables
| Symbol | Meaning |
|---|---|
| a | Centripetal acceleration (m/s²) |
| v | Linear velocity / speed of the object (m/s) |
| r | Radius of the circular path (meters) |
Alternative Form (using angular velocity)
Where ω (omega) is the angular velocity in radians per second. The two forms are equivalent because v = ωr.
Example 1
A car drives around a roundabout (radius 25 m) at 10 m/s
a = 10² / 25
a = 100 / 25
a = 4 m/s² (about 0.4 g)
Example 2
A satellite orbits Earth at 7,800 m/s at an altitude of 400 km (r = 6,771,000 m from Earth's center)
a = 7800² / 6,771,000
a = 60,840,000 / 6,771,000
a = 8.99 m/s² (nearly equal to g at that altitude)
Example 3
A cyclist rounds a curve (radius 10 m) at 6 m/s
a = 6² / 10
a = 36 / 10
a = 3.6 m/s²
When to Use It
- Determining how much grip a car needs on a curved road
- Calculating g-forces on roller coasters and in airplane turns
- Designing centrifuges for laboratory or industrial use
- Analyzing orbital mechanics for satellites and planets
- Understanding why objects feel heavier on spinning amusement rides
Key Relationships
- Doubling the speed quadruples the centripetal acceleration (because v is squared)
- Halving the radius doubles the acceleration
- Centripetal acceleration multiplied by mass gives centripetal force: F = ma = mv²/r
Key Notes
- Formula: a_c = v²/r = ω²r: Centripetal acceleration is always directed toward the center of the circular path. v is tangential speed, r is the radius, and ω is angular velocity (rad/s). The two forms are related by v = ωr.
- Centripetal force is not a new force: Newton's second law: F_net = ma_c = mv²/r. This net inward force is provided by whatever physical force exists — gravity for satellites, normal force for a car on a banked curve, tension for a spinning object on a string. "Centripetal force" names the direction, not the source.
- "Centrifugal force" is fictitious: In a rotating reference frame, objects appear to be pushed outward by a "centrifugal force." This is a pseudo-force arising from the non-inertial frame — it doesn't exist in an inertial (non-rotating) frame. The real effect is inertia: objects naturally travel in straight lines; a centripetal force makes them curve.
- Banked curve optimal angle: tan θ = v²/(rg): At this angle, no friction is needed to maintain circular motion — the horizontal component of the normal force provides exactly the centripetal force. Roads and racetracks are banked for this reason; the optimal angle depends on designed speed.
- Applications: Centripetal acceleration calculations govern satellite orbit design (gravity provides centripetal force), amusement park ride safety limits, centrifuge rotor specifications (where a_c can reach 10,000×g), car cornering limits, banked highway curve design, and astronaut training equipment.