Centripetal Acceleration
Simple Explanation
An object moving on a circular path is always accelerating toward the center of that circle, even at constant speed β this is centripetal acceleration, and it exists because a constantly-changing direction is itself a form of acceleration.
Why Do We Need It?
Centripetal acceleration is the reason curved motion needs a net inward force at all β without it, nothing could move in a circle, from a satellite's orbit to a car rounding a bend.
See It
An object on a circular path with an arrow pointing inward toward the center labelled ac, and a tangential velocity arrow perpendicular to it
Formula
Centripetal Acceleration
a_c = vΒ² / r = ΟΒ²r
The acceleration of an object moving on a circular path, always directed toward the centre of the circle β it changes the direction of motion, not the speed.
- a_c
- β centripetal acceleration, in metres per second squared (m/sΒ²)
- v
- β linear (tangential) speed, in metres per second (m/s)
- r
- β radius of the circular path, in metres (m)
- Ο
- β angular velocity, in radians per second (rad/s)
When to use it: Whenever an object moves on a circular path at constant speed and the acceleration responsible for curving that path is needed.
Worked Example
Find a centripetal acceleration
A car rounds a curve of radius 50 m at a constant speed of 20 m/s. Find its centripetal acceleration.
Why Does This Work?
Even at constant speed, an object moving in a circle constantly changes its direction of velocity β and acceleration is defined as any change in velocity, whether in magnitude or direction β so this continuous turning is itself an acceleration, and geometry shows it must point directly toward the center.
Real-Life Example
Why passengers feel pushed outward in a turning car
A car turning sharply around a corner requires a real inward (centripetal) acceleration to follow the curve, supplied by friction between the tires and the road.
Passengers feel "pushed" outward because their own bodies resist that inward acceleration (their inertia wants to keep them moving in a straight line) β the true physical acceleration is inward, toward the curve's center.
Practice
An object moves on a circular path of radius 2 m at angular velocity Ο=3 rad/s. Find its centripetal acceleration using a_c=ΟΒ²r.
MediumCommon mistake
Assuming "constant speed" means "no acceleration" β an object moving in a circle at constant SPEED is still accelerating, because its direction (part of velocity) is continuously changing.
Quick Review
- a_c = vΒ²/r = ΟΒ²r, always directed toward the center.
- Exists even at constant speed, because direction is changing.
- Requires a net inward force to sustain (see Chapter 2).