What Is Radial Acceleration?
Anything moving along a curved path — a car rounding a bend, a stone on a string, a satellite in orbit — is constantly accelerating, even if its speed never changes. That's because acceleration means any change in velocity, and velocity is speed and direction together. Turning changes direction, so turning is accelerating.
Radial acceleration, also called centripetal acceleration, is the piece of that acceleration that points straight toward the centre of the curve. It's what bends a straight-line path into a circle, and it's the number engineers reach for whenever something spins, orbits, or corners.
The Radial Acceleration Formula
There are two equivalent ways to write it, depending on what you already know:
[
a = \frac{v^2}{r} = \omega^2 r
]
Where:
- a is the radial (centripetal) acceleration, in metres per second squared (m/s²)
- v is the tangential (linear) velocity of the object, in metres per second (m/s)
- ω (omega) is the angular velocity, in radians per second (rad/s)
- r is the radius of the circular path, in metres (m)
The two formulas describe the same physical quantity because tangential and angular velocity are related by (v = \omega r). Substitute that into (a = v^2 / r) and the r's simplify to leave (a = \omega^2 r) — pick whichever formula matches the value you actually measured.
Worked Example 1: Using Tangential Velocity
A ball on a string moves at 10 m/s around a circle with a 5 metre radius:
[
a = \frac{v^2}{r} = \frac{10^2}{5} = \frac{100}{5} = 20 \text{ m/s}^2
]
Worked Example 2: Using Angular Velocity
The same ball's angular velocity is (\omega = v / r = 10 / 5 = 2) rad/s. Plugging that into the second formula:
[
a = \omega^2 r = 2^2 \times 5 = 4 \times 5 = 20 \text{ m/s}^2
]
Both routes land on exactly 20 m/s², confirming the two formulas are the same relationship viewed two ways. Try either pair of numbers in the calculator above and you'll get the identical result.
Radial vs Tangential Acceleration
Circular motion can involve two separate kinds of acceleration at once, and it's easy to mix them up:
| Radial (centripetal) acceleration | Tangential acceleration | |
|---|---|---|
| Direction | Toward the centre of the circle | Along the direction of travel |
| Changes | The direction of velocity | The magnitude (speed) of velocity |
| Present when | Any curved path, even at constant speed | Only when speed is increasing or decreasing |
| Formula | a = v² / r = ω²r | a = dv / dt |
A car holding a steady 50 km/h around a roundabout has radial acceleration but no tangential acceleration. The same car speeding up out of the roundabout has both at once, and the two combine at right angles to give its total acceleration.
How Radius Affects the Result
The two formulas respond to a changing radius in opposite directions, which trips a lot of students up:
- Constant speed (v fixed): a = v² / r means a bigger radius gives less acceleration — a gentle, sweeping curve pulls less than a tight one at the same speed.
- Constant spin rate (ω fixed): a = ω²r means a bigger radius gives more acceleration — a point near the rim of a spinning disc accelerates harder than a point near its centre, even though both complete a rotation in the same time.
That second case is why the outer edge of a flywheel or grinding wheel is under far more stress than the hub, despite every point on the wheel sharing the same angular velocity.
Quick Recap
- Radial acceleration points toward the centre of a curved path and always accompanies a change in direction.
- Use a = v² / r when you know the tangential (linear) velocity, or a = ω²r when you know the angular velocity — they always agree.
- At constant speed, a bigger radius means less radial acceleration; at constant spin rate, a bigger radius means more.
- Use the calculator above to check either formula and see the acceleration in m/s².
If you're studying circular motion further, the centripetal force calculator turns this acceleration into the actual force needed to sustain it.