What this calculator does
Anything moving in a circle at constant speed is still accelerating, because its direction is constantly changing even though its speed is not. That acceleration points toward the centre of the circle, and the force producing it, centripetal force, is what keeps the object on its circular path instead of travelling off in a straight line.
This calculator takes mass, speed and radius and returns the centripetal force, the centripetal acceleration in both metres per second squared and multiples of g, the angular velocity, and the time for one full revolution.
The formula
Centripetal force equals mass times speed squared, divided by radius. Centripetal acceleration is the same expression without the mass term, speed squared divided by radius. Angular velocity is speed divided by radius, and the period, the time for one revolution, is the circumference divided by speed.
| Term | Meaning |
|---|---|
| Centripetal force | F = mv²/r, directed toward the centre of the circular path. |
| Centripetal acceleration | a = v²/r, the rate of change of direction, not speed. |
| Angular velocity | ω = v/r, how fast the angle around the circle is changing, in radians per second. |
| Period | T = 2πr/v, the time taken to complete one full revolution. |
The inputs explained
| Field | What to enter |
|---|---|
| Mass (kg) | The mass of the object moving in the circle. |
| Speed (m/s) | Its speed, assumed constant around the circular path. |
| Radius (m) | The radius of the circular path. |
When to use it
Cornering at speed
A vehicle taking a curve of a given radius needs friction between tyre and road to supply the centripetal force; at a fixed speed, a tighter radius demands more force than a wider one.
Orbital and rotational mechanics problems
From a ball on a string to a satellite in orbit, the same relationship between force, speed and radius applies, only the source of the force changes, tension in one case, gravity in the other.
Checking whether a design is within limits
Centrifuges, amusement park rides and rotating machinery are all constrained by how much centripetal acceleration, expressed in g, a structure or occupant can safely handle at a given speed and radius.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How centripetal force changes with speed at a fixed radius
A fixed 2 kg mass on an 0.8 m radius, across a range of speeds.
| Speed | Centripetal force | Centripetal acceleration | In g |
|---|---|---|---|
| 2 m/s | 10.000 N | 5.000 m/s² | 0.510 g |
| 3 m/s | 22.500 N | 11.250 m/s² | 1.147 g |
| 5 m/s | 62.500 N | 31.250 m/s² | 3.187 g |
| 7 m/s | 122.500 N | 61.250 m/s² | 6.246 g |
| 10 m/s | 250.000 N | 125.000 m/s² | 12.746 g |
| 15 m/s | 562.500 N | 281.250 m/s² | 28.680 g |
How centripetal force changes with radius at a fixed speed
A fixed 2 kg mass moving at 5 m/s, across a range of radii.
| Radius | Centripetal force | Centripetal acceleration | Time for one revolution |
|---|---|---|---|
| 0.3 m | 166.667 N | 83.333 m/s² | 0.377 s |
| 0.5 m | 100.000 N | 50.000 m/s² | 0.628 s |
| 0.8 m | 62.500 N | 31.250 m/s² | 1.005 s |
| 1.2 m | 41.667 N | 20.833 m/s² | 1.508 s |
| 2 m | 25.000 N | 12.500 m/s² | 2.513 s |
| 4 m | 12.500 N | 6.250 m/s² | 5.027 s |
Questions
Is centripetal force a separate force of its own?
No. It is the name given to whatever net force is actually acting toward the centre, whether that is tension, friction, gravity or a normal force. There is no additional force beyond the ones already present.
What is the difference between centripetal and centrifugal force?
Centripetal force is real and points toward the centre. Centrifugal force is the apparent outward push felt inside a rotating frame of reference; it does not act on the object from an outside, non-rotating viewpoint.
Why does speed matter more than mass in the force calculation?
Because force depends on speed squared but only on mass to the first power, so doubling speed has a much larger effect on the force required than doubling mass does.
How is this related to angular velocity?
Speed and radius together determine angular velocity (ω = v/r), and centripetal force can equally be written as mω²r. Both forms describe the same physical quantity.
For the related spring and elastic energy relationship, see Hooke's law and spring energy.