Centripetal Force Calculator

What the centripetal force calculator does

Anything moving in a circle is being pulled toward the centre. This calculator works out that inward pull, the centripetal force, from the object's mass, the radius of its circular path, and how fast it is going. You can also turn it around and solve for the mass, the radius, or the speed when you know the force.

Below is what centripetal force is, where it comes from, the equations behind it, and a worked example.

How to use it

1. Choose what to calculate: the force, the mass, the radius, or the speed.
2. Pick how you are giving the speed, as an angular velocity in radians per second or a tangential velocity in metres per second, and enter the values the calculator asks for.
3. Press Calculate for the result, or Reset to clear it.

What centripetal force is

An object moving at a steady speed in a straight line feels no net force. But the moment its path curves, something has to be pushing or pulling it sideways, because its direction is changing, and changing direction is a kind of acceleration. Centripetal force is the name for that sideways pull when the path is a circle. The word means center-seeking, and it always points inward, toward the centre of the circle.

It is there in every circular motion, whether or not you notice it. The Moon circling the Earth, a car rounding a bend, a conker whirled on a string, a child on a roundabout: each is held in its circle by a force aimed at the centre. Take that force away and the object would not fly outward, it would simply carry straight on in a tangent, the way a stone leaves a sling. The centripetal force is what keeps bending the path back into a circle.

What provides it

Here is the key idea that often gets missed: centripetal force is not a new kind of force in its own right. It is a job, a role, and some ordinary force is always doing that job. The question is never just how big the centripetal force is, but what is actually supplying it.

For the Moon, gravity supplies it. For a car on a flat bend, the friction between tyres and road supplies it, which is why a car on ice slides straight off the curve. For a conker on a string, the tension in the string supplies it. For a bucket of water swung overhead, the bucket's bottom pushes the water inward. In each case a real, familiar force, gravity, friction, tension, or a push, happens to be pointing toward the centre, and so plays the centripetal role. This calculator gives the size of that inward force; what provides it depends on the situation.

The equations it uses

The centripetal force depends on the mass m, the radius r, and the speed, and it can be written two ways depending on how the speed is given. With the tangential speed v, the speed along the circle, it is:

F = m v² ÷ r

With the angular velocity ω, how fast the object sweeps around in radians per second, it takes the form:

F = m r ω²

The two are the same force, just expressed through different measures of how fast the object goes around, since the tangential speed equals the radius times the angular velocity. The calculator lets you enter whichever you have, and it rearranges the same relationship to solve for mass, radius, or speed when those are the unknowns.

Why it grows with speed and shrinks with radius

The speed enters the formula squared, which makes it the dominant influence. Double the speed and the centripetal force needed does not double, it quadruples. This is why taking a bend twice as fast demands four times the grip from your tyres, and why high-speed turns are so much more punishing than slow ones. It is the single biggest reason fast circular motion feels so forceful.

The radius works the other way. A larger radius means a gentler curve, so for the same speed a wider circle needs less inward force, while a tight circle needs more. Mass simply scales the whole thing: a heavier object on the same path needs proportionally more force to hold it there. Together these explain the feel of circular motion, the hard yank of a fast, tight turn against the easy drift of a slow, wide one.

Units and precision

The calculator works in SI units: mass in kilograms, radius in metres, tangential velocity in metres per second or angular velocity in radians per second, and the resulting force in newtons. The relationship is exact, so the result is as accurate as the numbers you put in; the formula assumes steady circular motion at a constant speed. Results are shown to several decimal places.

A worked example

Suppose a 2 kilogram ball is being whirled on a string in a circle of radius 1.5 metres at a tangential speed of 10 metres per second.

The centripetal force is F = mv²/r = (2 × 10²) ÷ 1.5 = 200 ÷ 1.5 ≈ 133 newtons, pulling inward along the string toward your hand. Swing it twice as fast, at 20 metres per second, and the force would leap to about 533 newtons, four times as much, because the speed is squared.

Questions people ask

How do you calculate centripetal force?

Multiply the mass by the square of the tangential speed and divide by the radius, F = mv²/r. If you have the angular velocity instead, use F = mrω².

What provides the centripetal force?

It is always some other force pointing toward the centre: gravity for an orbit, friction for a car on a bend, tension for an object on a string, a normal force for a wall of a spinning drum. It is a role, not a separate force.

Which way does centripetal force point?

Always toward the centre of the circular path, inward. That inward pull is what continually changes the object's direction and bends its path into a circle.

Why does going around faster need so much more force?

Because the speed appears squared in the formula. Doubling the speed quadruples the centripetal force needed, which is why fast, tight turns feel so much more forceful than slow ones.

References

A quick note on where the physics comes from. Centripetal force and acceleration in circular motion are standard mechanics, set out in OpenStax's University Physics and in Georgia State University's HyperPhysics. The SI units follow the US National Institute of Standards and Technology. The HyperPhysics link is worth a quick click to confirm it lands where you expect.

1. OpenStax, University Physics Volume 1, Section 6.3, Centripetal Force. https://openstax.org/books/university-physics-volume-1/pages/6-3-centripetal-force
2. HyperPhysics, Centripetal Force. http://hyperphysics.phy-astr.gsu.edu/hbase/cf.html
3. National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI). https://www.nist.gov/pml/special-publication-811