Centripetal Force Calculator – F = mv²/r

Compute Centripetal Force

Centripetal Force Formula

Formula

\[ F = \frac{mv^2}{r} \]

Where:

\(F\) — centripetal force (N)
\(m\) — mass of the object (kg)
\(v\) — tangential velocity of the object (m/s)
\(r\) — radius of the circular path (m)

This formula gives the inward force needed to keep an object of mass \(m\) moving at speed \(v\) along a circle of radius \(r\). The same equation can be rearranged to solve for any of the other variables:

\[ m = \frac{Fr}{v^2}, \quad v = \sqrt{\frac{Fr}{m}}, \quad r = \frac{mv^2}{F} \]

Centripetal Force – Calculation Example

Example: a 10 kg object moves in a circle of radius 2 m at a constant speed of 5 m/s. What centripetal force is required?

\[ F = \frac{mv^2}{r} = \frac{10 \times 5^2}{2} = \frac{10 \times 25}{2} = \frac{250}{2} = 125 \text{ N} \]

Result: the centripetal force is 125 N. This matches the calculator's default values, so you can reproduce it directly.

How to Use the Calculator

In the "I want to calculate" dropdown, select the variable you need: Centripetal Force (F), Mass (m), Velocity (v), or Radius (r). The calculator hides the field for the selected variable.
Enter the Mass (m) and choose the unit — kilograms or grams.
Enter the Velocity (v) and choose the unit — m/s or km/h.
Enter the Radius (r) and choose the unit — meters or centimeters.
If you are solving for mass, velocity, or radius, also enter the known Centripetal Force (F) in newtons.
The result appears instantly below the inputs. Values entered in grams, km/h, or centimeters are converted to SI units automatically.

What Is Centripetal Force?

Centripetal force is the net inward force that keeps an object moving along a circular path. Without it, the object would fly off in a straight line, as described by Newton's first law. Importantly, "centripetal" describes the direction of the force — toward the center of the circle — not a new type of force. In real situations it is supplied by something concrete: tension in a string, friction between car tires and the road, gravity for orbiting satellites, or the normal force from a loop-the-loop track.

The formula \( F = mv^2/r \) reveals two key relationships. First, the force grows with the square of the speed: doubling the speed requires four times the inward force. Second, a smaller radius demands more force — taking a tight turn is harder than a wide one at the same speed. This is why cars slow down for sharp curves and why high-speed turns on banked tracks are designed so carefully.

The result is expressed in newtons (N), the SI unit of force. To interpret it, compare it with the object's weight \(mg\): a centripetal force of 125 N on a 10 kg object (weight ≈ 98 N) means the circular motion pushes the object harder than gravity does — a passenger would feel noticeably heavier at the bottom of the loop. To express the same idea as a multiple of gravity, use the G-Force Calculator.

Assumptions and Limitations

Uniform circular motion: the formula assumes the object moves at constant speed. If the speed changes (non-uniform circular motion), there is also a tangential acceleration component, and the analysis becomes more complex.
Point mass: the object is treated as a point mass. For extended rotating bodies, use tools like the Rotational Kinetic Energy Calculator or the Angular Momentum Calculator.
Net force: the calculated value is the net inward force. Real-world sources of that force (friction, tension, gravity) must be able to supply it — if friction on a road is insufficient, the vehicle skids outward.
Speed, not angular velocity: the input is the linear (tangential) speed. If you know the angular velocity \(\omega\), convert first with \(v = \omega r\).
Unit conversions: grams, km/h, and centimeters are converted automatically, but the force must always be entered in newtons.

Practical Tips

For orbital problems around Earth or other planets, gravity acts as the centripetal force; pair this calculator with the Gravitational Force Calculator to check that the two are equal.
A quick sanity check: the centripetal force always points toward the center, and the centripetal acceleration is \(a = v^2/r\), independent of mass. Mass only scales the force, not the acceleration.
When analyzing banked curves or vehicle dynamics, keep in mind that friction limits are often the deciding factor rather than the theoretical force value.

Frequently Asked Questions

How do I calculate centripetal force by hand?

Multiply the mass by the square of the velocity, then divide by the radius: \( F = mv^2/r \). For a 1500 kg car taking a curve of radius 50 m at 20 m/s, the force is \( 1500 \times 20^2 / 50 = 12{,}000 \) N. Always convert to SI units first (kg, m/s, m) or your answer will be off by powers of ten.

What is a typical value of centripetal force?

There is no single "normal" value — it depends entirely on mass, speed, and radius. A useful benchmark is the object's weight \( mg \). If the centripetal force is a large fraction of the weight, passengers feel strong sideways or vertical forces. amusement park rides often reach 3–4 g, while a gentle highway curve stays well below 0.5 g.

How do I convert km/h to m/s for the formula?

Divide the speed in km/h by 3.6 to get m/s. For example, 90 km/h is 25 m/s. The calculator does this conversion automatically when you select km/h as the velocity unit, but you need it if computing by hand. Forgetting this step is one of the most common mistakes, and it makes the result wrong by a factor of 12.96.

What is the difference between centripetal and centrifugal force?

Centripetal force is the real inward force acting on an object in circular motion — provided by tension, friction, or gravity. Centrifugal force is a fictitious force that appears only when you analyze motion from a rotating reference frame; it feels like an outward push, but no real outward force acts on the object in an inertial frame.

Can I use angular velocity or RPM instead of linear speed?

Yes, but convert first. The calculator needs the tangential speed \( v \). If you know angular velocity \( \omega \) in rad/s, use \( v = \omega r \). If you have revolutions per minute, first convert to \( \omega = 2\pi \times \text{RPM} / 60 \), then multiply by the radius. Alternatively, use \( F = m\omega^2 r \) directly when computing by hand.

What are common mistakes when using the centripetal force formula?

The most frequent errors are using diameter instead of radius, mixing units (g with m, km/h with m/s), and squaring the radius instead of the velocity. Also remember that \( v \) is the tangential speed, not the angular velocity. Finally, the formula gives the net inward force — real suppliers like friction or tension must be able to provide it, or the circular path cannot be maintained.

How fast can a car go around a curve before it skids?

Set the maximum available friction equal to the required centripetal force: \( \mu mg = mv^2/r \), so \( v = \sqrt{\mu g r} \). On dry asphalt (\( \mu \approx 0.8 \)) with a 50 m curve, that is about 20 m/s or 71 km/h. This is an idealized estimate — real conditions, tire condition, and road banking lower the safe speed, so always obey posted speed limits.