Centripetal Force Equation, Derivation & Rearrangements

Core idea

Overview

Centripetal force is the net force required to keep an object moving in a curved or circular path, always directed toward the center of rotation. It is not a unique force itself but is provided by other physical interactions such as tension, gravity, or friction to counteract the object's inertia.

When to use: Apply this equation when an object travels at a constant speed along a circular trajectory. It assumes the force is perpendicular to the object's velocity and that the system is viewed from an inertial frame of reference.

Why it matters: This principle is fundamental in engineering safe highway curves, designing high-speed centrifuges for medical research, and calculating the orbits of satellites. It explains how forces must be balanced to prevent objects from flying off their intended paths due to inertia.

Symbols

Variables

F = Force, m = Mass, v = Velocity, r = Radius

F

Force

N

m

Mass

kg

v

Velocity

m/s

r

Radius

m

Walkthrough

Derivation

Understanding Centripetal Force

The resultant inward force required to keep an object moving in a circular path.

Motion is uniform circular motion.

1

Apply Newton's Second Law:

Resultant force equals mass times acceleration.

F = ma

2

Substitute Centripetal Acceleration:

Replace a with $v^{2}$ /r or $ω^{2}$ r.

F = \frac{m v ^{2}}{r} = m ω^{2} r

Result

F = \frac{m v ^{2}}{r} = m ω^{2} r

Source: AQA A-Level Physics — Circular Motion

Free formulas

Rearrangements

Solve for $F$

Make F the subject

F = \frac{m v ^{2}}{r}

F is already the subject of the formula.

Difficulty: 1/5

Solve for $m$

Make m the subject

m = \frac{F r}{v ^{2}}

Start from Centripetal Force. To make m the subject, clear r, then divide by $v^{2}$ .

Difficulty: 3/5

Solve for $v$

Make v the subject

v = \frac{F r}{m}

Start from Centripetal Force. To make v the subject, clear r, then make $v^{2}$ the subject, then take the square root.

Difficulty: 4/5

Solve for $r$

Make r the subject

r = \frac{m v ^{2}}{F}

Start from Centripetal Force. To make r the subject, clear r, then divide by F.

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph follows a parabolic curve opening upwards that starts from the origin, showing that force increases at an accelerating rate as velocity grows. For a student of physics, this means that small changes in velocity at high speeds require significantly more force to maintain circular motion than the same changes at low speeds. The most important feature of this curve is that the force is proportional to the square of the velocity, meaning that doubling the velocity results in a fourfold increase in the required force.

Graph type: parabolic

Why it behaves this way

Intuition

Imagine an object constantly being pulled or pushed towards the center of a circle, while its forward momentum simultaneously tries to carry it in a straight line, resulting in a continuous curved path.

m

Mass of the object undergoing circular motion.

More massive objects have greater inertia, so they require a proportionally larger centripetal force to achieve the same circular motion.

v

Speed of the object along the circular path.

The faster an object moves, the more forcefully it tries to move in a straight line, requiring a much larger (quadratically increasing) centripetal force to keep it turning.

r

Radius of the circular path.

A larger radius means a gentler curve, so less centripetal force is needed to keep the object on the path for a given speed and mass. Wider turns are easier to make.

Signs and relationships

v^2: The squared term indicates a non-linear, quadratic relationship: doubling the speed requires four times the centripetal force. This highlights the significant impact of speed on the required force.
1/r: The radius in the denominator signifies an inverse relationship: increasing the radius of the path decreases the required centripetal force. This means wider curves demand less force to navigate.

Free study cues

Insight

Canonical usage

In SI units, force is expressed in Newtons (N), mass in kilograms (kg), velocity in meters per second (m/s), and radius in meters (m).

Common confusion

A common mistake is mixing unit systems, such as using mass in grams while other quantities are in SI units, or forgetting to convert units to a consistent system before calculation.

Unit systems

$F$ N · The net centripetal force, directed towards the center of the circular path.

$m$ kg · The mass of the object undergoing circular motion.

$v$ m/s · The tangential speed of the object.

$r$ m · The radius of the circular path.

One free problem

Practice Problem

A 1200 kg race car travels around a circular track with a radius of 50 meters at a constant velocity of 20 m/s. What is the centripetal force exerted on the car by the track's friction?

Mass1200 kg

Velocity20 m/s

Radius50 m

Solve for: $F$

Hint: Square the velocity first, multiply by the mass, then divide by the radius.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Calculating tension in a swinging mass.

Study smarter

Tips

Always convert mass to kilograms and velocity to meters per second to ensure units remain consistent with Newtons.
The radius must be measured from the center of the circular path to the center of mass of the moving object.
Remember that velocity is squared, meaning doubling the speed quadruples the required centripetal force.

Avoid these traps

Common Mistakes

Using diameter instead of radius.
Using tangential speed incorrectly.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The resultant inward force required to keep an object moving in a circular path.

Apply this equation when an object travels at a constant speed along a circular trajectory. It assumes the force is perpendicular to the object's velocity and that the system is viewed from an inertial frame of reference.

This principle is fundamental in engineering safe highway curves, designing high-speed centrifuges for medical research, and calculating the orbits of satellites. It explains how forces must be balanced to prevent objects from flying off their intended paths due to inertia.

Using diameter instead of radius. Using tangential speed incorrectly.

Calculating tension in a swinging mass.

Always convert mass to kilograms and velocity to meters per second to ensure units remain consistent with Newtons. The radius must be measured from the center of the circular path to the center of mass of the moving object. Remember that velocity is squared, meaning doubling the speed quadruples the required centripetal force.

References

Sources

Halliday, Resnick, Walker, Fundamentals of Physics, 10th ed.
Wikipedia: Centripetal force
Halliday, Resnick, and Walker, Fundamentals of Physics, 10th ed.
NIST Guide for the Use of the International System of Units (SI)
Halliday, Resnick, Walker Fundamentals of Physics
Griffiths Introduction to Quantum Mechanics
AQA A-Level Physics — Circular Motion