Last Updated: September 5, 2026
approximately fixed center. In physics, circular motion formulas are used to calculate centripetal acceleration, centripetal force, speed, angular velocity, period, and frequency. These relationships help explain how objects move around curves, rotate in circular paths, travel through vertical loops, and follow approximately circular orbits.
This circular motion guide is designed for GCSE, IGCSE, A-Level, IB Physics, AP Physics, and introductory mechanics students, as well as learners studying engineering and other STEM subjects. Use these formulas when solving physics problems involving rotating objects, cars on curved roads, roller coasters, banked curves, and satellite motion. The guide includes the main circular motion formulas, uniform circular motion, real-world examples, and step-by-step worked examples.
Quick Summary
Circular motion describes the motion of an object along a circular path. Even when an object’s speed stays constant, its velocity changes because its direction is continuously changing. This change in velocity produces centripetal acceleration, which always points toward the center of the circle.
The most important circular motion formulas are:
- Centripetal acceleration: (a_c = \frac{v^2}{r} = \omega^2r)
- Centripetal force: (F_c = ma_c = \frac{mv^2}{r} = m\omega^2r)
- Tangential velocity: (v = \omega r)
- Angular velocity: (\omega = \frac{2\pi}{T} = 2\pi f)
- Period and frequency: (f = \frac{1}{T})
- Circular-path speed: (v = \frac{2\pi r}{T} = 2\pi rf)
There is not one single formula that applies to every circular motion problem. The correct formula depends on what quantities are given and what you need to calculate.
Key Takeaways
- Circular motion involves continuous changes in the direction of velocity.
- Uniform circular motion has constant speed but changing velocity.
- Centripetal acceleration always points toward the center of the circular path.
- The basic formula for centripetal acceleration is (a_c = \frac{v^2}{r}).
- The required net centripetal force is (F_c = \frac{mv^2}{r}).
- Centripetal force is not a separate type of physical force; it is the name given to the net inward force.
- Tension, friction, gravity, or normal force can provide the required centripetal force.
- In uniform circular motion, centripetal force does no work because it is perpendicular to instantaneous displacement.
- Vertical circular motion can have changing speed because gravity has a tangential component except at special points.
- Circular motion formulas are widely used in vehicle turns, roller coasters, rotating machinery, satellites, and other physics applications.
What Is Circular Motion?
Circular motion is the motion of an object along a circular path around a fixed or approximately fixed center. The radius (r) is the distance between the moving object and the center of the circular path.
A simple example is a ball attached to a string and swung in a circle. The ball’s direction changes continuously as it moves around the center.
Circular motion can be divided into two important cases:
- Uniform circular motion: speed remains constant while direction continuously changes.
- Non-uniform circular motion: both the direction and the speed can change.
The distinction is important because an object can have constant speed and still be accelerating. Acceleration occurs whenever velocity changes, and velocity includes both magnitude and direction.
For additional background on force and acceleration, see the Newton’s Laws of Motion Guide.
For problems involving constant linear acceleration, the SUVAT Equations Guide is useful. Circular motion requires additional radial relationships because the direction of velocity is continuously changing.
Examples of Circular Motion in Everyday Life
Examples of circular motion include a car turning around a roundabout, a ball attached to a string being swung in a circle, a roller coaster moving through a loop, a satellite moving around Earth, and the tip of a rotating fan blade.
Other physics circular motion examples include:
- A car travelling around a curved road
- A stone being whirled on a string
- The hands of an analog clock
- A point on a rotating wheel
- A roller coaster moving through a vertical loop
- A satellite in an approximately circular orbit
- Rotating components in engines and machines
- A rider moving around a circular amusement-park ride
In real systems, the path is not always a perfect circle. However, circular motion provides an important model for understanding curved motion and rotational dynamics.
Circular Motion Formulas
What Is the Formula for Circular Motion?
The main formula for circular motion depends on the quantity you need to find. For centripetal acceleration, use (a_c = \frac{v^2}{r}); for centripetal force, use (F_c = \frac{mv^2}{r}).
These are the most commonly used circular motion formulas:
| Quantity | Formula | SI Unit |
|---|---|---|
| Centripetal acceleration | (a_c = \frac{v^2}{r}) | m/s² |
| Centripetal acceleration | (a_c = \omega^2r) | m/s² |
| Centripetal force | (F_c = \frac{mv^2}{r}) | N |
| Centripetal force | (F_c = m\omega^2r) | N |
| Tangential velocity | (v = \omega r) | m/s |
| Angular velocity | (\omega = \frac{2\pi}{T}) | rad/s |
| Angular velocity | (\omega = 2\pi f) | rad/s |
| Frequency | (f = \frac{1}{T}) | Hz |
| Period | (T = \frac{1}{f}) | s |
| Linear speed around a circle | (v = \frac{2\pi r}{T}) | m/s |
These relationships are consistent with standard treatments of uniform circular motion, where centripetal acceleration is (v^2/r) and centripetal force is (mv^2/r).
What Do the Symbols Mean?
- (a_c) = centripetal acceleration
- (F_c) = centripetal force
- (v) = tangential or linear speed
- (r) = radius of the circular path
- (m) = mass
- (\omega) = angular velocity
- (T) = period
- (f) = frequency
- (\pi) ≈ 3.14159
The units must be consistent before substituting values into a formula. In most physics problems, use kilograms, meters, seconds, newtons, and radians per second.
Uniform Circular Motion Formula
What Is the Uniform Circular Motion Formula?
For uniform circular motion, the key formula is (a_c = \frac{v^2}{r}), while the required centripetal force is (F_c = \frac{mv^2}{r}).
Uniform circular motion occurs when an object moves around a circle with a constant speed and constant radius.
Although the speed is constant, the velocity is not constant because its direction changes at every point on the circular path.
For uniform circular motion:
[
a_c = \frac{v^2}{r}
]
and
[
a_c = \omega^2r
]
The two forms are equivalent because:
[
v = \omega r
]
Substituting (v = \omega r):
[
a_c = \frac{(\omega r)^2}{r}
]
[
a_c = \omega^2r
]
The centripetal acceleration points radially inward toward the center of the circular path. Standard physics treatments likewise distinguish this inward acceleration from tangential acceleration.
Centripetal Acceleration
Centripetal acceleration is the inward acceleration required to continuously change an object’s velocity direction while it follows a circular path.
The main formula is:
[
a_c = \frac{v^2}{r}
]
where:
- (a_c) is centripetal acceleration in m/s²
- (v) is tangential speed in m/s
- (r) is radius in m
Using angular velocity:
[
a_c = \omega^2r
]
How Does Speed Affect Centripetal Acceleration?
Because speed is squared:
[
a_c \propto v^2
]
If the speed doubles while the radius remains constant:
[
a_c’ = \frac{(2v)^2}{r} = 4a_c
]
So doubling the speed produces four times the centripetal acceleration.
If the radius doubles while speed remains constant:
[
a_c’ = \frac{v^2}{2r} = \frac{a_c}{2}
]
Therefore, increasing the radius reduces the centripetal acceleration required for a given speed.
Centripetal Force
Centripetal force is the net inward force required to keep an object moving along a circular path.
Using Newton’s Second Law:
[
F_c = ma_c
]
Substituting the centripetal acceleration formula:
[
F_c = \frac{mv^2}{r}
]
or:
[
F_c = m\omega^2r
]
Centripetal force is not a new independent force. Instead, it describes the role played by a real net force directed toward the center.
For example:
| Circular Motion Example | Force Providing Centripetal Force |
|---|---|
| Ball on a string | Tension |
| Car on a flat curved road | Static friction |
| Satellite around Earth | Gravity |
| Roller coaster loop | Gravity and/or normal force |
| Banked road | Component of normal force |
This is an important application of Newton’s Second Law: the actual forces in the free-body diagram must be resolved so their inward net component provides the required (F_c).
You can review the underlying force relationship in the F = ma Physics Guide.
Angular Velocity, Frequency, and Period
Circular motion can also be described using rotational quantities.
Angular Velocity
Angular velocity describes how quickly the angular position changes:
[
\omega = \frac{\Delta\theta}{\Delta t}
]
For one complete revolution:
[
\Delta\theta = 2\pi
]
Therefore:
[
\omega = \frac{2\pi}{T}
]
Because:
[
f = \frac{1}{T}
]
we can also write:
[
\omega = 2\pi f
]
Period
The period (T) is the time required for one complete revolution.
[
T = \frac{1}{f}
]
Frequency
Frequency (f) is the number of complete revolutions per second.
[
f = \frac{1}{T}
]
The unit of frequency is hertz (Hz).
Tangential Velocity
The relationship between tangential velocity and angular velocity is:
[
v = \omega r
]
Since:
[
\omega = \frac{2\pi}{T}
]
we obtain:
[
v = \frac{2\pi r}{T}
]
This equation is particularly useful when a problem gives the radius and time for one complete revolution.
How to Solve Circular Motion Problems
A reliable method for solving circular motion problems is:
Step 1: Identify the Radius
Find the radius (r) of the circular path.
Be careful not to use the diameter unless you first divide it by two:
[
r = \frac{d}{2}
]
Step 2: Identify the Given Motion Quantity
Determine whether the problem gives:
- Speed (v)
- Angular velocity (\omega)
- Period (T)
- Frequency (f)
- Mass (m)
- Radius (r)
Step 3: Select the Correct Formula
For acceleration:
[
a_c = \frac{v^2}{r}
]
For force:
[
F_c = \frac{mv^2}{r}
]
For angular velocity:
[
\omega = \frac{2\pi}{T}
]
For speed:
[
v = \omega r
]
Step 4: Check the Direction
Centripetal acceleration and the net centripetal force point toward the center.
Step 5: Check Units
Make sure:
- mass is in kg
- radius is in m
- speed is in m/s
- acceleration is in m/s²
- force is in N
- period is in s
- frequency is in Hz
Circular Motion Examples
The following worked examples show how the main circular motion formulas are applied in physics problems.
Example 1: Ball Moving in a Circle
A (0.5) kg ball is attached to a string and moves in a horizontal circle with a radius of (1.2) m at a speed of (4) m/s. Find the centripetal force.
Given
[
m = 0.5\text{ kg}
]
[
v = 4\text{ m/s}
]
[
r = 1.2\text{ m}
]
Formula
[
F_c = \frac{mv^2}{r}
]
Calculation
[
F_c = \frac{(0.5)(4^2)}{1.2}
]
[
F_c = \frac{8}{1.2}
]
[
\boxed{F_c = 6.67\text{ N}}
]
If the string provides the only inward force, its tension is (6.67) N.
Example 2: Finding Centripetal Acceleration
A car travels around a circular curve at (12) m/s. The radius of the curve is (40) m. Find its centripetal acceleration.
Formula
[
a_c = \frac{v^2}{r}
]
Calculation
[
a_c = \frac{12^2}{40}
]
[
a_c = \frac{144}{40}
]
[
\boxed{a_c = 3.6\text{ m/s}^2}
]
The acceleration points toward the center of the curve.
Example 3: Finding Circular Motion Speed From Period
An object moves around a circle with a radius of (2) m and completes one revolution every (4) seconds. Find its speed.
Formula
[
v = \frac{2\pi r}{T}
]
Calculation
[
v = \frac{2\pi(2)}{4}
]
[
v = \pi
]
[
\boxed{v \approx 3.14\text{ m/s}}
]
Example 4: Finding Centripetal Force From Angular Velocity
A (2) kg object moves in a circle of radius (0.5) m with an angular velocity of (6) rad/s. Find the centripetal force.
Formula
[
F_c = m\omega^2r
]
Calculation
[
F_c = (2)(6^2)(0.5)
]
[
F_c = (2)(36)(0.5)
]
[
\boxed{F_c = 36\text{ N}}
]
Example 5: Minimum Speed at the Top of a Vertical Loop
A roller coaster travels through a vertical loop with a radius of (8) m. Find the minimum speed at the top required to maintain contact with the track, assuming the ideal limiting case.
At the minimum-contact condition:
[
N = 0
]
so gravity alone provides the required centripetal force:
[
mg = \frac{mv^2}{r}
]
Cancel (m):
[
g = \frac{v^2}{r}
]
Therefore:
[
v_{\min} = \sqrt{gr}
]
Using (g = 9.8\text{ m/s}^2):
[
v_{\min} = \sqrt{(9.8)(8)}
]
[
v_{\min} = \sqrt{78.4}
]
[
\boxed{v_{\min} \approx 8.85\text{ m/s}}
]
This is the minimum speed at the top of the loop, not necessarily the speed at the bottom.
Who Is This Circular Motion Guide For?
This circular motion guide is for GCSE, IGCSE, A-Level, IB Physics, AP Physics, and introductory mechanics students who need to understand and apply circular motion formulas. It is also useful for engineering and STEM learners studying force, acceleration, rotation, vehicle dynamics, and orbital motion.
Use this guide when you need to:
- Identify the correct circular motion formula for a physics problem.
- Calculate centripetal acceleration or centripetal force.
- Solve uniform circular motion questions.
- Find speed from radius, period, frequency, or angular velocity.
- Analyze cars moving around curved roads.
- Solve vertical-loop and roller-coaster problems.
- Understand banked curves and the forces acting on vehicles.
- Apply circular motion principles to satellite and orbital motion.
- Practice numerical physics questions using worked examples.
The main decision in a circular motion problem is usually determining which quantity is known, which quantity must be found, and which force provides the required inward centripetal force. Once those are identified, the appropriate formula can be selected and applied.
Circular Motion on a Flat Road
When a car travels around a flat circular curve, static friction between the tires and road can provide the required centripetal force.
The maximum available static friction is:
[
f_s \leq \mu_sN
]
On a level road:
[
N = mg
]
so the maximum friction is:
[
f_{s,\max} = \mu_smg
]
For the car to remain on the circular path:
[
\frac{mv^2}{r} \leq \mu_smg
]
The mass cancels:
[
\frac{v^2}{r} \leq \mu_sg
]
Therefore, the maximum speed is:
[
v_{\max} = \sqrt{\mu_sgr}
]
This explains why a vehicle can lose its ability to follow a curve when friction is too small or speed is too high. OpenStax uses the same physical model for a car rounding a flat curve, where friction supplies the centripetal force.
Vertical Circular Motion
Vertical circular motion is different from ideal uniform circular motion because gravity can change the object’s speed as it moves around the loop.
At the top of a vertical loop, gravity points toward the center:
[
mg + N = \frac{mv^2}{r}
]
At the bottom, the direction toward the center is upward, so:
[
N – mg = \frac{mv^2}{r}
]
The speed therefore generally changes around a vertical loop.
For energy-based problems, the Kinetic Energy vs Potential Energy Guide can help with the relationship between height, speed, and mechanical energy.
Banked Curves
A banked curve is tilted so that the normal force has a horizontal component directed toward the center of the curve.
For an ideal banked curve with no friction:
[
N\cos\theta = mg
]
and:
[
N\sin\theta = \frac{mv^2}{r}
]
Dividing the equations gives:
[
\tan\theta = \frac{v^2}{rg}
]
Therefore:
[
\boxed{\theta = \tan^{-1}\left(\frac{v^2}{rg}\right)}
]
This relationship allows engineers to determine the ideal banking angle for a particular speed and curve radius.
Circular Motion and Work
Is Work Done by Centripetal Force in Uniform Circular Motion?
No. In uniform circular motion, the centripetal force does zero work because it is perpendicular to the instantaneous direction of motion.
Work is:
[
W = Fd\cos\theta
]
For the centripetal force:
[
\theta = 90^\circ
]
Therefore:
[
W = Fd\cos90^\circ = 0
]
So centripetal force changes the direction of velocity without changing the speed in uniform circular motion.
However, this statement should not be applied blindly to all circular motion. In non-uniform circular motion, a tangential component of force can change the object’s speed and therefore do work.
Circular Motion and Satellite Orbits
Satellites provide one of the most important physics circular motion examples.
For an ideal circular orbit, Earth’s gravitational force supplies the centripetal force:
[
\frac{GMm}{r^2} = \frac{mv^2}{r}
]
Canceling (m):
[
\frac{GM}{r^2} = \frac{v^2}{r}
]
Therefore:
[
\boxed{v = \sqrt{\frac{GM}{r}}}
]
where:
- (G) = gravitational constant
- (M) = mass of the central body
- (r) = orbital radius
- (v) = orbital speed
The satellite’s mass cancels from the equation, so the ideal circular orbital speed does not depend on the satellite’s own mass.
NASA explains the same basic idea: without gravity, an orbiting object would continue along a straight path, while gravity continuously pulls it toward Earth.
For further space-science information, see NASA’s official website.
Common Circular Motion Mistakes
Mistake 1: Thinking Centripetal Force Is a New Force
Centripetal force is not automatically an additional force that should be added to a free-body diagram.
Instead, identify the actual forces and determine which combination produces the required inward net force.
Mistake 2: Thinking Constant Speed Means Zero Acceleration
Constant speed does not mean constant velocity.
In uniform circular motion, velocity changes direction continuously, so acceleration is not zero.
Mistake 3: Using (v^2/r) Without Checking the Radius
The radius must be measured from the center of the circular path to the object.
Mistake 4: Confusing Centripetal and Centrifugal Force
In an inertial reference frame, the required real net force points inward. Centrifugal force is an apparent or fictitious outward force used when analyzing motion from a rotating reference frame.
Mistake 5: Assuming Every Circular Motion Problem Has Constant Speed
Vertical loops and other non-uniform circular motion problems can involve changing speed.
Frequently Asked Questions
What is circular motion?
Circular motion is motion in which an object follows a circular path around a center. The object’s velocity changes direction continuously, producing centripetal acceleration.
What is the formula for circular motion?
The most common circular motion formula for centripetal acceleration is (a_c = v^2/r). If you need the required centripetal force, use (F_c = mv^2/r).
What are the main circular motion formulas?
The main formulas are (a_c=v^2/r), (a_c=\omega^2r), (F_c=mv^2/r), (F_c=m\omega^2r), (v=\omega r), and (\omega=2\pi/T=2\pi f).
What is the uniform circular motion formula?
For uniform circular motion, the main acceleration formula is (a_c=v^2/r), and the corresponding centripetal force is (F_c=mv^2/r).
What are some examples of circular motion?
Examples of circular motion include a car turning around a roundabout, a ball moving on a string, a roller coaster moving through a loop, a satellite orbiting Earth, and a point on a rotating wheel.
Why does an object accelerate if its speed is constant?
An object accelerates whenever its velocity changes. In uniform circular motion, speed remains constant but the direction of velocity continuously changes, so the object has centripetal acceleration.
What provides centripetal force?
The actual force providing centripetal force depends on the physical situation. It may be tension, friction, gravity, normal force, or a combination of forces.
Does centripetal force do work?
Centripetal force does zero work in uniform circular motion because it is perpendicular to the instantaneous displacement. A tangential force in non-uniform circular motion can do work and change the object’s speed.
Related Physics Guides
For a broader understanding of the concepts used in circular motion, explore:
- Physics Formulas & Equations Library — quick reference for mechanics and core physics equations.
- Newton’s Laws of Motion Guide — understand (F=ma), force diagrams, and the laws behind centripetal force.
- SUVAT Equations Guide — review constant-acceleration kinematics.
- Projectile Motion Guide — study two-dimensional motion and velocity components.
- Kinetic Energy vs Potential Energy Guide — useful for vertical-loop and energy problems.
For an interactive way to explore forces and acceleration, use the PhET Forces and Motion simulation. The simulation allows students to experiment with force, friction, speed, and acceleration.
Conclusion
Circular motion is one of the most important applications of Newton’s Laws of Motion because it connects velocity, acceleration, force, and rotational quantities in a single physical model.
The most important formulas to remember are:
[
\boxed{a_c = \frac{v^2}{r}}
]
[
\boxed{F_c = \frac{mv^2}{r}}
]
[
\boxed{v = \omega r}
]
[
\boxed{\omega = \frac{2\pi}{T} = 2\pi f}
]
and:
[
\boxed{f = \frac{1}{T}}
]
When solving circular motion problems, first identify the radius, speed, angular velocity, period, frequency, and forces involved. Then choose the formula that matches the information given.
Understanding these relationships makes it easier to solve circular motion examples involving cars, rotating objects, roller coasters, banked curves, and satellites.