Centripetal Force Calculator
Calculate centripetal force for circular motion from mass, radius, and velocity, angular velocity, RPM, or period.
Calculator is for informational purposes only. Terms and Conditions
Choose what to solve for
Select the circular motion method and the unknown variable.
Enter the known values
All inputs are converted to SI units before the equation is solved.
Visual Check
Velocity is tangent to the path, while centripetal force points inward.
Solution
Live result, quick checks, warnings, and full solution steps.
Quick checks
- Centripetal acceleration—
Show solution steps See conversions, equation setup, substitution, assumptions, and checks
- Enter values to see the full calculation steps and checks.
Source, Standards, and Assumptions
Calculation basis, constants, assumptions, and limitations.
This calculator uses standard uniform circular motion equations and SI unit conversion constants.
- Assumptions will appear after a valid calculation.
On this page
Calculator Guide
How to Use the Centripetal Force Calculator
The Centripetal Force Calculator above finds the inward force needed to keep an object moving in a circular path. Enter mass, radius, and the motion value you know, such as tangential velocity, angular velocity, RPM, or period, then review the force, acceleration, g-force, and step-by-step math.
Centripetal force is a center-seeking net force. It is not usually a separate force by itself; it is the inward result of tension, friction, gravity, a normal force, or another physical interaction that keeps the object following a curve.
Quick Answer
To calculate centripetal force, use \(F_c = mv^2/r\). Convert mass to kilograms, velocity to meters per second, and radius to meters before substituting. If you know RPM instead of velocity, convert it to angular velocity with \(\omega = 2\pi RPM/60\), then use \(F_c = mr\omega^2\).
When not to rely on the simplified result
Do not use this simplified circular motion result as the only basis for high-speed machinery, rotating equipment containment, ride safety, vehicle design, structural design, or any application where fatigue, balance, material strength, code requirements, or manufacturer limits control the final decision.
Inputs and Outputs Used by the Calculator
The calculator uses the known circular motion values to solve for the unknown. The most common use is solving for centripetal force from mass, velocity, and radius, but the same relationships can also solve for mass, radius, velocity, angular velocity, RPM, or period.
| Value | What It Means | Common Units | How It Affects the Result |
|---|---|---|---|
| Mass | The amount of matter being constrained to move in a circular path. | kg, g, lbm, slug | Force increases directly with mass. |
| Radius | The distance from the center of rotation to the moving object. | m, cm, ft, in | For fixed tangential speed, a larger radius lowers the force. |
| Velocity | The tangential speed along the circular path. | m/s, ft/s, mph, km/h | Force changes with velocity squared, so this input dominates many results. |
| Angular velocity or RPM | The rotational speed of the object or rotating system. | rad/s, deg/s, rev/s, RPM | Force increases with the square of angular velocity. |
| Period | The time required for one complete revolution. | s, ms, min | A shorter period means faster rotation and higher force. |
| Force and acceleration | The inward force and inward acceleration required for circular motion. | N, lbf, m/s², g | These outputs help interpret the physical demand of the motion. |
Centripetal Force Formula
The main centripetal force formula is based on Newton’s Second Law and centripetal acceleration. For tangential velocity, use mass times velocity squared divided by radius.
Main Formula Using Velocity
This is the best formula when you know the object mass, tangential speed, and radius of the circular path.
Using Angular Velocity
This form is often better for rotating machinery, wheels, centrifuge-style problems, and RPM-based calculations.
Using Period
This form is useful when you know how long one complete revolution takes.
What the Variables Mean
Each variable describes either the object, the size of the circular path, or the speed of rotation. The formula only works correctly when the units match the equation being used.
\(F_c\): Centripetal Force
The inward net force required to keep the object moving in a circular path. In SI units, it is measured in newtons, where \(1\,N=1\,kg\cdot m/s^2\).
\(m\): Mass
The mass of the object in circular motion. Use kilograms for SI calculations. If using pounds-mass, the calculator converts internally before solving.
\(v\): Tangential Velocity
The speed along the circular path, not the speed toward the center. Velocity has a squared effect, so small speed changes can create large force changes.
\(r\): Radius
The distance from the center of rotation to the object. Do not enter diameter unless the calculator specifically asks for diameter.
\(\omega\): Angular Velocity
Rotational speed in radians per second. RPM must be converted before using \(F_c=mr\omega^2\).
\(T\): Period
The time for one complete revolution. Shorter periods mean faster rotation and higher centripetal force.
Mass vs weight in U.S. customary units
Use mass for \(m\), not weight. Pounds-mass, written as lbm, measure mass. Pounds-force, written as lbf, measure force. If you only know weight in lbf, it must be converted to an equivalent mass before using \(F_c=mv^2/r\) manually.
How to Use the Calculator
Use the calculator by choosing the input method that matches what you know. Then enter the known values, check the unit selectors, and compare the result with the formulas and sanity checks below.
| What You Know | Best Method | Typical Result |
|---|---|---|
| Mass, tangential velocity, and radius | Velocity and radius | Centripetal force or another rearranged variable |
| Mass, RPM, and radius | RPM and radius | Force, g-force, and angular velocity checks |
| Mass, angular velocity, and radius | Angular velocity and radius | Force using \(F_c=mr\omega^2\) |
| Mass, period, and radius | Period and radius | Force from time per revolution |
| Force, mass, and radius | Solve for selected motion input | Velocity, angular velocity, RPM, or period |
Select the motion method
Choose velocity and radius when you know linear speed, RPM and radius when you know rotational speed, angular velocity and radius when you know \(\omega\), or period and radius when you know time per revolution.
Choose the unknown
Most users solve for centripetal force, but the calculator can also help rearrange the relationship to solve for mass, radius, or a motion value.
Enter values and units
Use the exact unit selectors beside each input. This is especially important for mph, ft/s, inches, feet, RPM, and radians per second.
Review force, acceleration, and g-force
Check the main result and the quick checks. A high g-force, high RPM, or very large force may be mathematically correct but practically unsafe or unrealistic.
How to Interpret the Result
The result is the inward net force required for the selected circular motion. A larger value means the physical system must provide more inward force through tension, friction, gravity, structural restraint, or another force source.
What to do with the force
Use it to understand the inward load demand. For example, a string must supply tension, a tire must rely on friction, and a rotating machine part must be restrained by its structure.
What changes it most?
Speed usually dominates because \(v^2\) or \(\omega^2\) appears in the formula. Doubling speed makes the force four times larger.
Sanity check
If mass doubles, force should double. If velocity doubles, force should quadruple. If radius doubles while velocity stays fixed, force should be cut in half.
What if the force seems huge?
Large centripetal forces are common at high RPM or small radius. Before assuming the result is wrong, check whether speed was entered in RPM instead of rad/s, whether radius was entered as diameter, and whether mass units were selected correctly.
Input Checklist Before You Trust the Answer
Most wrong answers come from entering the right number in the wrong unit or using a circular motion formula for a motion that is not actually uniform.
Use radius, not diameter
The radius is the distance from the center to the object. If the circular path is 10 m across, the radius is 5 m. Entering 10 m instead of 5 m would cut the calculated force in half for the same mass and speed.
Check speed type
Tangential velocity, angular velocity, RPM, and period are not interchangeable without conversion. Select the method that matches your known value.
Watch squared speed
A small speed entry error becomes much larger because velocity or angular velocity is squared.
Confirm the force source
Identify what physically supplies the inward force: friction, tension, gravity, normal force, bearing reaction, or another restraint.
Worked Example
This example follows the most common use case: calculating centripetal force from mass, tangential velocity, and radius.
Formula
Substitution
Calculation
Final answer
The required centripetal force is 40 N. The acceleration check is \(a_c=v^2/r=10^2/5=20\,m/s^2\), so \(F_c=ma_c=2(20)=40\,N\). That reverse check confirms the result.
Car-turning example
For a car with \(m=1500\,kg\), \(v=20\,m/s\), and \(r=50\,m\), the inward force needed to follow the curve is:
On a flat curve, tire-road friction must provide this inward force. If that friction is not available, the car cannot safely follow the curve. For a related surface-resistance check, use the friction calculator.
RPM example check
If a \(0.25\,kg\) object rotates at \(1200\,RPM\) with a \(0.1\,m\) radius, convert RPM first:
This shows why high rotational speed can create large force even when the moving mass is small.
What the Formula Represents
The formula represents an inward force requirement. Velocity is tangent to the circular path, radius points from the center to the object, and centripetal force points back toward the center.
The labels use white background boxes and open spacing so the arrows do not overlap the text. Velocity is tangent to the path, while centripetal force points inward toward the center.
Reference Checks and Source Notes
Centripetal force does not have one universal “good” value because it depends on the mass, speed, and radius. A better reference check is to compare the calculated acceleration with gravity, \(g\), and to ask what physical force is providing the inward load.
| Check | What It Means | Why It Helps |
|---|---|---|
| \(a_c/g\) | Centripetal acceleration divided by \(9.80665\,m/s^2\) | Shows acceleration as a multiple of standard gravity. |
| Speed sensitivity | Double \(v\), force becomes four times larger. | Helps catch unrealistic high-speed entries. |
| Radius check | For fixed \(v\), larger \(r\) lowers force. | Helps catch diameter entered as radius. |
| Force provider | Tension, friction, gravity, normal force, or structure | Connects the calculated demand to the real physical system. |
Authoritative source note
For deeper physics context, OpenStax explains that centripetal force is perpendicular to velocity and points toward the center of curvature. The g-force check uses standard gravity, \(g_n=9.80665\,m/s^2\), which is the exact standard acceleration of gravity listed by NIST.
Design Notes and Practical Ranges
For classroom problems, the formula result is usually the final numerical answer. For engineering design, the result is only an inward force estimate that must be checked against real system limits.
Rotating machinery
For shafts, disks, flywheels, wheels, and rotating arms, also check balance, fatigue, material stress, fasteners, bearings, containment, and manufacturer speed limits.
Vehicles and curves
For a flat curve, friction often supplies the inward force. Road surface, tires, banking, speed, and weather can control whether the motion is safe.
Strings and cables
For a ball-on-string style problem, tension supplies the centripetal force. Real cords also need checks for strength, knots, wear, and dynamic effects.
Orbits and gravity
For orbital motion, gravity supplies the inward force. More complete orbital calculations may require gravitational parameters and non-circular motion assumptions.
Units and Conversions
Unit consistency is critical. The calculator can convert common units, but the physics formulas are easiest to verify in SI units: kilograms, meters, seconds, radians, and newtons.
RPM to Angular Velocity
Use this before substituting into \(F_c=mr\omega^2\).
Acceleration to g-Force
This expresses centripetal acceleration as a multiple of standard gravity.
Common unit trap
Do not enter RPM directly as \(\omega\) unless the selected unit is RPM and the calculator converts it. The angular velocity formula \(F_c=mr\omega^2\) expects \(\omega\) in radians per second. Also avoid mixing lbf and N, or inches and feet, without selecting the correct unit.
Centripetal Force vs Related Concepts
Centripetal force is closely related to centripetal acceleration, angular velocity, RPM, and centrifugal force. Keeping these ideas separate helps prevent common circular motion mistakes.
Force vs acceleration
Centripetal acceleration is \(a_c=v^2/r\). Centripetal force is \(F_c=ma_c\). Mass turns acceleration demand into force demand.
Velocity vs RPM
Velocity is linear speed along the path. RPM is revolutions per minute. Convert RPM to angular velocity before using angular-speed formulas.
Centripetal vs centrifugal
Centripetal force is the inward net force. Centrifugal force is commonly treated as an apparent outward effect in a rotating reference frame.
For a related viewpoint on the outward rotating-frame terminology, use the centrifugal force calculator. For the formula derivation and more examples, see the centripetal force equation guide.
Common Mistakes
Most centripetal force errors come from wrong units, wrong radius, or confusing linear and rotational speed. The do and don’t checks below catch the most common problems before they affect the result.
Do
- Use radius from the center of rotation to the object.
- Convert RPM to radians per second when using \(F_c=mr\omega^2\).
- Use tangential velocity, not radial velocity.
- Check acceleration in \(m/s^2\) and as a multiple of \(g\).
- Identify what force physically supplies the inward load.
Don’t
- Do not use diameter where radius is required.
- Do not mix mph, feet, kilograms, and seconds without conversion.
- Do not assume centripetal force points outward.
- Do not ignore the squared effect of speed.
- Do not use the simplified formula as a full machinery safety check.
Troubleshooting Unrealistic Results
If the result looks wrong, check the inputs before changing the formula. A mathematically valid centripetal force can still be unrealistic if the radius, speed, unit, or physical assumption is wrong.
Force is too high
Check whether velocity was entered too high, RPM was entered as rad/s, diameter was used as radius, or the radius is much smaller than intended.
Force is too low
Check whether speed was entered in the wrong unit, mass was too small, or radius was entered too large.
Acceleration is extreme
Review the g-force result. Very high values can occur in high-RPM equipment, but they may require balance, fatigue, and containment checks.
Direction seems confusing
Remember that velocity is tangent to the path and centripetal force points inward. The force changes direction of motion, not the speed magnitude in ideal uniform circular motion.
U.S. units look suspicious
Check whether force was entered in lbf while the selector is set to N, or whether a radius was entered in inches while the selector is set to feet.
Assumptions and Limitations
This calculator is best used for ideal uniform circular motion. It estimates the inward net force required for a mass moving around a circular path, but it does not model every real-world effect.
Uniform circular motion
The formulas assume constant radius and steady circular motion. Nonuniform speed can add tangential acceleration and additional forces.
Point-mass approximation
The calculation treats the moving mass as if it is located at a radius. Extended bodies may require stress, inertia, and distributed-mass analysis.
No loss or deformation model
The simplified formula does not include air drag, vibration, slip, flexibility, impact, bearing losses, or material deformation.
Not final design approval
For equipment, structures, vehicles, rides, or safety-critical systems, verify the result with manufacturer data, applicable standards, and qualified professional judgment.
Key Terms
These terms help connect the calculator inputs, formulas, and result interpretation.
Centripetal Force
The inward net force required to keep an object moving in a circular path.
Centripetal Acceleration
The inward acceleration of circular motion, usually calculated as \(a_c=v^2/r\).
Tangential Velocity
The linear speed along the circular path, perpendicular to the radius at any instant.
Angular Velocity
Rotational speed, typically expressed in radians per second.
Period
The time required for one complete revolution around the circular path.
RPM
Revolutions per minute. RPM must be converted before using angular velocity formulas manually.
FAQ
What is the centripetal force formula?
The most common centripetal force formula is \(F_c=mv^2/r\), where \(m\) is mass, \(v\) is tangential velocity, and \(r\) is radius. If angular velocity is known, use \(F_c=mr\omega^2\).
Can centripetal force be calculated from RPM?
Yes. Convert RPM to angular velocity using \(\omega=2\pi\cdot RPM/60\), then use \(F_c=mr\omega^2\). The calculator above can handle this directly when the RPM method is selected.
Does centripetal force point inward or outward?
Centripetal force points inward toward the center of the circular path. Tangential velocity points along the path, perpendicular to the inward force direction.
What happens to centripetal force when velocity doubles?
Because velocity is squared in \(F_c=mv^2/r\), doubling velocity makes the centripetal force four times larger if mass and radius stay the same.
Is centripetal force the same as centrifugal force?
No. Centripetal force is the real inward net force required for circular motion. Centrifugal force is commonly described as an apparent outward effect in a rotating reference frame.
What provides centripetal force in real life?
It depends on the situation. Tension can provide it for a ball on a string, friction can provide it for a car on a flat curve, gravity can provide it for orbital motion, and structural or bearing reactions can provide it in rotating equipment.