Centripetal Force Calculator finds the inward force on a mass in circular motion from speed, angular velocity, RPM or period and adds acceleration, a friction limit and bank angle.
Pick the velocity unit to enter speed, angular velocity, RPM or period.
Friction, Gravity, Units
Used by the friction and vertical circle cards. Clear a field to skip its card.
Step-by-Step Calculation
- Mass in kilograms: m = 1,000 kg
- Radius in meters: r = 50 m
- Tangential velocity: v = 15 m/s
- Formula: F = m × v² ÷ r
- F = 1,000 × 15² ÷ 50 = 4,500 N
Force at Other Speeds
| Speed (m/s) | Force (N) | g-force |
|---|---|---|
| 7.5 | 1,125 | 0.1147 |
| 11.25 | 2,531 | 0.2581 |
| 15 | 4,500 | 0.4589 |
| 18.75 | 7,031 | 0.717 |
| 22.5 | 10,125 | 1.032 |
| 30 | 18,000 | 1.835 |
Force at Other Radii
| Radius (m) | Force (N) | g-force |
|---|---|---|
| 25 | 9,000 | 0.9177 |
| 37.5 | 6,000 | 0.6118 |
| 50 | 4,500 | 0.4589 |
| 62.5 | 3,600 | 0.3671 |
| 75 | 3,000 | 0.3059 |
| 100 | 2,250 | 0.2294 |
Force Diagram
F = 4,500 N points to the center, v = 15 m/s runs along the tangent and r = 50 m. The dashed arrow F cf is the centrifugal force, which appears only in the rotating frame.
Centripetal Force Calculator for Physics Problems
The Centripetal Force Calculator finds the net inward force that keeps a mass on a circular path. With the default entries, a 1,000 kg car on a 50 m curve at 15 m/s needs 4,500 N toward the center. The same line also gives that force as 1,012 lbf and 458.9 kgf.
Solve For has five targets. Besides the force, it returns the centripetal acceleration, the mass, the radius or the velocity when the other values are known. That covers the usual physics homework layouts, where any one of the four quantities in the formula can be the unknown.
The velocity box takes more than a speed. Its unit list switches the entry between tangential velocity in m/s, km/h, ft/s or mph, angular velocity in rad/s, RPM or Hz, and a period in milliseconds, seconds, minutes, hours or days. The label on the box changes to match the choice.
Centripetal Force Formula
Centripetal force is mass times centripetal acceleration. OpenStax University Physics defines it as any net force causing uniform circular motion, and writes it with the tangential speed as follows.
$$F = \frac{m v^2}{r}$$
Here m is the mass in kilograms, v is the speed along the path in meters per second and r is the radius in meters. For the default car, the working is 1,000 × 15² ÷ 50, which is 1,000 × 225 ÷ 50 and comes to 4,500 N.
Speed is squared, so it matters most. Doubling the speed to 30 m/s on the same curve raises the force four times, to 18,000 N. Doubling the radius to 100 m halves it to 2,250 N. The Force at Other Speeds and Force at Other Radii tables in the Centripetal Force Calculator list these steps for your own entries.
Mass scales the force in a straight line. A second worked case is a 1,000 kg car at 20 m/s on the same 50 m curve, which needs 1,000 × 400 ÷ 50, or 8,000 N. That is 78% more force for a speed only 33% higher.
Formula With Angular Velocity and RPM
Many problems give a rotation rate and no speed. The centripetal force formula with angular velocity replaces v with ω times r, which leaves the angular velocity squared times the radius.
$$F = m \omega^2 r$$
Here ω is in radians per second. Notice that the radius now multiplies the force. At a fixed rotation rate, a point farther from the axis moves faster, so it needs more force. At a fixed speed, a larger radius means a gentler curve and less force.
To use the tool as a centripetal force calculator for RPM, pick Angular: RPM in the velocity unit list. One RPM is 0.10472 rad/s. A 0.5 kg mass on a 0.8 m string at 120 RPM turns at 12.57 rad/s, so the force is 0.5 × 12.57² × 0.8, or 63.17 N.
The RPM can also go straight into one line without converting to rad/s first. The factor 0.010966 is the square of 0.10472, m is in kilograms, r is in meters and N is the rotation rate in RPM.
$$F = 0.010966 \times m \times r \times N^2$$
The second result card then restates the motion every other way. For the 120 RPM example it shows a tangential velocity of 10.05 m/s and a period of 0.5 s. The Angular Velocity Calculator and the Angular Frequency Calculator convert between these rates on their own.
Centripetal Force Without Velocity
When a problem gives only the time for one lap, the speed is the circumference divided by the period. Putting that into the main formula gives the way to calculate centripetal force without velocity.
$$F = \frac{4 \pi^2 m r}{T^2}$$
Here T is the period in seconds. Choose Period: seconds in the velocity unit list and enter the lap time. The list also has milliseconds, minutes, hours and days, and the tool converts each one to seconds first. A 30 kg child sitting 2 m from the center of a merry-go-round that turns once every 4 s needs 39.48 × 30 × 2 ÷ 16, which is 148.0 N.
The table lists the form of the formula for each thing a problem might state. The Centripetal Force Calculator picks the matching row from the velocity unit, and the step-by-step working under the results prints the form it used.
| Known quantity | Formula for F | Formula for a |
|---|---|---|
| Tangential speed v (m/s) | m v² ÷ r | v² ÷ r |
| Angular velocity ω (rad/s) | m ω² r | ω² r |
| Rotation rate N (RPM) | 0.010966 m r N² | 0.010966 r N² |
| Frequency f (Hz) | 39.48 m r f² | 39.48 r f² |
| Period T (s) | 39.48 m r ÷ T² | 39.48 r ÷ T² |
The number 39.48 is 4π². A period works for orbits as well as playground rides. The Moon has a mass of 7.346 × 10²² kg, an orbit radius of 384,400 km and a period of 27.3217 days on NASA’s Moon fact sheet.
Enter the period as 27.3217 days and the radius as 384,400 km. The steps show the period as 2,360,590 s, and the result is 2.0 × 10²⁰ N, with a speed of 1,023 m/s that matches NASA’s mean orbital velocity of 1.022 km/s. Earth’s gravity supplies that force, and the Gravitational Force Calculator returns 1.98 × 10²⁰ N from the two masses and the distance, which agrees to within 1%.
Centripetal Force Unit
The centripetal force unit is the newton, the same as for any other force. One newton is one kilogram meter per second squared, which is exactly what m v² ÷ r produces when the mass is in kg, the speed in m/s and the radius in m.
The Centripetal Force Calculator converts other units to those three before it works. Mass can be entered in grams, metric tons, pounds or ounces, and radius in centimeters, kilometers, feet, inches or miles. The reverse modes accept the force in newtons, kilonewtons, pounds-force or kilograms-force.
| Unit | Equal to | The 4,500 N default |
|---|---|---|
| Newton (N) | 1 kg·m/s² | 4,500 N |
| Kilonewton (kN) | 1,000 N | 4.5 kN |
| Pound-force (lbf) | 4.448 N | 1,012 lbf |
| Kilogram-force (kgf) | 9.80665 N | 458.9 kgf |
Results are shown in SI units unless you change them. Open Friction, Gravity, Units and set Result Units to US, and the main answer, the four cards and both tables switch to lbf, lb, ft, ft/s and mph. The default car then reads 1,012 lbf and 14.76 ft/s², and it slides above 60.78 ft/s, which is 41.44 mph.
The Gravity box switches with it, from 9.80665 m/s² to 32.174 ft/s². The step-by-step working stays in SI, since that is how the formula is taught, and ends with one extra line that gives the answer in US units.
A pound entered as mass is a pound-mass, not a pound-force. The calculator turns each pound into 0.4536 kg first, so a 2,204.6 lb car gives the same 4,500 N as the 1,000 kg default. Mixing the two pounds is the usual reason a hand answer in US units comes out about 32 times too large or too small.
Centripetal Acceleration and g-Force
Set Solve For to Centripetal Acceleration and the tool works as a centripetal acceleration calculator. The acceleration points to the center of the circle and depends only on the speed and the radius.
$$a = \frac{v^2}{r} = \omega^2 r$$
The default car has 15² ÷ 50, or 4.5 m/s². The first card also gives that as 0.4589 g and 14.76 ft/s². Mass does not appear in the formula, so a truck and a bicycle on the same curve at the same speed share the same acceleration. Clear the mass box in this mode and only the acceleration is shown.
The g figure is the acceleration divided by standard gravity of 9.80665 m/s². It is a quick read on how hard a turn feels. The merry-go-round example comes to 4.935 m/s², which is 0.503 g, about half the rider’s weight pushing sideways.
Spinning lab equipment reaches thousands of g, and it is rated by that number instead of by force. The Centrifuge Calculator handles that case from rotor radius and RPM.
Solving for Velocity, Mass or Radius
The three reverse modes of the Centripetal Force Calculator rearrange the same formula. Each one takes the force as an input, and the default force in those modes is 8,000 N. The centripetal velocity formula gives the speed at which a circle needs exactly that force.
$$v = \sqrt{\frac{F r}{m}}$$
With 8,000 N, a 50 m radius and 1,000 kg, the speed is the square root of 400, or 20 m/s. The result line adds 72 km/h and 44.74 mph. If only the acceleration is known, the same idea gives v as the square root of a times r.
$$m = \frac{F r}{v^2}$$
The mass mode answers how much mass a given inward force can hold on the path. For 8,000 N on a 50 m circle at 15 m/s the answer is 1,778 kg, or 3,919 lb. Anything heavier at that speed needs more force than is available.
$$r = \frac{m v^2}{F}$$
The radius mode finds the tightest circle a force can hold. For 1,000 kg at 15 m/s with 8,000 N the radius is 28.13 m, or 92.27 ft. Both the mass and the radius modes also accept an angular velocity, RPM or period in place of the speed.
What Supplies the Centripetal Force
Centripetal force is not a separate kind of force. It is the name for whatever net force points to the center, and that can be friction, tension, gravity or a normal force. The last two result cards of the Centripetal Force Calculator test the calculated force against the two most common classroom cases.
| Motion | Force that points inward |
|---|---|
| Car on a flat curve | Static friction between tires and road |
| Car on a banked curve | Horizontal part of the normal force |
| Ball swung on a string | Tension in the string |
| Moon or satellite in orbit | Gravity |
| Roller coaster loop | Normal force from the track plus gravity |
Friction on a flat curve
On a level road, friction is the only sideways force. The most it can give is the friction coefficient times the weight, so the top speed on a flat curve does not depend on the mass.
$$v_{max} = \sqrt{\mu g r}$$
Open Friction, Gravity, Units to set μ and g. With μ of 0.7 the default car uses 65.6% of the friction limit, and it would slide above 18.53 m/s, which is 66.7 km/h. The card turns red when the entered speed is past that limit.
The g-force on the first card doubles as the smallest friction coefficient that holds the turn. OpenStax works a 900 kg car at 25 m/s on a 500 m curve. The Centripetal Force Calculator gives 1,125 N and 0.1275 g, which matches the textbook’s minimum coefficient of 0.13. The Friction Calculator covers the friction force itself.
Bank angle with no friction
A banked surface tilts the normal force so part of it points inward. At one ideal angle that part supplies the whole centripetal force, and the tires need no grip at all.
$$\tan \theta = \frac{v^2}{r g}$$
The default car needs a 24.65° bank. OpenStax gives 24.3 m/s as the ideal speed for a 100 m curve banked at 31.0°, and entering that speed and radius returns 31.05°. The Angle of Banking Calculator goes further and adds friction to a banked curve.
Vertical circle on a string or track
In a vertical circle, gravity helps at the top and works against the string at the bottom. The pull at the bottom is the centripetal force plus the weight. At the top it is the centripetal force minus the weight.
A 0.5 kg ball on a 1 m string at 5 m/s needs 12.5 N toward the center, and it weighs 4.903 N. The string pulls with 17.40 N at the bottom and 7.597 N at the top. The Tension Calculator handles strings in other setups.
$$v_{top} = \sqrt{g r}$$
Below that top speed the string goes slack and the ball leaves the circle. For the 1 m string it is 3.132 m/s. A ball swinging freely slows as it climbs, so it must pass the bottom at 7.002 m/s, which is the square root of 5 g r. The default car reads Slack at the top, because 15 m/s is under the 22.14 m/s a 50 m loop needs.
Centripetal Force vs Centrifugal Force
Centripetal force points inward and is a real net force on the moving object. It is what a force diagram drawn from the ground shows. Without it, the object would carry on in a straight line along the tangent, as Newton’s first law says.
Centrifugal force points outward and has the same size, m v² ÷ r. It only appears when the motion is described from inside the turning frame, such as by a passenger in the car. OpenStax calls it a fiction for that reason, since no object is pushing the passenger outward.
The force diagram in the Centripetal Force Calculator draws both. The solid arrow F points to the center, the velocity arrow runs along the tangent, and a dashed arrow marks the centrifugal force. For the outward view of the same numbers, the Centrifugal Force Calculator reports the force a rider or a rotating part feels.
A centripetal force that stays perpendicular to the velocity changes direction and not speed. It does no work, so the kinetic energy and the angular momentum of the object stay constant in uniform circular motion. The other tools for spinning systems are listed on the Rotational Motion Calculators hub.