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Walk-through

How to Use This Calculator

3 steps

1

Enter mass, velocity, and radius

Type the object's mass (kg), its speed along the circular path (m/s), and the radius of that circle (m) into the input card. The calculator updates instantly as you type — no button to press.

2

Pick Force or Acceleration

Use the Force / Acceleration tabs to switch which value is emphasized. Both read from the same three inputs — Force shows F = mv²/r as the headline number, Acceleration shows a = v²/r instead.

3

Or switch to Solve

Need mass, velocity, or radius instead of force? Click the Solve tab, choose the variable you're solving for from the dropdown, and enter the other three values. The chosen variable becomes the computed output.

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Reference

Formula & Methodology

2 formulas

Centripetal force

F = m·v² / r

F is the net inward force (newtons, N) required to keep an object of mass m (kg) moving at constant speed v (m/s) along a circular path of radius r (m). This is the same F in Newton's second law — it's not a separate kind of force, just the name for whatever force (tension, gravity, friction, normal force) happens to be pointing toward the center.

Centripetal acceleration

a = v² / r

a is the acceleration directed toward the center of the circle (m/s²), caused by the constant change in the direction of velocity even when speed itself is constant. Multiplying by mass (F = ma) recovers the force formula above.

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Glossary

Key Terms Explained

6 terms

Centripetal force The net force directed toward the center of a circular path that keeps an object moving in that circle instead of flying off in a straight line. Measured in newtons (N); F = mv²/r.

Centripetal acceleration The rate of change of velocity's direction (not necessarily its speed) for an object in circular motion, always pointed toward the center. Measured in m/s²; a = v²/r.

Circular motion Movement of an object along the circumference of a circle or circular arc. 'Uniform' circular motion means the speed is constant, even though velocity (a vector) is constantly changing direction.

Radius The distance from the center of the circular path to the object, in meters. A tighter turn (smaller radius) requires more centripetal force at the same speed, since force scales as 1/r.

Tangential velocity The object's linear speed at any instant, directed tangent to the circle (perpendicular to the radius). This is the 'v' in F = mv²/r — it's the everyday speed, not an angular rate.

Centrifugal force The outward-feeling push you notice in a turning car or spinning ride. It isn't a real force acting on the object — it's your body's inertia resisting the inward centripetal force, felt as an apparent (fictitious) force only inside the rotating frame of reference.

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Scenarios

Real-World Examples

3 worked examples

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Physics student

Baseline: 1 kg at 10 m/s on a 5 m radius

Mass 1 kg Velocity 10 m/s Radius 5 m

F = (1 × 10²) / 5 = 20 N. Acceleration a = 10² / 5 = 20 m/s². This is the calculator's default — a clean reference case worth memorizing: 1 kg, 10 m/s, 5 m radius always gives 20 N.

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Driver rounding a curve

Doubling speed on the same curve

Mass 1 kg Velocity 20 m/s Radius 5 m

F = (1 × 20²) / 5 = 80 N — four times the force from the baseline, even though velocity only doubled. Because force scales with v², doubling speed quadruples the grip (friction) a tire needs to hold a turn.

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Ride engineer

Solving for radius given a force limit

Force 20 N Mass 1 kg Velocity 10 m/s

Switching to the Solve tab with 'Radius' selected: r = mv²/F = (1 × 10²) / 20 = 5 m. Given a maximum allowable force, this rearrangement tells you the minimum safe turning radius at a given speed and mass.

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Deep Dive

What keeps things moving in a circle

3 chapters · 4 min read

Nothing pushes an object outward as it circles — it's the opposite. Centripetal force is the inward pull (tension, gravity, friction, or a normal force) that constantly redirects a moving object toward the center, and without it the object would simply travel in a straight line. This calculator turns the textbook formula F = mv²/r into an instant, editable tool for finding force, acceleration, or any of the three variables that produce them.

Why circular motion needs a constant inward force

Newton's first law says an object in motion stays in motion in a straight line unless acted on by a net force. A car rounding a curve, a ball on a string, and a satellite orbiting Earth are all constantly being pulled off that straight-line path — toward the center of the circle they're tracing. That inward pull is the centripetal force, and it has to be continuously supplied: friction between tires and road, tension in the string, or gravity between planet and satellite. Remove the force (cut the string, hit an icy patch) and the object flies off tangent to the circle, not radially outward.

Reading F = mv²/r

The force formula packs three intuitive relationships into one equation. Force is directly proportional to mass — a heavier object needs proportionally more force to hold the same circular path. Force is directly proportional to the square of velocity — doubling speed quadruples the force required, which is why a car that easily handles a curve at 30 mph can lose grip at 60 mph on the exact same curve. And force is inversely proportional to radius — tighter turns (smaller r) demand more force at a given speed, which is why highway on-ramps use wide, gradual curves rather than tight ones.

Centripetal force vs. the 'centrifugal' feeling

Sitting in a car that's turning hard, you feel pushed outward against the door — which seems to contradict an inward force. What's actually happening is your body's inertia: it wants to keep traveling in a straight line while the car (and the seat, and the door) accelerates you toward the center. The door pushes back on you, which is the real, inward centripetal force you feel as pressure. The 'outward' sensation is your inertia resisting that redirection, not a separate outward force — physicists call it a fictitious or apparent force because it only shows up when you analyze motion from inside the rotating frame.

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Questions

Frequently Asked Questions

6 questions

What is the formula for centripetal force?+

F = mv²/r, where F is the centripetal force in newtons, m is the object's mass in kilograms, v is its tangential (linear) speed in meters per second, and r is the radius of the circular path in meters. The related centripetal acceleration is a = v²/r, and F = ma ties the two together.

Why does centripetal force point inward instead of outward?+

An object moving at constant speed in a circle is still accelerating, because its velocity's direction keeps changing. Acceleration always points in the direction velocity is changing toward — which, for circular motion, is toward the center. By Newton's second law (F = ma), the force causing that acceleration must also point inward, toward the center of the circle.

Is centrifugal force real?+

Not as a force acting on the object from the outside — it's what physicists call a fictitious (or apparent) force. It's the sensation of your own inertia resisting the inward centripetal force, felt only by an observer inside the rotating system (like a passenger in a turning car). From an outside, non-rotating frame, only the real inward centripetal force exists.

What units does this calculator use?+

Mass in kilograms (kg), velocity in meters per second (m/s), radius in meters (m), force in newtons (N), and acceleration in meters per second squared (m/s²). Convert other units (pounds, mph, feet) to these before entering them for an accurate result.

Why does a tighter turn need more force?+

Force is inversely proportional to radius in F = mv²/r — halving the radius doubles the required force at the same mass and speed. Physically, a tighter circle means velocity's direction changes more rapidly over the same distance traveled, which demands more acceleration, and therefore more force, to achieve.

How is centripetal acceleration different from centripetal force?+

Acceleration (a = v²/r) describes how quickly velocity's direction is changing, independent of what's causing it. Force (F = mv²/r = ma) is that acceleration multiplied by the object's mass — it's the actual push or pull (tension, friction, gravity) needed to produce the acceleration. Two objects with different masses moving identically have the same acceleration but need different amounts of force.

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Centripetal Force Calculator

Circular Motion Inputs

Result

Force vs. Velocity

How to Use This Calculator

Enter mass, velocity, and radius

Pick Force or Acceleration

Or switch to Solve

Formula & Methodology

Key Terms Explained

Real-World Examples

Physics student

Driver rounding a curve

Ride engineer

What keeps things moving in a circle

Why circular motion needs a constant inward force

Reading F = mv²/r

Centripetal force vs. the 'centrifugal' feeling

Frequently Asked Questions

Get a branded PDF of your results

Centripetal Force Calculator

Circular Motion Inputs

Result

Force vs. Velocity

How to Use This Calculator

Enter mass, velocity, and radius

Pick Force or Acceleration

Or switch to Solve

Formula & Methodology

Key Terms Explained

Real-World Examples

What keeps things moving in a circle

Why circular motion needs a constant inward force

Reading F = mv²/r

Centripetal force vs. the 'centrifugal' feeling

Frequently Asked Questions

Get a branded PDF of your results

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