Centripetal Force Calculator

Free Online Centripetal Force Calculator with Steps

This centripetal force calculator instantly finds the inward force needed to keep any object moving along a circular path — a car rounding a curve, a ball on a string, a rider on an amusement park loop, or a satellite in orbit. Enter the object's mass, the radius of the circular path, and whatever you already know about its circular speed (linear velocity, angular velocity, period, or RPM), and the calculator instantly computes the centripetal force, centripetal acceleration, and every other circular-motion quantity — backed by a full step-by-step solution and a labeled diagram showing the radius, velocity, and force vectors with their real values marked directly on it.

Whether you're a physics student solving a circular motion problem for homework, an engineer checking a quick design number, or just curious how much force keeps a car from sliding off a curve, this tool accepts whichever speed value you have on hand and always shows exactly how the final answer was calculated — not just the number itself.

What Is Centripetal Force?

Centripetal force is the net force that acts on any object moving along a circular path, always directed inward, toward the center of the circle. It isn't a new or separate type of force — it's simply the name given to whatever combination of real forces (tension in a string, friction between tires and road, gravity holding a satellite in orbit, the normal force from a track) happens to be pointing toward the center and causing the circular motion. Without this inward force, an object would simply continue moving in a straight line, exactly as Newton's first law describes — circular motion only happens because something is constantly pulling the object off that straight path and toward the center.

Centripetal Force Formula

The centripetal force can be written in a few equivalent forms, depending on which circular-motion quantities are known:

• In terms of linear velocity: Fc = m × v² / r
• In terms of angular velocity: Fc = m × ω² × r
• Centripetal acceleration: ac = v² / r = ω² × r
• Relationship between linear and angular velocity: v = ω × r
• Angular velocity from period: ω = 2π / T
• Angular velocity from RPM: ω = rpm × 2π / 60

How to Calculate Centripetal Force — Step by Step

Every centripetal force problem, however the circular speed is given, can be solved using the same general approach:

• Step 1 — Identify the mass (m) and radius (r) of the circular path, converting both to kilograms and meters.
• Step 2 — Identify which speed value you know: linear velocity (v), angular velocity (ω), period (T), or rotational speed in RPM.
• Step 3 — Convert whatever you know into linear velocity using v = ω × r, where ω = 2π / T (from period) or ω = rpm × 2π / 60 (from RPM).
• Step 4 — Calculate the centripetal acceleration using ac = v² / r.
• Step 5 — Calculate the centripetal force using Newton's second law: Fc = m × ac.
• Step 6 — Convert the final force into your preferred unit (newtons, kilonewtons, pounds-force, or dynes).

Worked Example — Calculating Centripetal Force

Let's calculate the centripetal force on a 1,200 kg car rounding a curve of radius 50 m at 20 m/s.

• Step 1: m = 1,200 kg, r = 50 m, v = 20 m/s (already given directly).
• Step 2: Centripetal acceleration = ac = v² / r = 20² / 50 = 8 m/s².
• Step 3: Centripetal force = Fc = m × ac = 1,200 × 8 = 9,600 N.
• So the road must provide 9,600 N of inward friction force on the tires to keep the car on this curve at 20 m/s — if the available friction is less than that, the car will start to slide outward.

Understanding the Circular Motion Diagram

The diagram on this page draws the circular path as a dashed circle, with the object sitting on the path and three key elements labeled directly with your real numbers: the blue radius line running from the center to the object, the green tangential velocity arrow showing the direction and magnitude of the object's actual motion (always perpendicular to the radius), and the red centripetal force arrow showing the net inward force that bends that straight-line motion into a circle. The purple arc near the center marks the angular velocity, and the strip along the bottom shows the period, rotational speed, and centripetal acceleration — every quantity you need, visible in one place.

Why Velocity Is Tangent But Force Points Inward

One of the most common points of confusion in circular motion is that the object's velocity and the centripetal force point in completely different directions — velocity is always tangent to the circle (perpendicular to the radius), while the centripetal force is always directed straight inward, toward the center. This is exactly why the object doesn't fly off in a straight line: the inward force continuously changes the direction of the velocity vector (without necessarily changing its magnitude), curving the straight-line path into a circle. If that inward force suddenly disappeared, the object wouldn't fly outward along the radius — it would fly off tangentially, in whatever direction it was moving at that instant.

Linear Velocity, Angular Velocity, Period, and RPM Explained

These four quantities all describe the same circular motion from different angles, and this calculator can start from any one of them. Linear (tangential) velocity is the actual speed the object moves along the circular path, measured in distance per time (like m/s). Angular velocity is how fast the angle swept out changes, measured in radians per second — it's related to linear velocity by v = ω × r, so a point farther from the center moves faster in a straight-line sense even at the same angular velocity. Period is simply the time it takes to complete one full revolution, and RPM (revolutions per minute) is a everyday, practical way of expressing the same rotational speed, common on engines, motors, and machinery specifications.

Centripetal Force vs. Centrifugal Force

Centripetal force is a real, physical force pointing inward, and it's the actual force responsible for circular motion. Centrifugal force, by contrast, is what's called a fictitious or pseudo-force — it's the outward push you feel in a turning car or a spinning ride, but it only appears because you're observing motion from inside a rotating (non-inertial) reference frame. From an outside, stationary observer's point of view, there's no outward force at all — only the real, inward centripetal force acting on the object, and the object's own inertia trying to keep it moving in a straight line. Understanding this distinction is one of the most common conceptual hurdles in introductory circular motion physics.

Real-World Applications of Centripetal Force

Centripetal force calculations show up constantly in engineering, transportation, and everyday physics:

• Road and highway design — engineers use this formula to set safe speed limits and bank angles for curves, based on available tire friction.
• Amusement park rides — loops, spinning rides, and centrifuges are all designed around the centripetal force needed to keep riders safely on their circular path.
• Satellites and orbital mechanics — gravity itself supplies the centripetal force that keeps satellites and the Moon in orbit around Earth.
• Centrifuges in labs and industry — spinning samples at high angular velocity generates large centripetal force, used to separate particles by density.
• Washing machine spin cycles — the drum's centripetal force keeps water and clothes moving in a circle while excess water is forced outward through the drum's holes.
• Bicycle and motorcycle cornering — riders lean into turns to help their tires supply enough centripetal force to hold the curve without sliding.

Common Mistakes When Calculating Centripetal Force

A frequent mistake is confusing angular velocity (in radians per second) with RPM (revolutions per minute) and plugging one directly into a formula meant for the other — always convert to a common unit first. Another common error is forgetting to square the velocity in Fc = m × v² / r, which dramatically understates the required force since centripetal force grows with the square of speed, not linearly. It's also easy to mix up radius and diameter — the formula always requires the radius (distance from center to the object), not the full diameter across the circle. Finally, remember that centripetal force isn't a separate, additional force acting alongside gravity, friction, or tension — it's simply the name for whichever of those real forces happens to be supplying the net inward pull.

Why Use This Centripetal Force Calculator?

This tool accepts whichever circular-speed value you already have — velocity, angular velocity, period, or RPM — and instantly computes the centripetal force along with every related quantity, backed by a full step-by-step solution and a clearly labeled circular-motion diagram. Whether you're solving a textbook problem, checking a quick engineering estimate, or just trying to understand why velocity doubling quadruples the required force, this calculator gives you a clear, accurate answer in seconds.