Physics · Classical Mechanics · Dynamics & Forces
Centripetal Force Calculator
Calculates the centripetal force required to keep an object moving in a circular path given its mass, velocity, and radius.
Calculator
Formula
F_c is the centripetal force in Newtons (N), m is the mass of the object in kilograms (kg), v is the tangential speed in meters per second (m/s), and r is the radius of the circular path in meters (m). The force always points toward the center of the circular path.
Source: Newton's Second Law applied to uniform circular motion — Halliday, Resnick & Krane, Physics, 5th Ed., Chapter 6.
How it works
When an object moves along a circular path at constant speed, its velocity vector continuously changes direction — even though its magnitude stays the same. This change in direction represents acceleration, known as centripetal (or radial) acceleration, always directed toward the center of the circle. By Newton's Second Law (F = ma), a net inward force must be responsible for this acceleration. That force is the centripetal force.
The governing formula is Fc = mv²/r, where m is the object's mass in kilograms, v is its tangential speed in meters per second, and r is the radius of the circular path in meters. The force scales linearly with mass, quadratically with speed, and inversely with radius — meaning doubling the speed quadruples the required centripetal force, while doubling the radius halves it. This calculator also derives centripetal acceleration (ac = v²/r), angular velocity (ω = v/r), and the period of one full revolution (T = 2πr/v).
Practical applications span a wide range of disciplines. Civil engineers use it to design banked road curves that allow vehicles to navigate turns safely without relying entirely on friction. Aerospace engineers apply it when calculating orbital insertion forces for satellites. Mechanical engineers use it to analyze stresses on rotating machinery components such as flywheels, turbine blades, and centrifuges. Even roller-coaster designers depend on centripetal force calculations to ensure riders experience safe G-forces at loop peaks and valley transitions.
Worked example
Consider a car of mass 1,200 kg navigating a circular highway on-ramp with a radius of 50 m at a speed of 15 m/s.
Step 1 — Centripetal Force:
Fc = mv²/r = (1200 × 15²) / 50 = (1200 × 225) / 50 = 270,000 / 50 = 5,400 N
Step 2 — Centripetal Acceleration:
ac = v²/r = 225 / 50 = 4.5 m/s²
This is approximately 0.46 g — well within safe handling limits for a passenger car.
Step 3 — Angular Velocity:
ω = v/r = 15 / 50 = 0.3 rad/s
Step 4 — Period of Revolution:
T = 2πr/v = (2 × π × 50) / 15 ≈ 314.16 / 15 ≈ 20.94 s
This is the time to complete one full circular loop at that speed.
The required 5,400 N of inward force must be supplied by road friction (and banking, if present). If friction cannot supply this force, the car slides outward — illustrating why speed limits exist on curved roads.
Limitations & notes
This calculator assumes uniform circular motion — constant speed along a perfectly circular path. In reality, objects on curved paths often accelerate or decelerate (non-uniform circular motion), which introduces a tangential acceleration component not captured here. The formula also assumes a point mass; for extended rigid bodies, rotational inertia and distributed mass effects must be considered. Additionally, real-world scenarios involve friction, air resistance, gravity components, and structural compliance, none of which are modeled. At relativistic speeds (approaching the speed of light), classical centripetal force equations break down and relativistic mechanics must be applied. For banked curves, the normal force geometry must be included in the force balance, requiring a separate analysis beyond this basic formula.
Frequently asked questions
What provides centripetal force in real-world situations?
Centripetal force is not a new type of force — it is always supplied by an existing force in the system. For a car turning on a flat road, static friction between the tires and road provides it. For a satellite in orbit, gravity provides it. For a ball on a string, string tension provides it. The specific source depends entirely on the physical setup.
What is the difference between centripetal force and centrifugal force?
Centripetal force is a real inward-directed force that causes circular motion, observed from an inertial (non-rotating) reference frame. Centrifugal force is a fictitious or pseudo-force that appears to push outward when you analyze motion from within a rotating reference frame. In engineering and everyday language the two are often confused, but they are conceptually distinct — centrifugal force does not appear in inertial-frame equations of motion.
Why does centripetal force depend on the square of velocity?
Centripetal acceleration equals v²/r because both the magnitude of the velocity change and the rate at which direction changes are proportional to speed. Doubling speed means the direction changes twice as fast AND each directional change is twice as large, yielding a fourfold increase in acceleration — and therefore force. This quadratic relationship is why speed limits on curves are critically important for safety.
How do I convert RPM to tangential speed for this calculator?
First convert RPM to angular velocity in rad/s: ω = RPM × (2π / 60). Then multiply by the radius to get tangential speed: v = ω × r. For example, 300 RPM at a radius of 0.5 m gives ω = 31.42 rad/s and v = 15.71 m/s, which you can then enter directly into this calculator.
Can this calculator be used for orbital mechanics?
Yes, with care. For a circular orbit, gravity supplies the centripetal force: F = GMm/r² = mv²/r. You can use this calculator to find the required centripetal force for an orbiting satellite if you know its mass, orbital speed, and orbital radius. However, for elliptical orbits or precise mission planning, dedicated orbital mechanics tools accounting for Kepler's laws and perturbations are more appropriate.
Last updated: 2025-01-15 · Formula verified against primary sources.