Centripetal Force Calculator

The centripetal force calculator computes the inward force required to keep an object moving in a circular path at a given speed and radius, or any one of the other three variables (mass, velocity, radius) given the other three. It is used by physics and engineering students, by designers of roads, railways, and banked turns, by amusement-park engineers sizing roller-coaster loops, by pilots computing turn radius, by athletes and biomechanists analysing circular motion (hammer throw, discus, skating), and by orbital mechanics engineers computing satellite trajectories. The term "centripetal" literally means "centre-seeking", the force always points toward the centre of the circle, regardless of where the object is along its path.

How to Use the Centripetal Force Calculator

1. Pick the variable to solve for: force (F), mass (m), velocity (v), or radius (r).
2. Enter the other three values in the units shown (N, kg, m/s, m).
3. Click Calculate to see the result.
4. The result panel shows the solved value and, where relevant, related quantities such as angular velocity (ω = v/r) and centripetal acceleration (a = v²/r).
5. For complex problems involving multiple masses or non-uniform circular motion, you can switch to advanced mode (when available).

The Formula

The defining formula for centripetal force is:

F = m × v² / r

Where:

• F is the centripetal force (newtons, N)
• m is the mass of the object (kilograms, kg)
• v is the tangential speed (metres per second, m/s)
• r is the radius of the circular path (metres, m)

Equivalently, using angular velocity ω (radians per second) where v = ωr:

F = m × ω² × r

And using the period T (seconds) where ω = 2π/T:

F = (4π² × m × r) / T²

All three forms describe the same physics.

Worked Examples

Example 1, Car on a flat curve

A 1,500 kg car goes around a flat curve of radius 50 m at 15 m/s (about 54 km/h). What centripetal force is needed?

F = m × v² / r = 1,500 × (15)² / 50 = 1,500 × 225 / 50 = 6,750 N

This force is supplied by friction between the tyres and road. If the available friction is less than 6,750 N, the car skids outward.

Example 2, Banked curve

A 1,200 kg car travels at 20 m/s around a curve of radius 80 m. The curve is banked at angle θ. What is the minimum friction needed if the curve were not banked?

F = 1,200 × (20)² / 80 = 1,200 × 400 / 80 = 6,000 N

Banking the curve lets part of the normal force provide centripetal acceleration, reducing the required friction. The ideal banking angle θ for no friction at speed v is tan(θ) = v²/(g×r).

Example 3, Orbital motion

The International Space Station orbits Earth at roughly 400 km altitude with speed about 7.66 km/s. Its mass is about 420,000 kg. What centripetal force holds it in orbit?

Earth's radius is about 6,371 km, so r = 6,371 + 400 = 6,771 km = 6.771 × 10⁶ m.

F = 420,000 × (7,660)² / (6.771 × 10⁶) = 420,000 × 5.866 × 10⁷ / 6.771 × 10⁶ ≈ 3.64 × 10⁶ N

This force is provided by Earth's gravity alone, the ISS is in continuous free fall.

Example 4, Stopped car on a banked road

A car parks on a curve of radius 100 m, banked at 15°. The car does not move. What centripetal force is required?

Zero. A stationary object does not require centripetal force. The banking only matters when the car is moving.

Where Centripetal Force Shows Up

Roads and railways. Every curved road or track requires centripetal force, supplied by friction or by banking. The radius is designed so that for typical speeds, the required centripetal force does not exceed available friction (typically μ = 0.5-0.9 for rubber on dry asphalt).

Roller coasters and amusement rides. Vertical loops in roller coasters require large centripetal force at the bottom of the loop, often 3-4 g, to redirect riders' downward inertia into a circular path. Designers carefully balance speed and loop radius so that apparent weight at the bottom remains safe.

Athletics. Hammer throw, discus, shot put on a rotational platform, figure-skating spins, bicycle and motorcycle cornering, all involve centripetal acceleration. Athletes intuitively manage it by adjusting speed and grip.

Astronomy. Moons orbit planets, planets orbit stars, stars orbit galactic centres, all under centripetal force supplied by gravity. Kepler's laws of planetary motion are direct consequences of centripetal force combined with Newton's law of gravitation.

Centrifuges and washing machines. Spinning a wet load at high speed creates a centripetal force many times greater than gravity. Water droplets, following their inertia in straight lines, are flung outward, except the drum forces them into a circular path, pressing them through the drum holes.

Centripetal vs Centrifugal

In an inertial (non-rotating) reference frame, only centripetal force is real: it is the net inward force on the object.

In a rotating reference frame (e.g., inside a spinning carnival ride), passengers feel a "centrifugal" force pushing them outward. This is a fictitious force, it appears because the frame is accelerating. The fictitious force exactly balances the real centripetal force in magnitude. Both descriptions are valid; choose the one that matches your frame.

For calculation, always use the centripetal formula in an inertial frame. If you switch to a rotating frame, you must include fictitious forces.

Common Mistakes

Confusing centripetal and centrifugal. They have equal magnitudes but opposite directions, and only centripetal is "real" in an inertial frame. "Centrifugal" describes the apparent outward pull in a rotating frame.

Using diameter instead of radius. The formula requires r (radius), not d (diameter). If you measure across the circle, divide by 2 first.

Mixing units. Force must be in newtons, mass in kilograms, velocity in metres per second, radius in metres. Mixing imperial units (lb, mph, ft) with SI gives nonsensical answers.

Confusing angular velocity with tangential velocity. ω (rad/s) is not v (m/s). They are related by v = ω × r. The two centripetal formulas give the same answer only when you convert correctly.

Frequently Asked Questions

What is centripetal force? Centripetal force is the net inward force required to keep an object moving in a circular path. It is not a new kind of force, it is the resultant of whatever real forces are acting (gravity, tension, friction, normal force, electric force). What makes it "centripetal" is the direction: toward the centre of the circular motion.

What is the difference between centripetal force and gravity? Gravity is one specific force (the attraction between masses). Centripetal force is a description of any force that happens to point toward the centre of a circle. Gravity can act as the centripetal force for orbital motion (moons, planets), but centripetal force can equally well be supplied by tension (a ball on a string), friction (a car on a road), or the normal force (a banked curve).

What happens if centripetal force disappears? The object flies off in a straight line tangent to the circle, in the direction it was moving at the moment centripetal force vanished. This is Newton's first law, the natural state of motion is straight-line, not circular. Break a string, and a ball on it flies away tangentially.

Is centripetal force a "real" force? Yes and no. Centripetal force is the net of real forces (gravity, tension, friction). It is a useful bookkeeping concept to identify what is required for circular motion, but it is not itself a separate interaction.

What is the difference between centripetal and centrifugal force? Centripetal force is the inward force in an inertial (non-rotating) frame. Centrifugal force is the apparent outward force in a rotating frame, equal in magnitude to centripetal but opposite in direction. Both are valid in their respective frames.

Why do astronauts feel weightless in orbit? Astronauts are in continuous free fall, with gravity providing the centripetal force for their orbit. The floor of the spacecraft accelerates toward them at exactly g (gravity), so they feel no normal force, which is what we perceive as weight.

Can centripetal force do work? No. Centripetal force is always perpendicular to the velocity, and work is force times distance along the direction of force. Power = F·v·cos(θ) = 0 when θ = 90°. So centripetal force changes direction but not speed, and does no work.

What is the maximum speed on a banked curve with no friction? For a curve banked at angle θ and radius r, the design speed is v = √(g × r × tan(θ)). At this speed no friction is needed. Going faster requires inward friction; going slower requires outward friction (or the vehicle slides down the banking).

What is uniform circular motion? Motion at constant speed along a circular path. Speed is constant, but velocity changes direction continuously, so acceleration is non-zero and points toward the centre. Centripetal force supplies this acceleration.

**Q:**Can the Centripetal Force Calculator be used for professional or commercial purposes?A: Yes, the Centripetal Force Calculator The Centripetal Force Calculator provides mathematically correct results that are suitable for professional, commercial, and educational use. the Centripetal Force Calculator formulas used are well-established and validated against reference standards.

**Q:**How often are the formulas behind the Centripetal Force Calculator updated? When standards change (e.g., new physical constants, revised tax brackets, updated standards), the Centripetal Force Calculator is updated to reflect the current authoritative source. Each calculator's references section, including the Centripetal Force Calculator, lists the specific sources used.

References

• Halliday, D., Resnick, R., & Walker, J. Fundamentals of Physics, Wiley (any recent edition).
• Serway, R. A. & Jewett, J. W. Physics for Scientists and Engineers, Cengage.
• Tipler, P. A. & Mosca, G. Physics for Scientists and Engineers, W. H. Freeman.
• NIST reference constants used by the Centripetal Force Calculator: https://physics.gov/cuu/Constants/