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Centripetal Force Calculator

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How It Works

Centripetal force is the inward force required to keep an object of mass m moving in a circle of radius r at speed v: F = m v² / r. It's the net force directed toward the center, not a separate kind of force — it can be supplied by friction (a car on a curve), gravity (a satellite in orbit), tension (a swinging mass), or the normal force (a roller-coaster loop). Enter any three of force, mass, velocity, or radius and the calculator algebraically solves for the fourth, converting units automatically.

Example Problem

A 1,000 kg car rounds a circular curve of radius 50 m at 20 m/s. What centripetal force must friction supply to keep it on the road?

  1. Identify the formula: F = m v² / r.
  2. Square the velocity: v² = (20)² = 400 m²/s².
  3. Multiply mass by v²: 1,000 × 400 = 400,000 kg·m²/s².
  4. Divide by radius: 400,000 / 50 = 8,000 N.
  5. Interpret: friction between tires and pavement must supply about 8 kN inward — about 0.82 g of lateral grip per kg of car mass.

If friction can't supply enough force, the car slides outward along a tangent to the curve — Newton's first law in action.

Key Concepts

Centripetal force is a net force, not a new fundamental force. Any combination of gravity, normal force, friction, or tension can supply it as long as the net is directed toward the center. By Newton's second law, F = ma, so centripetal force equals mass times centripetal acceleration: F = m × (v²/r) = m v² / r. The faster the object or the tighter the radius, the more force is needed — speed enters quadratically, so doubling speed quadruples the required force at the same radius.

Applications

  • Automotive: sizing tire grip and banking angles for highway curves.
  • Aerospace: computing gravitational pull required to keep satellites in orbit.
  • Amusement parks: designing roller-coaster loops with structurally adequate track forces.
  • Industrial machinery: calculating centrifugal loads on spinning rotors, flywheels, and centrifuges.
  • Sports: estimating string tension on a hammer-throw or rope-spinning routine.

Common Mistakes

  • Confusing centripetal with centrifugal — centripetal force is real and inward; centrifugal is fictitious and appears only in rotating reference frames.
  • Forgetting to square the velocity. F scales as v², so a 50% speed increase requires 2.25× the force at the same radius.
  • Using diameter D instead of radius r. The formula uses r = D / 2.
  • Mixing mass units: kilograms must pair with m/s and meters to get newtons. The calculator handles unit conversion automatically when you set the correct dropdown.

Frequently Asked Questions

How do you calculate centripetal force?

Use F = m v² / r, where m is the mass, v is the tangential speed, and r is the radius of the circular path. The result is the inward (centripetal) force directed toward the center of the circle.

What is the formula for centripetal force?

F = m v² / r. Equivalently, since centripetal acceleration is a = v² / r, you can write F = m a — Newton's second law applied to circular motion.

What supplies the centripetal force for a car on a curve?

Friction between the tires and the road. If the curve is banked, a horizontal component of the normal force also contributes. When friction is insufficient (slick roads, excessive speed), the car cannot follow the curve and skids tangentially outward.

Is centripetal force the same as centrifugal force?

No. Centripetal force is real, inward, and required by Newton's laws to keep an object on a circle. Centrifugal force is an apparent (fictitious) outward force that appears only when you analyze motion from inside a rotating reference frame — it isn't a real interaction.

What centripetal force keeps a satellite in orbit?

Gravity. For a satellite in a circular orbit, the gravitational force F = G M m / r² equals the required centripetal force m v² / r. Solving gives the orbital speed v = √(G M / r) — only about 7.5 km/s at low Earth orbit.

Reference: Tipler, Paul A. 1995. Physics For Scientists and Engineers. Worth Publishers. 3rd ed.

Worked Examples

Automotive

What centripetal force does friction supply to a 1,000 kg car at 20 m/s on a 50 m curve?

  • F = m v² / r
  • F = 1,000 × (20)² / 50
  • F = 1,000 × 400 / 50
  • F = 8,000 N (≈ 1,798 lbf)

Roughly 0.82 g of lateral grip — near the friction limit of typical street tires.

Sports

What rope tension supports a 7.26 kg hammer spinning at 25 m/s on a 1.2 m wire?

  • F = m v² / r
  • F = 7.26 × (25)² / 1.2
  • F = 7.26 × 625 / 1.2
  • F ≈ 3,781 N (≈ 850 lbf)

Hammer-throw wires must withstand roughly half a metric ton of tension before release.

Industrial

What radius of curvature lets a 500 kg flywheel handle 50,000 N at 30 m/s?

  • r = m v² / F
  • r = 500 × (30)² / 50,000
  • r = 500 × 900 / 50,000
  • r = 9 m

Larger flywheels reduce the centripetal load on bearings for the same kinetic energy.

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