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Centripetal Acceleration Calculator

Centripetal acceleration equals velocity squared divided by radius

Solution

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How It Works

Centripetal acceleration is the inward acceleration of an object moving along a curved path: a = v² / r. It points toward the center of curvature and is what keeps the object on the circle even when speed is constant. Enter any two of acceleration, velocity, or radius and the calculator algebraically solves for the third, converting between m/s², ft/s², g, and other units automatically.

Example Problem

A car travels at 20 m/s around a circular track with a radius of 50 m. What is the centripetal acceleration the driver experiences?

  1. Identify the formula: a = v² / r.
  2. Square the velocity: v² = (20)² = 400 m²/s².
  3. Divide by the radius: 400 / 50 = 8 m/s².
  4. Express in g-units: 8 / 9.81 ≈ 0.82 g.
  5. Interpret: the driver feels about 0.82 g of lateral acceleration toward the center of the curve.

Centripetal acceleration depends only on speed and radius — it does not depend on the mass of the object. Mass enters when computing the centripetal force F = ma.

Key Concepts

Any object moving in a circle is continuously changing direction, which is itself a form of acceleration even when the speed is constant. The centripetal (center-seeking) component of acceleration always points along the radius toward the center of the circle, perpendicular to the velocity. By Newton's second law, this acceleration requires a centripetal force F = ma directed inward — 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).

Applications

  • Automotive engineering: computing lateral g-force on cars rounding curves to set safe speed limits.
  • Aerospace: determining the inward acceleration required to maintain orbit at a given radius and speed.
  • Amusement parks: designing centrifuge rides and roller-coaster loops within human g-tolerance.
  • Industrial machinery: calculating loads on spinning shafts, flywheels, and centrifuges.
  • Sports science: measuring the g-force on hammer-throw weights and discus.

Common Mistakes

  • Confusing centripetal and centrifugal — centripetal is the real inward acceleration; centrifugal is an apparent outward force felt only in a rotating frame.
  • Forgetting to square the velocity. Doubling speed quadruples the centripetal acceleration at the same radius.
  • Treating it as a force directly — a = v²/r is the acceleration. The corresponding force is F = ma = mv²/r.
  • Using diameter instead of radius. The formula uses the radius r; convert from diameter as r = D / 2.

Frequently Asked Questions

How do you calculate centripetal acceleration?

Use a = v² / r, where v is the tangential speed and r is the radius of the circular path. Square the speed, divide by the radius — the result is the inward acceleration in m/s².

What is the formula for centripetal acceleration?

a = v² / r. Equivalently, since v = ωr, the formula can also be written a = ω² r, where ω is the angular velocity in rad/s.

What is the difference between centripetal and centrifugal acceleration?

Centripetal acceleration is real: it points inward and is what curves the path of a moving object. Centrifugal force is fictitious — it appears only in a rotating reference frame because the frame itself is accelerating. The inertia you feel pulling you outward in a turning car is your body's tendency to move in a straight line, not a real outward force.

Does centripetal acceleration depend on mass?

No. The formula a = v² / r contains no mass term. Mass only enters when you compute the centripetal force F = ma = mv²/r needed to maintain the motion.

What is the centripetal acceleration of a car going around a sharp curve?

For a car at 20 m/s (≈ 45 mph) on a 50 m radius curve, a = (20)²/50 = 8 m/s² ≈ 0.82 g — close to the lateral grip limit of typical street tires. Sharper curves or higher speeds quickly exceed the friction limit and the car begins to skid.

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

Worked Examples

Automotive

What centripetal acceleration does a car experience at 20 m/s on a 50 m curve?

  • a = v² / r
  • a = (20)² / 50
  • a = 400 / 50
  • a = 8 m/s² (≈ 0.82 g)

Close to the lateral-grip limit of typical street tires (≈ 0.85 g) — sharper curves at this speed would slip.

Aerospace

What speed is required for a satellite at 6,778 km radius to feel 9.0 m/s² of centripetal acceleration?

  • v = √(a · r)
  • v = √(9.0 × 6,778,000)
  • v = √61,002,000
  • v ≈ 7,810 m/s

Matches typical low-Earth-orbit orbital speed at ISS altitude (~400 km above sea level).

Amusement Park

What loop radius keeps a roller-coaster rider at 4 g when traveling 25 m/s?

  • r = v² / a
  • r = (25)² / 39.24 (where 4 g = 39.24 m/s²)
  • r = 625 / 39.24
  • r ≈ 15.9 m

4 g is near the upper edge of acceptable instantaneous g-loading for amusement rides; tighter loops or higher speeds rapidly approach blackout territory.

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