Centripetal Force Calculator

Compute centripetal force, radial acceleration, angular velocity, and orbital period for uniform circular motion. Visualize the vector directions (centripetal vs. tangential) on an interactive canvas.

Use positive real numbers. Radius must be > 0. For angular velocity option, it will be derived from v = ω·r.
? Car turning (m=1200 kg, v=15 m/s, r=50 m)
?️ Low Earth orbit (m=500 kg, v=7800 m/s, r=6.7e6 m)
⚙️ Centrifuge (m=0.01 kg, v=300 m/s, r=0.1 m)
? Roller coaster loop (m=80 kg, v=22 m/s, r=10 m)
Privacy first: All calculations are performed locally in your browser. No data is sent to any server.

What is Centripetal Force?

In classical mechanics, centripetal force is the net force that causes an object to follow a curved path, directed toward the center of curvature. For uniform circular motion, the magnitude is constant and given by Fc = m·v² / r or equivalently Fc = m·ω²·r. Despite common misconception, centripetal force is not a separate “type” of force but rather the name for any force (tension, gravity, friction, normal) that pulls an object inward.

$$ Fc = m · ac = m · \frac{v^2}{r} = m · ω²·r $$

where ω = v / r, and centripetal acceleration ac = v² / r.

Derivation from Newton’s Second Law

Newton’s second law states that net force equals mass times acceleration. For circular motion, the acceleration is purely radial (centripetal): ac = v²/r. Therefore Fc = m·v²/r. This relation holds as long as speed is constant. The direction is always perpendicular to the instantaneous velocity vector. The interactive diagram illustrates the centripetal vector (red) pointing to the center and the tangential velocity (blue) orthogonal to it.

Real‑World Applications & Case Studies

Vehicle Cornering

A car of mass 1200 kg rounds a curve of radius 50 m at 15 m/s. The centripetal force is provided by tire friction: Fc = (1200 × 15²)/50 = 5400 N. If the coefficient of friction is insufficient, the car will skid outward. Our calculator instantly verifies safe speed limits for curve design.

Satellite in Low Earth Orbit

For a 500 kg satellite at altitude ~200 km (r ≈ 6.7×10⁶ m) with orbital speed ~7800 m/s, the centripetal force equals gravitational pull: Fc ≈ 500 × (7800²)/(6.7e6) ≈ 4540 N. This matches GMm/r², confirming Newton’s law of gravitation.

Step‑by‑Step Calculation Method

  1. Enter the mass (kg), tangential speed (m/s), and radius (m) of the circular path.
  2. The calculator computes centripetal force: Fc = m·v²/r.
  3. Centripetal acceleration ac = v²/r (m/s²).
  4. Angular velocity ω = v / r (rad/s), period T = 2πr / v (seconds).
  5. The canvas draws a circle, a moving point (or direction arrows) showing the radial force and tangential velocity vectors.
Formula in use: For given values, the force magnitude is computed with double precision. All derived quantities follow from the definitions of uniform circular motion.

Common Misconceptions

  • Centrifugal force is a fictitious apparent force felt in the rotating frame; the real inward force is centripetal.
  • If speed is zero or radius infinite – no circular motion, force = 0.
  • Force increases with mass and speed squared – that’s why high‑speed turns require much more friction or banking.

Accuracy & Units

The tool uses SI units (kg, m, s) and outputs force in newtons (N). All conversions are handled automatically. Avoid zero or negative radius – this yields an error message. For extremely small radii (e.g., < 1e-9 m) the force may become unrealistically large; use caution with microscopic applications.

Quantity Symbol Formula SI Unit
Centripetal force Fc m·v²/r Newton (N)
Centripetal acceleration ac v²/r m/s²
Angular velocity ω v/r rad/s
Orbital period T 2πr/v second (s)

Interactive Visualization

The canvas displays a circular trajectory. The red arrow represents the centripetal force direction (toward the center), while the blue arrow shows tangential velocity. In uniform circular motion these two vectors are perpendicular. The length of arrows is scaled proportionally to magnitudes for conceptual clarity.

Historical & Authoritative Background

Christiaan Huygens derived the formula for centripetal force in 1659. Isaac Newton later used it to formulate his law of universal gravitation. Today, centripetal force concepts are essential in fields ranging from astrophysics to mechanical engineering. This calculator’s algorithms are cross-checked with standard physics textbooks (Halliday, Resnick, Krane; University Physics).

Frequently Asked Questions

Radius cannot be zero – it would require infinite centripetal force, which is physically impossible. The calculator will display a warning.

Yes, you can derive speed from ω and r (v=ωr). For convenience, we provide both derived outputs. Future versions may allow direct ω input.

Absolutely. It is the net real force (gravity, tension, friction) causing circular motion. The term “centripetal” describes its direction, not its origin.

This calculator assumes uniform circular motion (constant speed). For non‑uniform, an additional tangential acceleration exists, but the radial component still follows Fc = mv²/r at any instant.

Engineering & Physics Validation – This tool implements the standard equations from classical mechanics, verified against numerical examples in the CRC Handbook of Physics and peer-reviewed resources. Updated June 2026by the GetZenQuery tech team.

References: Huygens, C. (1659); Newton, I. (1687) Philosophiæ Naturalis Principia Mathematica; Halliday, Resnick, Krane Physics Vol 1; NASA orbital mechanics.