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Circular Motion Calculator

Physics

Input Parameters

Formula Reference

Angular Velocity

ω = v / r

Centripetal Accel.

a = v² / r

Centripetal Force

F = m·v² / r

Period

T = 2π / ω

Frequency

f = 1 / T

Kinetic Energy

KE = ½mv²

About This Tool

🔄 Circular Motion Calculator – Centripetal Force, Acceleration & More

The Circular Motion Calculator is a free online physics tool that computes all key quantities for an object in uniform circular motion: centripetal force, centripetal acceleration, angular velocity, period, frequency, and kinetic energy — from just three inputs: mass, radius, and tangential velocity. It is designed for physics students, engineers, educators, and anyone studying rotational or orbital mechanics.

This free online centripetal force calculator supports mass in kg, g, and lb; radius in m, cm, and ft; and velocity in m/s, km/h, and ft/s. All values are automatically converted to SI units before calculation, so mixing unit systems is perfectly safe.

📘 What Is Uniform Circular Motion?

Uniform circular motion occurs when an object travels along a circular path at constant speed. Although the speed is constant, the velocity vector continuously changes direction — meaning the object is always accelerating. This acceleration, directed toward the centre of the circle, is called centripetal acceleration (from the Latin centrum, centre, and petere, to seek). The force that produces this acceleration is the centripetal force.

Real-world examples of uniform circular motion include satellites in circular orbits, cars rounding a banked curve, electrons in a magnetic field, and clothes in a spinning washing machine drum.

⚙️ How the Circular Motion Calculator Works

The calculator uses the following formulas, all derived from Newton's second law applied to circular paths:

  • Angular velocity: ω = v / r (rad/s) — how fast the angle changes.
  • Centripetal acceleration: a = v² / r (m/s²) — the inward acceleration needed to maintain the circular path.
  • Centripetal force: F = m × v² / r (N) — the net inward force that keeps the object on its circular path.
  • Period: T = 2π / ω = 2πr / v (s) — time to complete one full revolution.
  • Frequency: f = 1 / T (Hz) — number of revolutions per second.
  • Kinetic energy: KE = ½mv² (J) — translational kinetic energy of the object (constant throughout the motion).

The calculator offers two modes: Basic Mode for a single object with step-by-step derivations, and Multi-Object Mode for side-by-side comparison of several objects with different masses, radii, and speeds.

🧮 Practical Examples

Example 1 — Car on a roundabout: A 1 200 kg car drives at 10 m/s around a roundabout of radius 25 m.
ω = 10 / 25 = 0.4 rad/s, a = 10² / 25 = 4 m/s², F = 1200 × 4 = 4 800 N, T = 2π / 0.4 ≈ 15.7 s.

Example 2 — Satellite orbit: A 500 kg satellite orbits at 7 800 m/s at a radius of 6 700 km (6 700 000 m).
a = 7800² / 6700000 ≈ 9.08 m/s², F ≈ 4 540 N, T = 2π × 6700000 / 7800 ≈ 5 397 s (≈ 90 min).

Example 3 — Ball on a string: A 0.2 kg ball swings in a horizontal circle of radius 0.5 m at 3 m/s.
a = 9 / 0.5 = 18 m/s², F = 0.2 × 18 = 3.6 N (provided by the string tension), f = 3 / (2π × 0.5) ≈ 0.955 Hz.

💡 Tips and Best Practices

  • Centripetal vs centrifugal: The centripetal force is real and directed inward. The centrifugal "force" felt by a passenger is a pseudo-force in the rotating reference frame — it does not appear in an inertial frame calculation.
  • Speed vs angular velocity: Two objects on the same rotating platform have the same angular velocity (ω) but different tangential speeds — the outer object moves faster. Use ω when discussing rotation rates; use v when comparing linear speeds.
  • Force scales with v²: Doubling the speed quadruples the centripetal force required. This is why highway on-ramps and banked turns are so important for vehicle safety at high speeds.
  • Multi-object comparison: Use the Multi-Object tab to compare how objects at different orbital radii need different forces even at the same speed — essential for understanding planetary systems or gear trains.

🔗 Related Concepts

Circular motion is closely connected to rotational kinetic energy (KE = ½Iω²), explored in the Rotational Kinetic Energy Calculator. The moment of inertia (I) determines how the mass distribution affects angular acceleration via the Moment of Inertia Calculator. Torque (τ = F × r) is the rotational analogue of force and is covered by the Torque Calculator. For translational motion leading into circular paths, the Momentum Calculator and Kinetic Energy Calculator are complementary tools. Unit conversions for speed (m/s ↔ km/h) and frequency (Hz ↔ rpm) are available via the Speed Converter and Frequency Converter.

Frequently Asked Questions

Is the Circular Motion Calculator free?

Yes, Circular Motion Calculator is totally free :)

Can I use the Circular Motion Calculator offline?

Yes, you can install the webapp as PWA.

Is it safe to use Circular Motion Calculator?

Yes, any data related to Circular Motion Calculator only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

What is uniform circular motion and how is it calculated?

Uniform circular motion describes an object moving at constant speed along a circular path. The key quantities are centripetal acceleration (a = v²/r), centripetal force (F = m·v²/r), angular velocity (ω = v/r), period (T = 2π/ω), and frequency (f = 1/T). Enter mass, radius, and tangential velocity into this calculator to instantly derive all of these values.

What is the difference between angular velocity and tangential velocity?

Tangential velocity (v, in m/s) is the linear speed of the object along the circular path — the speed you'd measure on a speedometer. Angular velocity (ω, in rad/s) describes how fast the angle changes and is related by ω = v/r. A larger radius means a higher tangential speed for the same angular velocity, which is why the outer edge of a spinning disk moves faster than the centre.

Why must a centripetal force exist for circular motion?

According to Newton's first law, an object moving in a straight line continues to do so unless a net force acts on it. For circular motion, a centripetal ('centre-seeking') force constantly redirects the velocity toward the centre of the circle without changing the speed. This force can be provided by gravity (planetary orbits), tension (ball on a string), normal force (car on a banked curve), or friction.

How does changing the radius affect centripetal force?

Centripetal force F = m·v²/r is inversely proportional to radius when speed is held constant — halving the radius doubles the force. Conversely, it is directly proportional to the square of velocity — doubling speed quadruples the force. This is why tighter turns at high speed demand significantly more force, making them dangerous in vehicles.

What units does this calculator support?

Mass can be entered in kg, g, or lb. Radius supports m, cm, and ft. Tangential velocity accepts m/s, km/h, or ft/s. All values are internally converted to SI units (kg, m, m/s) before calculation. Results are always displayed in SI: rad/s for angular velocity, m/s² for acceleration, N for force, s for period, Hz for frequency, and J for kinetic energy.

What is the kinetic energy of an object in circular motion?

Even though the speed is constant in uniform circular motion, the object still possesses kinetic energy KE = ½mv². This energy doesn't change over time (constant speed), but it represents the energy that would need to be supplied or dissipated to bring the object to rest. The calculator displays kinetic energy alongside the other circular motion parameters.