Calcyx
Physics

Circular Motion Calculator

Compute centripetal acceleration, angular speed, period, frequency, and force from any radius plus one motion input.

Distance from the center of rotation to the object.

Used to convert acceleration into required centripetal force.

Optional. If left blank, it will be derived from period or frequency.

Time for one full revolution. Leave empty if you know speed or frequency.

Revolutions per second. Leave empty if you know speed or period.

Give radius and mass, then provide either speed, period, or frequency. The calculator fills in the missing motion values automatically.

Summary

Acceleration is 20 m/s². Required centripetal force is 20 N.

Centripetal acceleration

20 m/s²

Centripetal force

20 N

Angular velocity

2 rad/s

Linear speed

10 m/s

Period

3.142 s

Frequency

0.318 Hz

Key equations

  • a_c = v^2 / r
  • v = 2πr / T
  • v = 2πr · f
  • f = 1 / T

Assumes uniform circular motion: constant speed, fixed radius, and a single object mass.

Circular Motion Calculator

Overview

Circular motion shows up everywhere: satellites orbiting Earth, a stone on a string, the rotating drum of a washing machine, even the tires that keep your bike moving. In every case the object constantly changes direction while keeping (roughly) the same speed, which means it needs a sideways pull toward the center. That pull creates centripetal acceleration and the corresponding centripetal force. This calculator turns radius plus one motion value—speed, period, or frequency—into the full set of circular motion numbers so you do not have to rearrange formulas by hand.

How to use the tool

  1. Enter the radius of the path in meters. This is the distance from the center to the object.
  2. Enter the mass of the object in kilograms. Force requires mass; without it you would only get acceleration.
  3. Provide one motion input:
    • Linear speed v (m/s), or
    • Period T (seconds per revolution), or
    • Frequency f (revolutions per second, Hz).
  4. Leave the other motion fields blank; the calculator back-fills them automatically.
  5. Read the results: centripetal acceleration, required force, angular velocity, and the derived period/frequency pair.

If you edit any one motion field, the others update instantly based on the standard relationships.

Math behind the scenes

Uniform circular motion relies on a few compact equations:

  • Speed to acceleration: a_c = v^2 / r (sideways acceleration needed to keep turning).
  • Speed from period: v = 2πr / T (one circumference every period).
  • Speed from frequency: v = 2πr · f (frequency is revolutions per second).
  • Period and frequency: f = 1 / T.
  • Force: F_c = m · a_c (Newton’s second law applied to the sideways acceleration).

The calculator enforces positive values for radius, mass, and motion inputs to avoid undefined divisions.

Interpreting the outputs

  • Centripetal acceleration (m/s²): How hard the motion bends inward. Higher speed or smaller radius increases it.
  • Centripetal force (N): The pull the string, road, track, or gravity must supply to produce that acceleration for the given mass.
  • Angular velocity (rad/s): How quickly the object sweeps through angle; it equals speed divided by radius.
  • Period and frequency: Two ways to describe the timing of each revolution. A short period means high frequency.

Values are formatted with your chosen language locale so commas and decimal points feel familiar.

Worked examples

  • Car in a curve: A 1,200 kg car takes a 50 m radius bend at 15 m/s. Acceleration is 15^2 / 50 = 4.5 m/s²; force is about 5,400 N, roughly half the weight of the car. If the tires cannot supply that sideways friction, the car will skid.
  • Swinging a bucket: A 2 kg bucket moves in a vertical circle of radius 0.8 m with a period of 1.1 s. Speed is 2π·0.8 / 1.1 ≈ 4.57 m/s, acceleration about 26.1 m/s², and force about 52 N at the top (gravity adds or subtracts from the handle tension depending on position).
  • Small satellite: A 750 kg cubesat orbiting 400 km above Earth has radius about 6.77×10^6 m and period 5,600 s. The required centripetal force equals gravity at that altitude, about 7,400 N, and the orbital speed comes out near 7.7 km/s.

Practical tips and limitations

  • The equations assume uniform speed and a fixed radius. They do not model spiraling paths or changing speeds.
  • For banked curves and friction limits, you still need free-body diagrams; this tool only gives the pure circular motion numbers.
  • Use SI units (meters, seconds, kilograms) for consistent results. Converting miles per hour to m/s or inches to meters before entering avoids mistakes.
  • Extremely small periods or very large radii can produce huge forces—helpful when stress-testing designs but unrealistic for everyday objects.

FAQ

Is centripetal force a new kind of force? No. It is the net inward force required to produce the inward acceleration. Tension, gravity, friction, or the normal force can provide it.

What happens if speed drops to zero? Without speed there is no circular motion, so acceleration and centripetal force both become zero.

How does this differ from centrifugal force? Centrifugal force is a convenient “apparent” force used in a rotating frame of reference. In an inertial frame only the inward (centripetal) force is needed.