About the Circular Motion Calculator
An object moving at constant speed around a circle is still accelerating, because its velocity vector keeps changing direction. That inward-pointing acceleration is called centripetal ("center-seeking") acceleration, and by Newton's second law it requires a net inward force — the centripetal force — to sustain it. This calculator takes an object's mass, the radius of its circular path, and its tangential (linear) speed and computes the centripetal acceleration, the centripetal force, the angular velocity, and the period of one full revolution.
The physics behind the calculation
- Angular velocity: ω = v / r, measured in radians per second. It describes how fast the angle around the circle is sweeping out, independent of the circle's size.
- Centripetal acceleration: a = v² / r (equivalently a = ω²r). This acceleration always points from the object toward the center of the circle and is what continuously "bends" a straight-line path into a circular one.
- Centripetal force: applying F = ma radially gives F = mv² / r = mω²r. This is the net inward force — supplied in practice by tension in a string, gravity for an orbit, friction between tires and road, or the normal force on a banked curve.
- Period and frequency: the period T = 2πr / v = 2π / ω is the time for one complete revolution; frequency f = 1 / T counts revolutions per second (hertz).
Getting accurate results
- Centripetal force is not an extra force you add to a free-body diagram — it is the label for whatever real net inward force already exists. Do not add "centripetal force" on top of tension, gravity, or friction; those forces sum to produce it.
- Keep radius and speed in consistent units before converting — this calculator handles the conversion once you pick the units, but double-check that the radius you enter is measured to the center of the circular path, not to an edge.
- This calculator assumes uniform circular motion (constant speed). If the object is speeding up or slowing down along the circle, there is also a tangential acceleration component that this tool does not compute.
Frequently Asked Questions
What is centripetal force and where does it come from?
Centripetal force is the net inward force needed to keep an object moving along a circular path; it is not a new physical force but whatever real force (tension, gravity, friction, or a normal force) supplies that inward pull. Its magnitude is F = mv²/r, where m is mass, v is speed, and r is the radius of the circle.
What is the difference between centripetal force and centrifugal force?
Centripetal force is real and points toward the center of the circle — it is what actually causes the curved path. Centrifugal force is a fictitious, outward-pointing force that only appears if you analyze the motion from inside the rotating object's own (non-inertial) reference frame; in the stationary lab frame, only centripetal force exists.
How do I find angular velocity, period, or frequency from speed and radius?
Angular velocity is ω = v/r (in radians per second). The period, or time for one full revolution, is T = 2πr/v = 2π/ω. Frequency is the reciprocal of the period, f = 1/T, measured in hertz (revolutions per second).
Does uniform circular motion mean the velocity is constant?
No. In uniform circular motion the speed (magnitude of velocity) stays constant, but the direction of the velocity vector is always changing as the object moves around the circle. That continuous change in direction is itself an acceleration — the centripetal acceleration — even though speed never changes.