Circular Motion Calculator
Anything moving in a circle is accelerating toward the centre, whether it feels like it or not. Enter the speed, the radius and the mass, and this calculator returns the centripetal acceleration and force, the angular speed, and the period and frequency of the revolution.
Acceleration Without Speeding Up
Circular motion's founding paradox is that the speed can be perfectly constant while the acceleration never sleeps: velocity is a vector, its direction rotates with the motion, and a rotating velocity is an acceleration — directed at the centre, of size a = v² ÷ r. The force that must therefore act, F = m v² ÷ r, is not a new species of force but a job description: something in the world (a string's tension, a road's friction, the Sun's gravity) must be found performing it, and if nothing is, the object departs on the tangent, as every hammer thrower and every skidding car demonstrates. The calculator's five outputs are one motion read five ways: the acceleration and force (the dynamics), the angular speed ω = v ÷ r, and the revolution's period and frequency (the bookkeeping) — a complete identity card for uniform circular motion.
Worked Example: Ten Metres per Second on a Five-Metre Circle
The defaults run a playground-scale case: v = 10 m/s, r = 5 m, m = 2 kg. Acceleration = 100 ÷ 5 = 20.0 m/s² — twice gravity, from a modest speed, because the radius is tight — and the centre-seeking force is 40.0 N. The angular speed is 2.00 rad/s, the period 3.14 s (the circumference 31.4 m at 10 m/s — π making its scheduled appearance), and the frequency 0.318 Hz. Scale the intuition outward: a car at the same 10 m/s on a 50 m bend needs only 2 m/s² of friction (comfortable), while a racing line at 50 m/s on that bend needs 50 m/s² — five g, beyond tyres, into wings-and-downforce territory. The v² is the tyrant in every version of this story, and the calculator exists to feel its weight before the bend does.
Where the Centripetal Force Comes From
Since the force is a job description, the professional question is always "who is hired?" — and the candidates form a short, examinable list. A string (tension, up to its breaking point), a surface (the normal force on a banked track or a looping rollercoaster's seat — at the loop's top, gravity and the track split the wage), friction (the flat-cornering car, capped by μ — the hub's friction tools price that ceiling), and gravity itself, alone, for orbits: set m v² ÷ r equal to GMm ÷ r² and the mass cancels into the orbital speed formula v = √(GM ÷ r), which is why astronauts and their station share an orbit regardless of mass. The calculator gives the demand; matching it against a supplier's capacity — string strength, friction coefficient, available gravity — is engineering, and every failure of circular motion ever filmed is a demand notice the supplier could not meet.
The Rotor's Confession: Stress from Speed
Engineers fear this formula where it meets materials: a spinning disc's rim stress grows with the square of its rim speed (the same v², wearing a density), so turbine and flywheel designers quote tip speed the way this page quotes velocity — as the number that must not grow. The best flywheels turn the relation into storage (energy per kilogram rises with v² as well — the hub's kinetic tools carry that ledger), and the containment rings around jet-engine fan discs exist because a blade released at full speed departs with the energy this page's numbers predict. Circular motion's five outputs look like kinematics homework until a rotor is involved; then they are a safety case, priced in advance.
The Speed of the Tip of the Earth
Stand on the equator and this calculator takes your measure: the planet's radius there is 6,378 km and its period one sidereal day, so your speed is 2πr ÷ T ≈ 465 m/s — faster than a cruising airliner's half — with a centripetal acceleration of 0.034 m/s² quietly reducing your apparent weight by a third of a percent (one reason g is measurably smaller at the equator, and launch sites hug it for the free velocity). Move to latitude 45° and the circle shrinks by cos 45°, the speed to 329 m/s. It is the same five outputs, computed for the reader personally: you are, at this moment, a worked example that has been running since birth at a third of a kilometre per second.
The Wall of Death, Audited
The fairground's wall of death is this calculator performed without a net: a motorcycle circling a vertical wooden cylinder stays up because the wall's normal force, hired as the centripetal supplier, also generates the friction that holds the machine against gravity — and the minimum speed falls straight out of the two ledgers: the required m v² ÷ r must be large enough that friction (μ times that same normal force) covers m g, giving v ≥ √(g r ÷ μ). For a 6 m radius and μ near 0.5, the threshold is about 10.8 m/s — call it 39 km/h of sustained, deliberate circling, with the rider's body angled by exactly the resultant the calculator's force implies. The act looks like defiance of physics; it is, like most good acts, obedience with style.
Related Tools
More from the Physics Calculators hub: the Velocity Calculator and the Friction Calculator — the supplier whose ceiling decides most real corners.
Frequently Asked Questions
Is centrifugal force real?
As a bookkeeping term in a rotating frame, yes — engineers designing centrifuges and pilots describing turns use it daily, and in that frame it balances the books exactly (mass × v²/r, pointing outward). In the inertial frame this page uses, there is no outward force: there is only the inward force and the object's insistence on travelling straight, which the rotating observer experiences as an outward push. The felt "force" slinging you against the car door is your body's inertia auditioning for the tangent while the door supplies the centripetal correction. Both descriptions predict identically; the calculator prints the inertial one, whose F = mv²/r must be supplied by something identifiable — the discipline that keeps the physics honest.
Why does the Moon not fall into the Earth?
It is falling, perpetually, at the calculator's rate: the Moon's centripetal acceleration toward Earth is about 0.0027 m/s², which over each second curves its path by exactly the amount its tangential speed (1.02 km/s) would otherwise carry it away — the falling and the fleeing balance into an orbit, Newton's own cannonball argument. "Falling around" is the correct summary of every orbit: the ground (or the primary) curves away as fast as the faller falls toward it. Enter the Moon's figures into this page — v = 1,020 m/s, r = 384,400 km — and its acceleration line returns the 0.0027 that ties the tides' master to its string of gravity.
What sets the maximum speed on a flat corner?
Friction's budget: the required centripetal force m v² ÷ r cannot exceed μ × m × g, so vmax = √(μ g r) — mass cancelling, which is why a truck and a sports car share the same geometric limit on the same rubber (the truck's disadvantage lives elsewhere). At μ = 0.7 and r = 50 m, the cap is √(0.7 × 9.81 × 50) ≈ 18.5 m/s, about 67 km/h — the calculator's force line, divided by the friction ceiling, is a speed limit written by physics and only enforced, occasionally, by police. Banking the corner hires the normal force as a co-supplier and raises the cap, which is the entire rationale of the banked oval.
How do centrifuges use this?
By shrinking r and spinning fast: a laboratory ultracentrifuge at 30,000 rpm on a 10 cm radius has ω = 3,142 rad/s and a = ω²r ≈ 987,000 m/s² — about 100,000 g, which is the "RCF" figure protocols quote and this page's formula (in its ω dress, a = ω²r) computes directly. Sedimentation that gravity would take a week to settle finishes in minutes at a hundred thousand gravities; blood separates, uranium enriches (in gas centrifuges at even wilder figures), and astronauts train at the gentle end of the same dial. The calculator's steady-speed assumption is the centrifuge's whole operating point.
What is the difference between period and frequency here?
Two tellings of the revolution rate: the period T is seconds per turn (the stopwatch's figure), the frequency f is turns per second (the engineer's hertz), each the other's reciprocal — the widget prints both because different trades refuse to convert. Angular speed ω = 2πf = 2π ÷ T converts either into the radians-per-second the dynamics prefer, and the worked example shows the family agreeing: T = 3.14 s, f = 0.318 Hz, ω = 2.00 rad/s are one fact in three dialects, and fluency among them is most of what "understanding circular motion" means in practice.
Does the formula change for vertical circles?
The demand doesn't (m v² ÷ r toward the centre, always), but the speed stops being uniform — gravity speeds the object on the way down and taxes it on the way up — and the suppliers' shares rotate: at the top, gravity helps the track; at the bottom, the track must supply the demand plus support the weight, which is why loop-the-loop calculations are done at the top (minimum speed √(g r) to stay on the rail) and structures are rated at the bottom. This page prices the uniform case exactly; the vertical circle is the same equation evaluated moment by moment, with the energy ledger (the hub's potential and kinetic tools) tracking the speed's variation around the loop.