Circular Motion & Centripetal Force Calculator

Circular Motion Guide

What do I need to know about Centripetal Force and Acceleration?

For circular motion, a centripetal force acts toward the centre. F = mv2/r = m(omega)2r. Centripetal acceleration: a = v2/r. Centripetal force is not a new type of force — it is provided by an existing force. String tension for a ball on a string. Gravity for a satellite or moon. Friction for a car turning a corner. Normal reaction on a banking road. Example: 2kg ball on 5m string at 10 m/s. F = 2 x 100/5 = 40N inward.

What's the key thing to understand about Angular Velocity and Period?

Angular velocity omega = 2pi/T = 2pif. Linear speed v = omega x r. Period T = 2pir/v. Earth rotation: T = 86400s. omega = 2pi/86400 = 7.27 x 10 to the -5 rad/s. Speed at equator (r = 6.37 x 10 to the 6 m): v = 463 m/s.

What should I know about Vertical Circles and Minimum Speed?

At the top of a vertical circle: T + mg = mv2/r. Minimum speed for string to remain taut (T=0): v_min = sqrt(gr). Below this speed the string goes slack. At the bottom: T = mv2/r + mg. Tension is always greatest at the bottom of the circle. Applications: roller coaster loops, bucket of water demonstration.

What's the key thing to understand about Satellites and Orbits?

Gravity provides centripetal force: GMm/r2 = mv2/r. Orbital speed v = sqrt(GM/r). Period T = 2pi x sqrt(r3/GM). Geostationary orbit: T = 86400s, altitude 35786km above Earth. Low Earth orbit (ISS at 400km): T approx 92 minutes. Kepler third law: T2 is proportional to r3 follows directly from these equations.

What's the difference between Centripetal and Centrifugal Force?

Centripetal force is the real, physical net force that acts toward the centre of a circular path and is exactly what keeps an object moving in a circle rather than flying off in a straight line — it's provided by whatever physical mechanism applies the inward pull or push, such as tension in a string, gravity for an orbiting satellite, friction between tyres and road, or the normal force from a curved track. Centrifugal force, by contrast, is not a real force at all in an inertial (non-rotating) reference frame — it's a fictitious or 'pseudo' force that only appears to act if you analyse the motion from inside the rotating frame itself, such as a passenger's perspective inside a turning car, who feels pushed outward even though no actual outward force is acting on them. What the passenger is really experiencing is their own inertia (Newton's first law): their body wants to continue in a straight line while the car turns beneath them, creating the sensation of being flung outward. Understanding this distinction is essential for correctly applying F = mv²/r — the centripetal force equation always describes the one real, inward-acting net force, never an outward one.

What should I know about Banked Curves?

On a flat road, the entire centripetal force needed to turn a corner must come from friction between the tyres and the road surface, which has a limit — exceed it and the vehicle skids outward. Banking a curve (tilting the road surface so the outer edge is higher than the inner edge) allows part of the normal force from the road to contribute toward the centre of the turn, reducing the reliance on friction. For a frictionless banked curve, the ideal banking angle θ satisfies tan θ = v²/(rg), where v is speed, r is the radius of the turn, and g is gravitational acceleration — at exactly this speed, the banked curve provides all the necessary centripetal force without any friction at all. This is exactly why racetracks like NASCAR ovals use steep banking angles (allowing very high cornering speeds) and why highway on-ramps and railway curves are banked to a lesser degree — each is engineered around a target speed, and travelling significantly faster or slower than that design speed on a banked curve shifts the reliance back onto friction, in the direction of sliding up or down the bank respectively.

What's the key thing to understand about Angular Momentum in Circular Motion?

Angular momentum L = mvr (or L = Iω for an extended body) is conserved whenever no external torque acts on a rotating system — the rotational analogue of linear momentum conservation. This is why a skater spinning with arms outstretched speeds up dramatically when pulling their arms in: reducing r (or moment of inertia I) forces ω to increase to keep L constant. The same principle, applied to orbiting planets, is Kepler's second law: a planet sweeps out equal areas in equal times, moving fastest at its closest approach to the Sun (perihelion) and slowest at its farthest point (aphelion).

Worked Example: A Car Rounding a Curve

A 1,200 kg car rounds a flat curve of radius 50 m at a constant speed of 15 m/s. The centripetal acceleration required is a_c = v²/r = 15² / 50 = 225/50 = 4.5 m/s². The centripetal force needed to produce that acceleration is F_c = m × a_c = 1,200 × 4.5 = 5,400 N, and on a flat road this force is supplied entirely by friction between the tires and the pavement. If the available friction force (μ × m × g) is less than 5,400 N, the car will skid outward — which is exactly why icy or wet roads (low μ) force drivers to slow down on the same curve. Doubling the speed to 30 m/s would require a_c = 30²/50 = 18 m/s², four times as much — a direct consequence of the v² term, and the reason highway curves posted for higher speeds need either a much larger radius or a banked surface to supply part of the force through the normal force instead of friction alone.

Table: Centripetal Acceleration Across Common Scenarios

Centripetal acceleration scales with v²/r, so very different situations — a spinning drum, a car on a curve, a satellite in orbit — can be compared on the same footing once you know speed and radius. The table below gives typical orders of magnitude.

Tangential Acceleration and Non-Uniform Circular Motion

Not all circular motion happens at constant speed. When an object speeds up or slows down while moving along a circular path — a car accelerating out of a curve, a ball on a string being swung faster — there are two acceleration components acting at once. Centripetal (radial) acceleration, a_c = v²/r, always points toward the center and is responsible for changing the direction of motion. Tangential acceleration, a_t = dv/dt (the rate of change of speed), points along the direction of travel and is responsible for changing the speed. The total acceleration is the vector sum of the two, and because they're perpendicular to each other, its magnitude is a_total = √(a_c² + a_t²). For example, a cyclist going around a circular track of radius 20 m at 8 m/s while accelerating at 1.5 m/s² tangentially experiences a centripetal acceleration of 8²/20 = 3.2 m/s², giving a total acceleration magnitude of √(3.2² + 1.5²) = √(10.24 + 2.25) = √12.49 ≈ 3.53 m/s² — noticeably larger than either component alone.

Table: Period and Frequency for Common Rotating Objects

Period (T, the time for one full revolution) and frequency (f, revolutions per second) are reciprocals of each other, f = 1/T, and relate to angular velocity through ω = 2πf = 2π/T. The table below shows typical values across a range of familiar rotating systems.

Circular Motion in Washing Machines and Centrifuges

The spin cycle of a washing machine and a laboratory centrifuge both use circular motion to separate materials by density, relying on exactly the same centripetal physics covered in this calculator. As the drum spins, water (less dense, or rather, more easily squeezed through the drum's perforations under the pressure difference created by spinning) is forced outward and through small holes in the drum wall, while the heavier, denser clothing fabric is held in a tighter circular path by the drum itself. Spin speed matters enormously here because the centripetal force needed scales with the square of angular velocity: doubling a washing machine's spin speed from 600rpm to 1200rpm doesn't double the water extraction force, it roughly quadruples it, which is why higher-spin-speed washing machines extract noticeably more water and leave clothes needing less tumble-dryer time. Laboratory centrifuges apply the same principle at far higher speeds to separate even finer differences in density, such as separating blood plasma from blood cells in a matter of minutes.