About

Uniform circular motion errors cascade. A miscalculated centripetal acceleration a_c leads to incorrect force requirements, which in engineered systems (centrifuges, roller coasters, orbital mechanics) means structural failure or trajectory deviation. This calculator solves the full kinematic chain: provide any two of radius r, tangential speed v, period T, or frequency f, and it derives the remaining quantities plus centripetal acceleration, angular velocity ω, and centripetal force F_c when mass m is supplied. All formulas follow standard Newtonian mechanics with SI unit output.

The tool assumes a rigid, frictionless circular path with constant speed (uniform circular motion). It does not model non-uniform cases where tangential acceleration is nonzero. For banked curves or conical pendulums, additional geometric decomposition of forces is required beyond the scope of this calculator. Pro tip: when working with rotating machinery, always convert RPM to rad/s before applying these formulas. Remember that 1 RPM = 2π÷60 rad/s.

Formulas

The fundamental relation for uniform circular motion links tangential speed v, radius r, and period T:

v = 2πrT

Frequency f is the reciprocal of period:

f = 1T

Angular velocity ω in rad/s:

ω = 2πf = 2πT = vr

Centripetal acceleration a_c always points radially inward:

a_c = v²r = ω²r

When mass m is provided, centripetal force F_c follows Newton's second law:

F_c = m ⋅ a_c = mv²r

Where: v = tangential speed (m/s), r = radius of circular path (m), T = period of one revolution (s), f = frequency (Hz), ω = angular velocity (rad/s), a_c = centripetal acceleration (m/s²), m = mass of the object (kg), F_c = centripetal force (N).

Reference Data

System	Radius	Velocity	Period	Centripetal Acceleration	Notes
Earth orbit (Sun)	1.496 × 10¹¹ m	29,783 m/s	365.25 days	5.93 × 10⁻³ m/s²	Nearly circular orbit
Moon orbit (Earth)	3.844 × 10⁸ m	1,022 m/s	27.32 days	2.72 × 10⁻³ m/s²	Sidereal period
ISS orbit	6.771 × 10⁶ m	7,660 m/s	92.68 min	8.66 m/s²	~408 km altitude
Geostationary orbit	4.216 × 10⁷ m	3,075 m/s	23.93 hr	0.224 m/s²	Sidereal day
Car on highway curve	200 m	27.8 m/s	45.2 s	3.86 m/s²	~100 km/h
Centrifuge (lab)	0.10 m	314 m/s	0.002 s	9.87 × 10⁵ m/s²	~30,000 RPM
Washing machine spin	0.25 m	26.2 m/s	0.06 s	2,742 m/s²	~1,000 RPM
Ferris wheel	50 m	2.62 m/s	120 s	0.137 m/s²	2 min rotation
Bicycle wheel	0.35 m	8.33 m/s	0.264 s	198 m/s²	~30 km/h
Hammer throw (athletics)	1.22 m	29 m/s	0.264 s	689 m/s²	At release
DVD at edge	0.06 m	6.60 m/s	0.057 s	726 m/s²	~1,050 RPM (1x speed)
Proton in LHC	4,243 m	~3 × 10⁸ m/s	8.9 × 10⁻⁵ s	2.12 × 10¹³ m/s²	Relativistic regime
Electron in H atom	5.29 × 10⁻¹¹ m	2.19 × 10⁶ m/s	1.52 × 10⁻¹⁶ s	9.05 × 10²² m/s²	Bohr model (classical approx.)
Ceiling fan tip	0.60 m	12.6 m/s	0.30 s	264 m/s²	~200 RPM
Earth surface (equator)	6.371 × 10⁶ m	465 m/s	24 hr	0.034 m/s²	Axial rotation

Frequently Asked Questions

Acceleration describes how the velocity vector changes direction, not how position changes. In uniform circular motion, the speed is constant but the direction of velocity continually rotates. The rate of this directional change is always perpendicular to the velocity and aimed at the center. Without this inward acceleration, the object would travel in a straight line tangent to the circle (Newton's first law). The magnitude is a_c = v²÷r.

Multiply RPM by 2π÷60. For example, 3000 RPM = 3000 × 2π ÷ 60 ≈ 314.16 rad/s. This calculator accepts period or frequency directly. Convert RPM to frequency first: f = RPM ÷ 60.

No. This tool models ideal uniform circular motion on a flat, frictionless plane. The centripetal force computed is the net radial force required. On a banked curve, this force is supplied by the horizontal component of the normal force (N sinθ). On a flat curve, static friction provides the centripetal force. You must verify that the available friction (μ_smg) exceeds the calculated F_c to confirm the object won't skid.

As r → 0, centripetal acceleration a_c = v²÷r diverges to infinity for any nonzero speed. Physically, an infinitely tight turn at finite speed requires infinite force, which is impossible. The calculator will flag radius values at or below zero as invalid. In real systems, minimum turn radius is constrained by material strength and friction limits.

Only as an approximation at a specific point. Vertical circular motion is non-uniform because gravitational potential energy converts to kinetic energy. The speed varies around the loop. At the top, the minimum speed to maintain contact is v_min = √gr. You can input the speed at a specific point to find the instantaneous centripetal acceleration there, but the tool does not model the energy variation around the loop.

Mass has zero effect on centripetal acceleration. The formula a_c = v²÷r depends only on geometry and speed. However, centripetal force scales linearly with mass: F_c = m ⋅ a_c. A 2000 kg car on the same curve at the same speed needs exactly twice the friction force of a 1000 kg car.