Centripetal Acceleration Calculator

Definition of centripetal acceleration

Centripetal acceleration a is the acceleration that changes the direction of velocity when a body moves in a circle. In uniform circular motion the speed v stays constant, and the velocity vector turns with the radius. The magnitude is a = v²/r, directed toward the center of the circle and perpendicular to the tangential velocity. The SI unit is the meter per second squared.

This is not the acceleration that speeds a body up or slows it down along the path. Tangential acceleration changes the speed and equals dv/dt. At constant v the straight-line change in speed is zero, and the direction still changes. That change of direction is what a = v²/r describes. The calculator reports the inward magnitude, not the x and y components.

History: Huygens and Newton

Christiaan Huygens derived the magnitude v²/r in Horologium oscillatorium in 1673, describing the centrifugal force felt in a frame that rotates with the body. Isaac Newton introduced the name centripetal force, vis centripeta, for the inward force that curves the path in an inertial frame. He published it in Philosophiae Naturalis Principia Mathematica in 1687. The magnitude of the acceleration is the same in both accounts. What changes is the frame in which the reader describes it.

Introductory mechanics (Halliday, Resnick, and Walker; Young and Freedman) uses Newton’s account: acceleration toward the center, and a force F = m·a that keeps the body on the circle. The feeling of being thrown outward in a car on a curve is a description in a non-inertial frame, and it is not a separate result of this calculator.

Derivation of a = v²/r

Step 1. In a short time Δt the body travels an arc of length v·Δt. The radius turns through an angle Δθ = v·Δt/r, in radians. Step 2. The velocity vector has a fixed length v and turns through the same angle Δθ. The change in that vector therefore has length |Δv| = v·Δθ = v²·Δt/r, when the angle is small. Step 3. Acceleration is |Δv|/Δt, so a = v²/r. The direction of Δv, as the velocity vector turns, points toward the center.

Step 4. For v = 14 m/s and r = 40 m: v² = 196 (m/s)² and a = 196/40 = 4.9 m/s². Step 5. The angular velocity of this motion is ω = v/r = 0.35 rad/s, so the same magnitude is a = ω²·r = 0.35²·40 = 4.9 m/s². The time for one lap is T = 2πr/v = 17.95 s, and from that a = 4π²r/T². Twice the speed, 28 m/s, gives a = 19.6 m/s², because the numerator grows by a factor of four.

Assumptions: constant speed and path radius

The formula is exact for uniform motion on a circle, or on an arc of constant radius of curvature, when the speed does not change. If a car on a curve is still speeding up, the full acceleration has two parts: the centripetal part v²/r and the tangential part dv/dt. The calculator computes only the first. The radius has to come from the geometry of the path. Divide a roundabout’s diameter by two before it goes into the radius field.

The model does not include road banking, tire friction, or the slope of the track. It also does not replace a local measurement of gravity. The constant g₀ = 9.80665 m/s², adopted by the 3rd General Conference on Weights and Measures in 1901, is used only to express the result in units of g. It is not the gravity at the reader’s location.

Angular velocity, period, and force

The same pair v and r yields two more kinematic quantities, with no extra measurement. Angular velocity ω = v/r is in radians per second. The lap period T = 2πr/v is the time for one full circuit. At v = 0 it is not defined, because the body never closes a circle. Frequency is f = 1/T, and ω = 2πf. A separate angular-velocity calculator finds ω from the period, and the calculator v = ω·r reverses the step ω = v/r.

Centripetal force has magnitude F = m·a = m·v²/r. This calculator has no mass field and does not report newtons. At the bottom of a vertical loop the track must supply more than m·a, because the weight mg points away from the center when the center is above the body. At the top of the loop, weight helps to curve the path. The calculator adds neither of those forces to a.

Orders of magnitude: from a roundabout to orbit

On a roundabout of radius 15 m at 8 m/s the acceleration is 4.267 m/s², or 0.4351 g. A car at 28 m/s on a 120 m curve has 6.533 m/s². A carousel seat, 8 m/s and 6 m from the axis, exceeds one standard g: 10.67 m/s². A stone on a 1.2 m string at 5 m/s has 20.83 m/s². The bottom of a loop, 20 m/s and a 10 m radius, gives 40 m/s² of path acceleration alone.

A laboratory centrifuge at 30 m/s and a radius of 0.10 m reaches 9,000 m/s², or 917.7 g. A rounded low orbit, 7,800 m/s on a radius of 6,771,000 m (6,371 km plus 400 km), has a centripetal acceleration of 8.985 m/s², or 0.9163 g. In a circular orbit that value equals the local gravitational acceleration: the craft and the passenger fall at the same a, so the passenger feels no normal force from the floor. Orbital acceleration is not zero.

Sources

The definition, the derivation, and the history come from mechanics textbooks and metrology sources: C. Huygens, Horologium oscillatorium, 1673; I. Newton, Philosophiae Naturalis Principia Mathematica, 1687; D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics, the chapter on motion in a circle; H. D. Young and R. A. Freedman, University Physics, uniform circular motion; R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. I, Chapter 12; BIPM, The International System of Units (SI), 9th edition; the 1901 declaration of the 3rd CGPM on standard gravity, g₀ = 9.80665 m/s². This page is for educational purposes.