Circular-Motion Force Magnitude Calculator

This calculator evaluates the force magnitude associated with uniform circular motion from mass, angular speed, and radius. The same magnitude is described as an inward centripetal net force in an inertial frame and as an outward apparent centrifugal force when a rotating frame is used. Naming the reference frame prevents a correct number from being paired with an incorrect physical explanation.

Method and formula

For mass m in kilograms, angular speed ω in radians per second, and radius r in metres:

F = m × ω² × r

The result is in newtons because kg × (rad/s)² × m reduces to kg·m/s²; the radian is dimensionless in SI. The formula returns a magnitude, so the interface accepts non-negative inputs. Direction must be supplied by the physical model and chosen reference frame.

The equivalent linear-speed form is F = m × v² ÷ r when v = ωr and the radius is positive. This page uses angular speed directly and does not calculate stresses, torque, bearing loads, or material safety factors.

Worked example

For a 2 kg mass moving with angular speed 3 rad/s at a radius of 1.5 m:

F = 2 × 3² × 1.5 = 27 N

In an inertial laboratory frame, a 27 N resultant force directed toward the centre is required for that circular path. In a frame rotating with the mass, an apparent 27 N outward centrifugal term can be introduced so Newton's equations can be used in that non-inertial frame.

What the result means

The square on angular speed matters: doubling angular speed while mass and radius remain fixed multiplies the required force magnitude by four. Doubling mass or radius only doubles it.

“Centripetal force” is not necessarily a separate type of force. Tension, gravity, friction, a normal force, or a combination may provide the inward net force. The calculator cannot determine which interaction provides it.

Reference frames and engineering limits

• Centripetal describes the inward net force or acceleration needed for circular motion in an inertial frame. Centrifugal describes an apparent outward force introduced in a rotating frame; it is not an additional outward interaction to add to the inertial-frame force balance.
• The equation assumes a known instantaneous radius and angular speed. Nonuniform rotation may also involve tangential acceleration and forces that this result omits.
• Real rotating equipment requires geometry, imbalance, vibration, material strength, fatigue, fasteners, bearings, operating transients, and safety factors to be assessed by a qualified engineer.
• Measurements and units must be consistent. A speed in revolutions per minute must first be converted to radians per second.

Sources

• NASA, Circular Motion instructor material — the central-force relationship expressed as mω²r.
• NASA Goddard, Frames of Reference: The Centrifugal Force — explanation of inertial and rotating frames and the apparent centrifugal term.

Editorial record

• Author: SoupCalc Editorial Team
• Method review: SoupCalc Engineering Review
• Last reviewed: August 10, 2026
• Review scope: equation, SI units, reference-frame terminology, example, validation behavior, and engineering boundaries