Calculate Coriolis Force in a Rotating Reference Frame

State which observer measures the velocity

In a reference frame rotating with angular velocity vector Ω, the apparent Coriolis force is F_C = −2m(Ω × v_rel). The velocity v_rel must be measured relative to that rotating frame. It is not automatically the object's velocity seen by a stationary laboratory observer. The cross product also means direction matters: using only two scalar speeds loses essential information. Start by drawing the rotation axis, choosing coordinate directions and identifying the relative velocity. The formula describes one term in a rotating-frame equation of motion, not every force acting on the object.

Calculate a perpendicular-motion example

Suppose a turntable rotates at 0.50 rad/s about the positive z axis. A 2 kg object has instantaneous relative velocity 3 m/s along positive x, perpendicular to that axis. The Coriolis-force magnitude is 2 × 2 × 0.50 × 3 = 6 N. The cross product z × x points along positive y, so the minus sign makes F_C point along negative y. The corresponding apparent acceleration is −3 m/s² along y. Enter 2*2*0.5*3 in the linked Physics Calculator to verify the magnitude after determining direction separately.

Check the sign with an explicit component calculation

Using Ω = (0, 0, 0.50) and v_rel = (3, 0, 0), the cross product is (0, 1.50, 0). Multiplication by −2m = −4 gives F_C = (0, −6, 0) N. Reversing the relative velocity to (−3, 0, 0) changes the force to (0, +6, 0) N. Reversing the rotation direction also reverses the force. These checks are useful because memorizing “left” or “right” without defining the observer and axes can produce an incorrect direction for an otherwise correct magnitude.

Use the angle between the two vectors

For arbitrary orientation, the magnitude is 2mΩv_rel sin θ, where θ is the angle between the rotation-axis vector and relative velocity. With m = 1.5 kg, Ω = 2 rad/s, speed 4 m/s and θ = 30°, the magnitude is 24 × 0.5 = 12 N. Parallel or antiparallel velocity gives zero because sin θ is zero. Perpendicular motion gives the maximum magnitude for fixed m, Ω and speed. The local expression evaluator uses radians for trigonometric functions, so enter sin(pi/6) for the 30° example.

Convert revolutions per minute before substitution

A rotation rate of 30 revolutions per minute equals 30 × 2π/60 = π rad/s. For a 0.50 kg mass moving at 2 m/s perpendicular to that rotation axis, the force magnitude is 2 × 0.50 × π × 2 = 2π N, approximately 6.283 N. Substituting 30 directly as radians per second would overstate the result. Radians are dimensionless in SI dimensional bookkeeping, and mΩv has units kg × 1/s × m/s, giving newtons. Keep the conversion and the chosen vector angle visible with the calculation.

Separate Coriolis and centrifugal terms

An object stationary relative to the turntable has v_rel = 0, so its Coriolis term is zero even if it is far from the rotation axis. The centrifugal term instead depends on position through −mΩ × (Ω × r). For a 2 kg object at radius 1 m with Ω = 0.50 rad/s, its magnitude is mΩ²r = 0.50 N. That is a different calculation from the earlier 6 N Coriolis example. If the rotation rate changes, an Euler term is also needed; translation of the frame's origin can introduce another apparent-force term.

Use the result as a component of a model

The Coriolis force is perpendicular to relative velocity, so its instantaneous dot product with that velocity is zero. It can change the direction of motion without directly contributing that dot-product power term. This does not mean every rotating system conserves mechanical energy or follows a path determined by Coriolis force alone. The local Physics Calculator evaluates the explicit numerical products and sine values here; it does not transform reference frames, compute cross products from vector inputs or integrate a trajectory. Use the full force balance and initial conditions for those broader tasks.

Related guides

Calculate Vector Components, Magnitude and Direction

Add physics vectors by components, recover direction with atan2, and distinguish vector operations from scalar calculations using checked examples.

Calculate Net Force with Mass, Acceleration and Directions

Use F = ma for a chosen object, combine signed forces correctly, and check Newtons, kilograms and acceleration with worked examples.

Connect Acceleration to Kinetic Energy with Time or Distance

Distinguish acceleration from energy, derive energy change for constant acceleration, and check time-based and distance-based examples.