Rotating reference frame

An everyday example of a rotating reference frame is the surface of the Earth.

(This article considers only frames rotating about a fixed axis.

If Earth were to rotate many times faster, these fictitious forces could be felt by humans, as they are when on a spinning carousel.

Instead, centrifugal force originates in the rotation of the frame of reference within which observations are made.

[2][3][4][5][6][7] The mathematical expression for the Coriolis force appeared in an 1835 paper by a French scientist Gaspard-Gustave Coriolis in connection with hydrodynamics, and also in the tidal equations of Pierre-Simon Laplace in 1778.

Early in the 20th century, the term Coriolis force began to be used in connection with meteorology.

Perhaps the most commonly encountered rotating reference frame is the Earth.

Moving objects on the surface of the Earth experience a Coriolis force, and appear to veer to the right in the northern hemisphere, and to the left in the southern.

Movements of air in the atmosphere and water in the ocean are notable examples of this behavior: rather than flowing directly from areas of high pressure to low pressure, as they would on a non-rotating planet, winds and currents tend to flow to the right of this direction north of the equator, and to the left of this direction south of the equator.

This article is restricted to a frame of reference that rotates about a fixed axis.

[10][11] The following is a derivation of the formulas for accelerations as well as fictitious forces in a rotating frame.

Then, by taking time derivatives, formulas are derived that relate the velocity of the particle as seen in the two frames, and the acceleration relative to each frame.

Using these accelerations, the fictitious forces are identified by comparing Newton's second law as formulated in the two different frames.

To derive these fictitious forces, it's helpful to be able to convert between the coordinates

representing standard unit basis vectors in the rotating frame.

Thus the time derivative of these vectors, which rotate without changing magnitude, is

, now representing standard unit basis vectors in the general rotating frame.

denotes the transformation taking basis vectors of the inertial- to the rotating frame, with matrix columns equal to the basis vectors of the rotating frame, then the cross product multiplication by the rotation vector is given by

and we want to examine its first derivative then (using the product rule of differentiation):[12][13]

This result is also known as the transport theorem in analytical dynamics and is also sometimes referred to as the basic kinematic equation.

Applying the result of the previous subsection to the displacement

the velocities in the two reference frames are related by the equation where subscript

Carrying out the differentiations and re-arranging some terms yields the acceleration relative to the rotating reference frame,

is the apparent acceleration in the rotating reference frame, the term

, is the Euler acceleration and is zero in uniformly rotating frames.

When the expression for acceleration is multiplied by the mass of the particle, the three extra terms on the right-hand side result in fictitious forces in the rotating reference frame, that is, apparent forces that result from being in a non-inertial reference frame, rather than from any physical interaction between bodies.

is the mass of the object being acted upon by these fictitious forces.

Notice that all three forces vanish when the frame is not rotating, that is, when

Thus the particle in addition to the real force is acted upon by a fictitious force...The particle will move according to Newton's second law of motion if the total force acting on it is taken as the sum of the real and fictitious forces.This equation has exactly the form of Newton's second law, except that in addition to F, the sum of all forces identified in the inertial frame, there is an extra term on the right...This means we can continue to use Newton's second law in the noninertial frame provided we agree that in the noninertial frame we must add an extra force-like term, often called the inertial force.

It is convenient to consider magnetic resonance in a frame that rotates at the Larmor frequency of the spins.