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Arotating frame of reference is a special case of anon-inertial reference frame that isrotating relative to aninertial reference frame. An everyday example of a rotating reference frame is the surface of theEarth. (This article considers only frames rotating about a fixed axis. For more general rotations, seeEuler angles.)

Allnon-inertial reference frames exhibitfictitious forces; rotating reference frames are characterized by three:^[1]

• thecentrifugal force,
• theCoriolis force,

and, for non-uniformly rotating reference frames,

• theEuler force.

Scientists in a rotating box can measure therotation speed andaxis of rotation by measuring these fictitious forces. For example,Léon Foucault was able to show the Coriolis force that results from Earth's rotation using theFoucault pendulum. If Earth were to rotate many times faster, these fictitious forces could be felt by humans, as they are when on a spinningcarousel.

Inclassical mechanics,centrifugal force is an outward force associated withrotation. Centrifugal force is one of several so-calledpseudo-forces (also known asinertial forces), so named because, unlikereal forces, they do not originate in interactions with other bodies situated in the environment of the particle upon which they act. 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 scientistGaspard-Gustave Coriolis in connection withhydrodynamics, and also in thetidal equations ofPierre-Simon Laplace in 1778. Early in the 20th century, the term Coriolis force began to be used in connection withmeteorology.

Perhaps the most commonly encountered rotating reference frame is theEarth. Moving objects on the surface of the Earth experience a Coriolis force, and appear to veer to the right in theNorthern Hemisphere, and to the left in theSouthern Hemisphere. 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 theequator, and to the left of this direction south of the equator. This effect is responsible for the rotation of largecyclones (seeCoriolis effects in meteorology).

Inclassical mechanics, theEuler acceleration (named forLeonhard Euler), also known asazimuthal acceleration^[8] ortransverse acceleration^[9] is anacceleration that appears when a non-uniformly rotating reference frame is used for analysis of motion and there is variation in theangular velocity of thereference frame's axis. This article is restricted to a frame of reference that rotates about a fixed axis.

TheEuler force is afictitious force on a body that is related to the Euler acceleration byF = ma, wherea is the Euler acceleration andm is the mass of the body.^[10][11]

The following is a derivation of the formulas for accelerations as well as fictitious forces in a rotating frame. It begins with the relation between a particle's coordinates in a rotating frame and its coordinates in an inertial (stationary) 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${\displaystyle \left(x^{\prime },y^{\prime },z^{\prime }\right)}$ of the rotating reference frame and the coordinates${\displaystyle (x,y,z)}$ of aninertial reference frame with the same origin.^{[note 1]} If the rotation is about the${\displaystyle z}$ axis with a constantangular velocity${\displaystyle \mathrm{\Omega }}$ (so${\displaystyle z^{\prime }=z}$ and${\displaystyle \frac{\mathrm{d}\theta }{\mathrm{d}t}\equiv \mathrm{\Omega },}$ which implies${\displaystyle \theta (t)=\mathrm{\Omega }t+{\theta }_0}$ for some constant${\displaystyle {\theta }_0}$ where${\displaystyle \theta (t)}$ denotes the angle in the${\displaystyle x-y}$-plane formed at time${\displaystyle t}$ by${\displaystyle \left(x^{\prime },y^{\prime }\right)}$ and the${\displaystyle x}$-axis), and if the two reference frames coincide at time${\displaystyle t=0}$ (meaning${\displaystyle \left(x^{\prime },y^{\prime },z^{\prime }\right)=(x,y,z)}$ when${\displaystyle t=0,}$ so take${\displaystyle {\theta }_0=0}$ or some other integer multiple of${\displaystyle 2\pi }$), the transformation from rotating coordinates to inertial coordinates can be written$${\displaystyle x=x^{\prime }\cos (\theta (t))-y^{\prime }\sin (\theta (t))}$$$${\displaystyle y=x^{\prime }\sin (\theta (t))+y^{\prime }\cos (\theta (t))}$$whereas the reverse transformation is$${\displaystyle x^{\prime }=x\cos (-\theta (t))-y\sin (-\theta (t))}$$$${\displaystyle y^{\prime }=x\sin (-\theta (t))+y\cos (-\theta (t)).}$$

This result can be obtained from arotation matrix.

Introduce the unit vectors${\displaystyle \hat{\mathit{ı}},\hat{\mathit{ȷ}},\hat{\mathit{k}}}$ representing standard unit basis vectors in the rotating frame. The time-derivatives of these unit vectors are found next. Suppose the frames are aligned at${\displaystyle t=0}$ and the${\displaystyle z}$-axis is the axis of rotation. Then for a counterclockwise rotation through angle${\displaystyle \mathrm{\Omega }t}$:$${\displaystyle \hat{\mathit{ı}}(t)=(\cos \theta (t),\sin \theta (t))}$$where the${\displaystyle (x,y)}$ components are expressed in the stationary frame. Likewise,$${\displaystyle \hat{\mathit{ȷ}}(t)=(-\sin \theta (t),\cos \theta (t)).}$$

Thus the time derivative of these vectors, which rotate without changing magnitude, is$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\hat{\mathit{ı}}(t)=\mathrm{\Omega }(-\sin \theta (t),\cos \theta (t))=\mathrm{\Omega }\hat{\mathit{ȷ}};}$$$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\hat{\mathit{ȷ}}(t)=\mathrm{\Omega }(-\cos \theta (t),-\sin \theta (t))=-\mathrm{\Omega }\hat{\mathit{ı}},}$$where${\displaystyle \mathrm{\Omega }\equiv \frac{\mathrm{d}}{\mathrm{d}t}\theta (t).}$ This result is the same as found using avector cross product with the rotation vector${\displaystyle \mathbf{\Omega }}$ pointed along the z-axis of rotation${\displaystyle \mathbf{\Omega }=(0,0,\mathrm{\Omega }),}$ namely,$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\hat{\mathit{u}}=\mathbf{\Omega }\mathbf{\times }\hat{\mathit{u}},}$$where${\displaystyle \hat{\mathit{u}}}$ is either${\displaystyle \hat{\mathit{ı}}}$ or${\displaystyle \hat{\mathit{ȷ}}.}$

Introduce unit vectors${\displaystyle \hat{\mathit{ı}},\hat{\mathit{ȷ}},\hat{\mathit{k}}}$, now representing standard unit basis vectors in the general rotating frame. As they rotate they will remain normalized and perpendicular to each other. If they rotate at the speed of${\displaystyle \mathrm{\Omega }(t)}$ about an axis along the rotation vector${\displaystyle \mathbf{\Omega }(t)}$ then each unit vector${\displaystyle \hat{\mathit{u}}}$ of the rotating coordinate system (such as${\displaystyle \hat{\mathit{ı}},\hat{\mathit{ȷ}},}$ or${\displaystyle \hat{\mathit{k}}}$) abides by the following equation:$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\hat{\mathit{u}}=\mathbf{\Omega }\times \hat{\mathit{u}}.}$$So if${\displaystyle R(t)}$ 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${\displaystyle \mathbf{\Omega }\times =R^{\prime }(t)\cdot R(t{)}^T}$.

If${\displaystyle \mathit{f}}$ is a vector function that is written as^{[note 2]}$${\displaystyle \mathit{f}(t)=f_1(t)\hat{\mathit{ı}}+f_2(t)\hat{\mathit{ȷ}}+f_3(t)\hat{\mathit{k}},}$$and we want to examine its first derivative then (using theproduct rule of differentiation):^[12][13]where${\displaystyle {\left(\frac{\mathrm{d}\mathit{f}}{\mathrm{d}t}\right)}_{\mathrm{r}}}$ denotes the rate of change of${\displaystyle \mathit{f}}$ as observed in the rotating coordinate system. As a shorthand the differentiation is expressed as:$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\mathit{f}=\left[{\left(\frac{\mathrm{d}}{\mathrm{d}t}\right)}_{\mathrm{r}}+\mathbf{\Omega }\times \right]\mathit{f}.}$$

This result is also known as thetransport theorem in analytical dynamics and is also sometimes referred to as thebasic kinematic equation.^[14]

A velocity of an object is the time-derivative of the object's position, so

${\displaystyle \mathbf{v}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}.}$

The time derivative of a position${\displaystyle \mathit{r}(t)}$ in a rotating reference frame has two components, one from the explicit time dependence due to motion of the object itself in the rotating reference frame, and another from the frame's own rotation. Applying the result of the previous subsection to the displacement${\displaystyle \mathit{r}(t),}$ thevelocities in the two reference frames are related by the equation

${\displaystyle {\mathbf{v}}_{\mathbf{i}}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\left(\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}\right)}_{\mathrm{i}}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}=\left[{\left(\frac{\mathrm{d}}{\mathrm{d}t}\right)}_{\mathrm{r}}+\mathbf{\Omega }\times \right]\mathit{r}={\left(\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}\right)}_{\mathrm{r}}+\mathbf{\Omega }\times \mathbf{r}={\mathbf{v}}_{\mathrm{r}}+\mathbf{\Omega }\times \mathbf{r},}$

where subscript${\displaystyle \mathrm{i}}$ means the inertial frame of reference, and${\displaystyle \mathrm{r}}$ means the rotating frame of reference.

Acceleration is the second time derivative of position, or the first time derivative of velocity

${\displaystyle {\mathbf{a}}_{\mathrm{i}}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\left(\frac{{\mathrm{d}}^2\mathbf{r}}{\mathrm{d}{t}^2}\right)}_{\mathrm{i}}={\left(\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}\right)}_{\mathrm{i}}=\left[{\left(\frac{\mathrm{d}}{\mathrm{d}t}\right)}_{\mathrm{r}}+\mathbf{\Omega }\times \right]\left[{\left(\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}\right)}_{\mathrm{r}}+\mathbf{\Omega }\times \mathbf{r}\right],}$

where subscript${\displaystyle \mathrm{i}}$ means the inertial frame of reference,${\displaystyle \mathrm{r}}$ the rotating frame of reference, and where the expression, again,${\displaystyle \mathbf{\Omega }\times }$ in the bracketed expression on the left is to be interpreted as anoperator working onto the bracketed expression on the right.

As${\displaystyle \mathbf{\Omega }\times \mathbf{\Omega }=\mathbf{0}}$, the first time derivatives of${\displaystyle \mathbf{\Omega }}$ inside either frame, when expressed with respect to the basis of e.g. the inertial frame, coincide.Carrying out thedifferentiations and re-arranging some terms yields the accelerationrelative to the rotating reference frame,${\displaystyle {\mathbf{a}}_{\mathrm{r}}}$

${\displaystyle {\mathbf{a}}_{\mathrm{r}}={\mathbf{a}}_{\mathrm{i}}-2\mathbf{\Omega }\times {\mathbf{v}}_{\mathrm{r}}-\mathbf{\Omega }\times (\mathbf{\Omega }\times \mathbf{r})-\frac{\mathrm{d}\mathbf{\Omega }}{\mathrm{d}t}\times \mathbf{r}}$

where${\displaystyle {\mathbf{a}}_{\mathrm{r}}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\left(\frac{{\mathrm{d}}^2\mathbf{r}}{\mathrm{d}{t}^2}\right)}_{\mathrm{r}}}$ is the apparent acceleration in the rotating reference frame, the term${\displaystyle -\mathbf{\Omega }\times (\mathbf{\Omega }\times \mathbf{r})}$ representscentrifugal acceleration, and the term${\displaystyle -2\mathbf{\Omega }\times {\mathbf{v}}_{\mathrm{r}}}$ is theCoriolis acceleration. The last term,${\displaystyle -\frac{\mathrm{d}\mathbf{\Omega }}{\mathrm{d}t}\times \mathbf{r}}$, is theEuler 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 infictitious forces in the rotating reference frame, that is, apparent forces that result from being in anon-inertial reference frame, rather than from any physical interaction between bodies.

UsingNewton's second law of motion${\displaystyle \mathbf{F}=m\mathbf{a},}$ we obtain:^{[1][12][13][15][16]}

• theCoriolis force$${\displaystyle {\mathbf{F}}_{\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{s}}=-2m\mathbf{\Omega }\times {\mathbf{v}}_{\mathrm{r}}}$$
• thecentrifugal force$${\displaystyle {\mathbf{F}}_{\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{u}\mathrm{g}\mathrm{a}\mathrm{l}}=-m\mathbf{\Omega }\times (\mathbf{\Omega }\times \mathbf{r})}$$
• and theEuler force$${\displaystyle {\mathbf{F}}_{\mathrm{E}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{r}}=-m\frac{\mathrm{d}\mathbf{\Omega }}{\mathrm{d}t}\times \mathbf{r}}$$

where${\displaystyle m}$ is the mass of the object being acted upon by thesefictitious forces. Notice that all three forces vanish when the frame is not rotating, that is, when${\displaystyle \mathbf{\Omega }=0.}$

For completeness, the inertial acceleration${\displaystyle {\mathbf{a}}_{\mathrm{i}}}$ due to impressed external forces${\displaystyle {\mathbf{F}}_{\mathrm{i}\mathrm{m}\mathrm{p}}}$ can be determined from the total physical force in the inertial (non-rotating) frame (for example, force from physical interactions such aselectromagnetic forces) usingNewton's second law in the inertial frame:$${\displaystyle {\mathbf{F}}_{\mathrm{i}\mathrm{m}\mathrm{p}}=m{\mathbf{a}}_{\mathrm{i}}}$$Newton's law in the rotating frame then becomes

${\displaystyle {\mathbf{F}}_{\mathrm{r}}={\mathbf{F}}_{\mathrm{i}\mathrm{m}\mathrm{p}}+{\mathbf{F}}_{\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{u}\mathrm{g}\mathrm{a}\mathrm{l}}+{\mathbf{F}}_{\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{s}}+{\mathbf{F}}_{\mathrm{E}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{r}}=m{\mathbf{a}}_{\mathrm{r}}.}$

In other words, to handle the laws of motion in a rotating reference frame:^[16][17][18]

Treat the fictitious forces like real forces, and pretend you are in an inertial frame.

Obviously, a rotating frame of reference is a case of a non-inertial frame. 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 toF, 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 frameprovided we agree that in the noninertial frame we must add an extra force-like term, often called theinertial force.

It is convenient to considermagnetic resonance in a frame that rotates at theLarmor frequency of the spins. This is illustrated in the animation below. Therotating wave approximation may also be used.

• Absolute rotation
• Centrifugal force (rotating reference frame) Centrifugal force as seen from systems rotating about a fixed axis
• Coriolis force The effect of the Coriolis force on the Earth and other rotating systems
• Inertial frame of reference
• Non-inertial frame
• Fictitious force A more general treatment of the subject of this article

• Animation clip showing scenes as viewed from both an inertial frame and a rotating frame of reference, visualizing the Coriolis and centrifugal forces.

• Frames of reference
• Classical mechanics
• Astronomical coordinate systems
• Rotation

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