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<title>Rotating reference frame</title>

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Rotating reference frame

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks cm-sidebar"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle cm-sidebar-title"><a href="Classical_mechanics" title="Classical mechanics">Classical mechanics</a></th></tr><tr><td class="sidebar-image"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}</annotation>

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</math><img src="./c2ad0a6d6780c3abc5247abd82bd8a2249d56ff3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.318ex; height:5.509ex;" alt="{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}" loading="lazy"><div class="sidebar-caption"><a href="Second_law_of_motion" class="mw-redirect" title="Second law of motion">Second law of motion</a></div></td></tr><tr><th class="sidebar-heading cm-sidebar-above">

<ul><li><a href="History_of_classical_mechanics" title="History of classical mechanics">History</a></li>

<li><a href="Timeline_of_classical_mechanics" title="Timeline of classical mechanics">Timeline</a></li>

<li><a href="List_of_textbooks_on_classical_mechanics_and_quantum_mechanics" title="List of textbooks on classical mechanics and quantum mechanics">Textbooks</a></li></ul>

</div></th></tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Branches</div></div><div class="sidebar-list-content mw-collapsible-content plainlist"><div class="hlist">

<ul><li><a href="Applied_mechanics" title="Applied mechanics">Applied</a></li>

<li><a href="Celestial_mechanics" title="Celestial mechanics">Celestial</a></li>

<li><a href="Continuum_mechanics" title="Continuum mechanics">Continuum</a></li>

<li><a href="Dynamics_(mechanics)" title="Dynamics (mechanics)">Dynamics</a></li>

<li><a href="Classical_field_theory" title="Classical field theory">Field theory</a></li>

<li><a href="Kinematics" title="Kinematics">Kinematics</a></li>

<li><a href="Kinetics_(physics)" title="Kinetics (physics)">Kinetics</a></li>

<li><a href="Statics" title="Statics">Statics</a></li>

<li><a href="Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a></li></ul>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Fundamentals</div></div><div class="sidebar-list-content mw-collapsible-content plainlist"><div class="hlist">

<ul><li><a href="Acceleration" title="Acceleration">Acceleration</a></li>

<li><a href="Angular_momentum" title="Angular momentum">Angular momentum</a></li>

<li><a href="Couple_(mechanics)" title="Couple (mechanics)">Couple</a></li>

<li><a href="D'Alembert's_principle" title="D'Alembert's principle">D'Alembert's principle</a></li>

<li><a href="Energy" title="Energy">Energy</a>

<ul><li><a href="Kinetic_energy#Newtonian_kinetic_energy" title="Kinetic energy">kinetic</a></li>

<li><a href="Potential_energy" title="Potential energy">potential</a></li></ul></li>

<li><a href="Force" title="Force">Force</a></li>

<li><a href="Frame_of_reference" title="Frame of reference">Frame of reference</a></li>

<li><a href="Inertial_frame_of_reference" title="Inertial frame of reference">Inertial frame of reference</a></li>

<li><a href="Impulse_(physics)" title="Impulse (physics)">Impulse</a></li>

<li><a href="Inertia" title="Inertia">Inertia</a> / <a href="Moment_of_inertia" title="Moment of inertia">Moment of inertia</a></li>

<li> <a href="Mechanical_power_(physics)" class="mw-redirect" title="Mechanical power (physics)">Mechanical power</a></li>

<li><a href="Work_(physics)" title="Work (physics)">Mechanical work</a></li>

<li> <a href="Moment_(physics)" title="Moment (physics)">Moment</a></li>

<li><a href="Momentum" title="Momentum">Momentum</a></li>

<li><a href="Space" title="Space">Space</a></li>

<li><a href="Speed" title="Speed">Speed</a></li>

<li><a href="Torque" title="Torque">Torque</a></li>

<li><a href="Velocity" title="Velocity">Velocity</a></li>

<li><a href="Virtual_work" title="Virtual work">Virtual work</a></li></ul>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Formulations</div></div><div class="sidebar-list-content mw-collapsible-content plainlist">

<ul><li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="Newton's_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></div></li>

<li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a> <div class="plainlist"><ul><li><a href="Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a></li><li><a href="Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a></li><li><a href="Routhian_mechanics" title="Routhian mechanics">Routhian mechanics</a></li><li><a href="Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a></li><li><a href="Appell's_equation_of_motion" title="Appell's equation of motion">Appell's equation of motion</a></li><li><a href="Koopman%E2%80%93von_Neumann_classical_mechanics" title="Koopman–von Neumann classical mechanics">Koopman–von Neumann mechanics</a></li></ul></div></div></li></ul></div></div></td>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Core topics</div></div><div class="sidebar-list-content mw-collapsible-content plainlist"><div class="hlist">

<ul><li><a href="Damping" title="Damping">Damping</a></li>

<li><a href="Displacement_(geometry)" title="Displacement (geometry)">Displacement</a></li>

<li><a href="Equations_of_motion" title="Equations of motion">Equations of motion</a></li>

<li><a href="Euler's_laws_of_motion" title="Euler's laws of motion">Euler's laws of motion</a></li>

<li><a href="Fictitious_force" title="Fictitious force">Fictitious force</a></li>

<li><a href="Friction" title="Friction">Friction</a></li>

<li><a href="Harmonic_oscillator" title="Harmonic oscillator">Harmonic oscillator</a></li></ul>

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<ul><li><a href="Inertial_frame_of_reference" title="Inertial frame of reference">Inertial</a> / <a href="Non-inertial_reference_frame" title="Non-inertial reference frame">Non-inertial reference frame</a></li></ul>

<ul><li><a href="Motion" title="Motion">Motion</a> (<a href="Linear_motion" title="Linear motion">linear</a>)</li>

<li><a href="Newton's_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a></li>

<li><a href="Newton's_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></li>

<li><a href="Relative_velocity" title="Relative velocity">Relative velocity</a></li>

<li><a href="Rigid_body" title="Rigid body">Rigid body</a>

<ul><li><a href="Rigid_body_dynamics" title="Rigid body dynamics">dynamics</a></li>

<li><a href="Euler's_equations_(rigid_body_dynamics)" title="Euler's equations (rigid body dynamics)">Euler's equations</a></li></ul></li>

<li><a href="Simple_harmonic_motion" title="Simple harmonic motion">Simple harmonic motion</a></li>

<li><a href="Vibration" title="Vibration">Vibration</a></li></ul>

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<div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Rotation_around_a_fixed_axis" title="Rotation around a fixed axis">Rotation</a></div></div><div class="sidebar-list-content mw-collapsible-content plainlist"><div class="hlist">

<ul><li><a href="Circular_motion" title="Circular motion">Circular motion</a></li>

<li><a href="Centripetal_force" title="Centripetal force">Centripetal force</a></li>

<li><a href="Centrifugal_force" title="Centrifugal force">Centrifugal force</a>

<ul><li><a href="Reactive_centrifugal_force" title="Reactive centrifugal force">reactive</a></li></ul></li>

<li><a href="Coriolis_force" title="Coriolis force">Coriolis force</a></li>

<li><a href="Pendulum_(mechanics)" title="Pendulum (mechanics)">Pendulum</a></li>

<li><a href="Tangential_speed" title="Tangential speed">Tangential speed</a></li>

<li><a href="Rotational_frequency" title="Rotational frequency">Rotational frequency</a></li></ul>

</div>

<ul><li><a href="Angular_acceleration" title="Angular acceleration">Angular acceleration</a> / <a href="Angular_displacement" title="Angular displacement">displacement</a> / <a href="Angular_frequency" title="Angular frequency">frequency</a> / <a href="Angular_velocity" title="Angular velocity">velocity</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content plainlist"><div class="hlist">

<ul><li><a href="Johannes_Kepler" title="Johannes Kepler">Kepler</a></li>

<li><a href="Galileo_Galilei" title="Galileo Galilei">Galileo</a></li>

<li><a href="Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li>

<li><a href="Isaac_Newton" title="Isaac Newton">Newton</a></li>

<li><a href="Jeremiah_Horrocks" title="Jeremiah Horrocks">Horrocks</a></li>

<li><a href="Edmond_Halley" title="Edmond Halley">Halley</a></li>

<li><a href="Pierre_Louis_Maupertuis" title="Pierre Louis Maupertuis">Maupertuis</a></li>

<li><a href="Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a></li>

<li><a href="Johann_Bernoulli" title="Johann Bernoulli">Johann Bernoulli</a></li>

<li><a href="Leonhard_Euler" title="Leonhard Euler">Euler</a></li>

<li><a href="Jean_le_Rond_d'Alembert" title="Jean le Rond d'Alembert">d'Alembert</a></li>

<li><a href="Alexis_Clairaut" title="Alexis Clairaut">Clairaut</a></li>

<li><a href="Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange</a></li>

<li><a href="Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a></li>

<li><a href="Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li>

<li><a href="William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</a></li>

<li><a href="Carl_Gustav_Jacob_Jacobi" title="Carl Gustav Jacob Jacobi">Jacobi</a></li>

<li><a href="Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a></li>

<li><a href="Edward_Routh" title="Edward Routh">Routh</a></li>

<li><a href="Joseph_Liouville" title="Joseph Liouville">Liouville</a></li>

<li><a href="Paul_%C3%89mile_Appell" title="Paul Émile Appell">Appell</a></li>

<li><a href="Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li>

<li><a href="Bernard_Koopman" title="Bernard Koopman">Koopman</a></li>

<li><a href="John_von_Neumann" title="John von Neumann">von Neumann</a></li></ul>

</div></div></div></td>

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<ul><li> <a href="Portal%3APhysics" title="Portal:Physics">Physics portal</a></li>

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A rotating frame of reference is a special case of a <a href="Non-inertial_reference_frame" title="Non-inertial reference frame">non-inertial reference frame</a> that is <a href="Rotation" title="Rotation">rotating</a> relative to an <a href="Inertial_reference_frame" class="mw-redirect" title="Inertial reference frame">inertial reference frame</a>. An everyday example of a rotating reference frame is the surface of the <a href="Earth" title="Earth">Earth</a>. (This article considers only frames rotating about a fixed axis. For more general rotations, see <a href="Euler_angles#Vehicles_and_moving_frames" title="Euler angles">Euler angles</a>.)

<div class="mw-heading mw-heading2"><h2 id="Fictitious_forces">Fictitious forces</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Fictitious_force" title="Fictitious force">Fictitious force</a></div>

All <a href="Non-inertial_reference_frame" title="Non-inertial reference frame">non-inertial reference frames</a> exhibit <a href="Fictitious_force" title="Fictitious force">fictitious forces</a>; rotating reference frames are characterized by three:<a href="#cite_note-Arnold-1">[1]</a>

<ul><li>the <a href="Centrifugal_force_(fictitious)" class="mw-redirect" title="Centrifugal force (fictitious)">centrifugal force</a>,</li>

<li>the <a href="Coriolis_force" title="Coriolis force">Coriolis force</a>,</li></ul>

and, for non-uniformly rotating reference frames,

<ul><li>the <a href="Euler_force" title="Euler force">Euler force</a>.</li></ul>

Scientists in a rotating box can measure the <a href="Rotation_speed" class="mw-redirect" title="Rotation speed">rotation speed</a> and <a href="Axis_of_rotation" class="mw-redirect" title="Axis of rotation">axis of rotation</a> by measuring these fictitious forces. For example, <a href="L%C3%A9on_Foucault" title="Léon Foucault">Léon Foucault</a> was able to show the Coriolis force that results from Earth's rotation using the <a href="Foucault_pendulum" title="Foucault pendulum">Foucault pendulum</a>. If Earth were to rotate many times faster, these fictitious forces could be felt by humans, as they are when on a spinning <a href="Carousel" title="Carousel">carousel</a>.

<div class="mw-heading mw-heading3"><h3 id="Centrifugal_force">Centrifugal force</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Centrifugal_force" title="Centrifugal force">Centrifugal force</a></div>

In <a href="Classical_mechanics" title="Classical mechanics">classical mechanics</a>, centrifugal force is an outward force associated with <a href="Rotation" title="Rotation">rotation</a>. Centrifugal force is one of several so-called <a href="Pseudo-force" class="mw-redirect" title="Pseudo-force">pseudo-forces</a> (also known as <a href="Inertial_force" class="mw-redirect" title="Inertial force">inertial forces</a>), so named because, unlike <a href="Fundamental_interaction" title="Fundamental interaction">real forces</a>, 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.<a href="#cite_note-2">[2]</a><a href="#cite_note-Marsden-3">[3]</a><a href="#cite_note-Taylor_A-4">[4]</a><a href="#cite_note-Marion-5">[5]</a><a href="#cite_note-6">[6]</a><a href="#cite_note-7">[7]</a>

<div class="mw-heading mw-heading3"><h3 id="Coriolis_force">Coriolis force</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Coriolis_force" title="Coriolis force">Coriolis force</a></div>

The mathematical expression for the Coriolis force appeared in an 1835 paper by a French scientist <a href="Gaspard-Gustave_Coriolis" class="mw-redirect" title="Gaspard-Gustave Coriolis">Gaspard-Gustave Coriolis</a> in connection with <a href="Hydrodynamics" class="mw-redirect" title="Hydrodynamics">hydrodynamics</a>, and also in the <a href="Theory_of_tides" title="Theory of tides">tidal equations</a> of <a href="Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> in 1778. Early in the 20th century, the term Coriolis force began to be used in connection with <a href="Meteorology" title="Meteorology">meteorology</a>.

Perhaps the most commonly encountered rotating reference frame is the <a href="Earth" title="Earth">Earth</a>. Moving objects on the surface of the Earth experience a Coriolis force, and appear to veer to the right in the <a href="Northern_hemisphere" class="mw-redirect" title="Northern hemisphere">northern hemisphere</a>, and to the left in the <a href="Southern_hemisphere" class="mw-redirect" title="Southern hemisphere">southern</a>. 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 <a href="Equator" title="Equator">equator</a>, and to the left of this direction south of the equator. This effect is responsible for the rotation of large <a href="Cyclone#Structure" title="Cyclone">cyclones</a> (see <a href="Coriolis_effect" class="mw-redirect" title="Coriolis effect">Coriolis effects in meteorology</a>).

<div class="mw-heading mw-heading3"><h3 id="Euler_force">Euler force</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Euler_force" title="Euler force">Euler force</a></div>

In <a href="Classical_mechanics" title="Classical mechanics">classical mechanics</a>, the Euler acceleration (named for <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>), also known as azimuthal acceleration<a href="#cite_note-Morin-8">[8]</a> or transverse acceleration<a href="#cite_note-Fowles-9">[9]</a> is an <a href="Acceleration" title="Acceleration">acceleration</a> that appears when a non-uniformly rotating reference frame is used for analysis of motion and there is variation in the <a href="Angular_velocity" title="Angular velocity">angular velocity</a> of the <a href="Frame_of_reference" title="Frame of reference">reference frame</a>'s axis. This article is restricted to a frame of reference that rotates about a fixed axis.

The Euler force is a <a href="Fictitious_force" title="Fictitious force">fictitious force</a> on a body that is related to the Euler acceleration by F  = ma, where a is the Euler acceleration and m is the mass of the body.<a href="#cite_note-Battin-10">[10]</a><a href="#cite_note-11">[11]</a>

<div class="mw-heading mw-heading2"><h2 id="Relating_rotating_frames_to_stationary_frames">Relating rotating frames to stationary frames</h2></div>

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.

<div class="mw-heading mw-heading3"><h3 id="Relation_between_positions_in_the_two_frames">Relation between positions in the two frames</h3></div>

To derive these fictitious forces, it's helpful to be able to convert between the coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x',y',z'\right)}">

<mrow>

<mrow>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(x',y',z'\right)}</annotation>

</semantics>

</math><img src="./491a1d5bc1874171326bfe40ff5cc9494d85c522.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.512ex; height:3.009ex;" alt="{\displaystyle \left(x',y',z'\right)}" loading="lazy"> of the rotating reference frame and the coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x,y,z)}</annotation>

</semantics>

</math><img src="./22a8c93372e8f8b6e24d523bd5545aed3430baf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}" loading="lazy"> of an <a href="Inertial_reference_frame" class="mw-redirect" title="Inertial reference frame">inertial reference frame</a> with the same origin.<a href="#cite_note-12">[note 1]</a>

If the rotation is about the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> axis with a constant <a href="Angular_velocity" title="Angular velocity">angular velocity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"> (so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z'=z}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z'=z}</annotation>

</semantics>

</math><img src="./80bfd939a15c0857a6b1df928f061d0e8973c342.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.962ex; height:2.509ex;" alt="{\displaystyle z'=z}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} \theta }{\mathrm {d} t}}\equiv \Omega ,}">

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} \theta }{\mathrm {d} t}}\equiv \Omega ,}</annotation>

</semantics>

</math><img src="./f46ac08a2d4ba6e06cf07e290e06fcc5310a689e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.642ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} \theta }{\mathrm {d} t}}\equiv \Omega ,}" loading="lazy"> which implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta (t)=\Omega t+\theta _{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta (t)=\Omega t+\theta _{0}}</annotation>

</semantics>

</math><img src="./d1f59cb96dd608b541bf7adf8196153df459e2c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.341ex; height:2.843ex;" alt="{\displaystyle \theta (t)=\Omega t+\theta _{0}}" loading="lazy"> for some constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta _{0}}</annotation>

</semantics>

</math><img src="./18b67de6bf25dba7a24e66967ff6319858798734.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.145ex; height:2.509ex;" alt="{\displaystyle \theta _{0}}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta (t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta (t)}</annotation>

</semantics>

</math><img src="./e92e767769c09cc676d3b32facf194677e467fc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.739ex; height:2.843ex;" alt="{\displaystyle \theta (t)}" loading="lazy"> denotes the angle in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x-y}</annotation>

</semantics>

</math><img src="./3129cb3620bd9f38d0304a0fca719644d7d2d265.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle x-y}" loading="lazy">-plane formed at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x',y'\right)}">

<mrow>

<mrow>

<msup>

</msup>

<msup>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(x',y'\right)}</annotation>

</semantics>

</math><img src="./ccb6a0c24edc64b1884b8d120e2d0fe05c30524a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.703ex; height:3.009ex;" alt="{\displaystyle \left(x',y'\right)}" loading="lazy"> and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">-axis),

and if the two reference frames coincide at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=0}</annotation>

</semantics>

</math><img src="./43469ec032d858feae5aa87029e22eaaf0109e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}" loading="lazy"> (meaning <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x',y',z'\right)=(x,y,z)}">

<mrow>

<mrow>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(x',y',z'\right)=(x,y,z)}</annotation>

</semantics>

</math><img src="./7f1512ae7122f3fccdef51d448755491e884f289.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.061ex; height:3.009ex;" alt="{\displaystyle \left(x',y',z'\right)=(x,y,z)}" loading="lazy"> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=0,}</annotation>

</semantics>

</math><img src="./55ff4c2b109c38fe7038da6238ae875f4d37e643.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.747ex; height:2.509ex;" alt="{\displaystyle t=0,}" loading="lazy"> so take <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{0}=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta _{0}=0}</annotation>

</semantics>

</math><img src="./392d824088476f13d5645ac73b8a738dd76d4c67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.406ex; height:2.509ex;" alt="{\displaystyle \theta _{0}=0}" loading="lazy"> or some other integer multiple of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

</semantics>

</math><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy">), the transformation from rotating coordinates to inertial coordinates can be written

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=x'\cos(\theta (t))-y'\sin(\theta (t))}</annotation>

</semantics>

</math>

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=x'\sin(\theta (t))+y'\cos(\theta (t))}</annotation>

</semantics>

</math>

whereas the reverse transformation is

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x'=x\cos(-\theta (t))-y\sin(-\theta (t))}</annotation>

</semantics>

</math>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y'=x\sin(-\theta (t))+y\cos(-\theta (t))\ .}</annotation>

</semantics>

</math>

This result can be obtained from a <a href="Rotation_matrix" title="Rotation matrix">rotation matrix</a>.

Introduce the unit vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\imath }}},\ {\hat {\boldsymbol {\jmath }}},\ {\hat {\boldsymbol {k}}}}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\imath }}},\ {\hat {\boldsymbol {\jmath }}},\ {\hat {\boldsymbol {k}}}}</annotation>

</semantics>

</math><img src="./227dda128fc49046fc65b7fec4331b9434a087a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.255ex; height:3.176ex;" alt="{\displaystyle {\hat {\boldsymbol {\imath }}},\ {\hat {\boldsymbol {\jmath }}},\ {\hat {\boldsymbol {k}}}}" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=0}</annotation>

</semantics>

</math><img src="./43469ec032d858feae5aa87029e22eaaf0109e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}" loading="lazy"> and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">-axis is the axis of rotation. Then for a counterclockwise rotation through angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega t}</annotation>

</semantics>

</math><img src="./ced27e18f08bd81f7261548e77c068ccbd0ab1de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.518ex; height:2.176ex;" alt="{\displaystyle \Omega t}" loading="lazy">:

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\imath }}}(t)=(\cos \theta (t),\ \sin \theta (t))}</annotation>

</semantics>

</math>

where the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation>

</semantics>

</math><img src="./41cf50e4a314ca8e2c30964baa8d26e5be7a9386.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}" loading="lazy"> components are expressed in the stationary frame. Likewise,

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\jmath }}}(t)=(-\sin \theta (t),\ \cos \theta (t))\ .}</annotation>

</semantics>

</math>

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

<mfrac>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\hat {\boldsymbol {\imath }}}(t)=\Omega (-\sin \theta (t),\ \cos \theta (t))=\Omega {\hat {\boldsymbol {\jmath }}}\ ;}</annotation>

</semantics>

</math>

<mfrac>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\hat {\boldsymbol {\jmath }}}(t)=\Omega (-\cos \theta (t),\ -\sin \theta (t))=-\Omega {\hat {\boldsymbol {\imath }}}\ ,}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \equiv {\frac {\mathrm {d} }{\mathrm {d} t}}\theta (t).}">

<mfrac>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega \equiv {\frac {\mathrm {d} }{\mathrm {d} t}}\theta (t).}</annotation>

</semantics>

</math><img src="./afe40424a0237ed8ff3f8d03a02699eafc8fe477.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.131ex; height:5.509ex;" alt="{\displaystyle \Omega \equiv {\frac {\mathrm {d} }{\mathrm {d} t}}\theta (t).}" loading="lazy">

This result is the same as found using a <a href="Vector_cross_product" class="mw-redirect" title="Vector cross product">vector cross product</a> with the rotation vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\Omega }}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\Omega }}}</annotation>

</semantics>

</math><img src="./9c95a28793dac7413de04992d822822f7e2dba16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\Omega }}}" loading="lazy"> pointed along the z-axis of rotation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\Omega }}=(0,\ 0,\ \Omega ),}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\Omega }}=(0,\ 0,\ \Omega ),}</annotation>

</semantics>

</math><img src="./a91dd3d3b17a24fa0af3eea45f4ff1e1672abcd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.718ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {\Omega }}=(0,\ 0,\ \Omega ),}" loading="lazy"> namely,

<mfrac>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\hat {\boldsymbol {u}}}={\boldsymbol {\Omega \times }}{\hat {\boldsymbol {u}}}\ ,}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {u}}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {u}}}}</annotation>

</semantics>

</math><img src="./2a90aba65a92ed5d98ec22961b0ce7165060c0c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.343ex;" alt="{\displaystyle {\hat {\boldsymbol {u}}}}" loading="lazy"> is either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\imath }}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\imath }}}}</annotation>

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</math><img src="./d4bc189ac595a50b9975ac58220c7ff0295b37ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.237ex; height:2.343ex;" alt="{\displaystyle {\hat {\boldsymbol {\imath }}}}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\jmath }}}.}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\jmath }}}.}</annotation>

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</math><img src="./800d15185ce3543d5180a1f14cc4c4d34e131ca1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.032ex; height:2.676ex;" alt="{\displaystyle {\hat {\boldsymbol {\jmath }}}.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Time_derivatives_in_the_two_frames">Time derivatives in the two frames</h3></div>

Introduce unit vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\imath }}},\ {\hat {\boldsymbol {\jmath }}},\ {\hat {\boldsymbol {k}}}}">

<mover>

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\imath }}},\ {\hat {\boldsymbol {\jmath }}},\ {\hat {\boldsymbol {k}}}}</annotation>

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</math><img src="./227dda128fc49046fc65b7fec4331b9434a087a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.255ex; height:3.176ex;" alt="{\displaystyle {\hat {\boldsymbol {\imath }}},\ {\hat {\boldsymbol {\jmath }}},\ {\hat {\boldsymbol {k}}}}" loading="lazy">, 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega (t)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Omega (t)}</annotation>

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</math><img src="./dcd81d597f937f23da35708c4b1e9d58b80fc87f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.327ex; height:2.843ex;" alt="{\displaystyle \Omega (t)}" loading="lazy"> about an axis along the rotation vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\Omega }}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\Omega }}(t)}</annotation>

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</math><img src="./1ac5c426e970e45d0f10acbbda7f4ad7554ff5ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.58ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {\Omega }}(t)}" loading="lazy"> then each unit vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {u}}}}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {u}}}}</annotation>

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</math><img src="./2a90aba65a92ed5d98ec22961b0ce7165060c0c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.343ex;" alt="{\displaystyle {\hat {\boldsymbol {u}}}}" loading="lazy"> of the rotating coordinate system (such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\imath }}},\ {\hat {\boldsymbol {\jmath }}},}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\imath }}},\ {\hat {\boldsymbol {\jmath }}},}</annotation>

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</math><img src="./278105d0c7c82a65cf9de05d5ab04da9726257f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.883ex; height:2.676ex;" alt="{\displaystyle {\hat {\boldsymbol {\imath }}},\ {\hat {\boldsymbol {\jmath }}},}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {k}}}}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {k}}}}</annotation>

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</math><img src="./67b4a9ff689b66b872eff5546f951902864ff519.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.404ex; height:2.843ex;" alt="{\displaystyle {\hat {\boldsymbol {k}}}}" loading="lazy">) abides by the following equation:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\hat {\boldsymbol {u}}}={\boldsymbol {\Omega }}\times {\boldsymbol {\hat {u}}}\ .}</annotation>

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So if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(t)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle R(t)}</annotation>

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</math><img src="./83f5825ff7b37103bb1978b2cecddf8423a0d3f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.413ex; height:2.843ex;" alt="{\displaystyle R(t)}" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\Omega }}\times =R'(t)\cdot R(t)^{T}}">

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\Omega }}\times =R'(t)\cdot R(t)^{T}}</annotation>

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</math><img src="./ef391a9b625dc5ea4b2b498f97d3ba913b4dc084.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.417ex; height:3.176ex;" alt="{\displaystyle {\boldsymbol {\Omega }}\times =R'(t)\cdot R(t)^{T}}" loading="lazy">.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {f}}}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {f}}}</annotation>

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</math><img src="./9637dfaeb3214b577efe019ad53b8269f042e306.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.45ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {f}}}" loading="lazy"> is a vector function that is written as<a href="#cite_note-13">[note 2]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {f}}(t)=f_{1}(t){\hat {\boldsymbol {\imath }}}+f_{2}(t){\hat {\boldsymbol {\jmath }}}+f_{3}(t){\hat {\boldsymbol {k}}}\ ,}</annotation>

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</math>

and we want to examine its first derivative then (using the <a href="Product_rule" title="Product rule">product rule</a> of differentiation):<a href="#cite_note-Lanczos-14">[12]</a><a href="#cite_note-Taylor-15">[13]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {f}}&={\frac {\mathrm {d} f_{1}}{\mathrm {d} t}}{\hat {\boldsymbol {\imath }}}+{\frac {\mathrm {d} {\hat {\boldsymbol {\imath }}}}{\mathrm {d} t}}f_{1}+{\frac {\mathrm {d} f_{2}}{\mathrm {d} t}}{\hat {\boldsymbol {\jmath }}}+{\frac {\mathrm {d} {\hat {\boldsymbol {\jmath }}}}{\mathrm {d} t}}f_{2}+{\frac {\mathrm {d} f_{3}}{\mathrm {d} t}}{\hat {\boldsymbol {k}}}+{\frac {\mathrm {d} {\hat {\boldsymbol {k}}}}{\mathrm {d} t}}f_{3}\\&={\frac {\mathrm {d} f_{1}}{\mathrm {d} t}}{\hat {\boldsymbol {\imath }}}+{\frac {\mathrm {d} f_{2}}{\mathrm {d} t}}{\hat {\boldsymbol {\jmath }}}+{\frac {\mathrm {d} f_{3}}{\mathrm {d} t}}{\hat {\boldsymbol {k}}}+\left[{\boldsymbol {\Omega }}\times \left(f_{1}{\hat {\boldsymbol {\imath }}}+f_{2}{\hat {\boldsymbol {\jmath }}}+f_{3}{\hat {\boldsymbol {k}}}\right)\right]\\&=\left({\frac {\mathrm {d} {\boldsymbol {f}}}{\mathrm {d} t}}\right)_{\mathrm {r} }+{\boldsymbol {\Omega }}\times {\boldsymbol {f}}\end{aligned}}}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\mathrm {d} {\boldsymbol {f}}}{\mathrm {d} t}}\right)_{\mathrm {r} }}">

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {\mathrm {d} {\boldsymbol {f}}}{\mathrm {d} t}}\right)_{\mathrm {r} }}</annotation>

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</math><img src="./f2585e9e2e3a09190297c66e8cb7ce8be91295af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.877ex; height:6.176ex;" alt="{\displaystyle \left({\frac {\mathrm {d} {\boldsymbol {f}}}{\mathrm {d} t}}\right)_{\mathrm {r} }}" loading="lazy"> denotes the rate of change of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {f}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {f}}}</annotation>

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</math><img src="./9637dfaeb3214b577efe019ad53b8269f042e306.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.45ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {f}}}" loading="lazy"> as observed in the rotating coordinate system. As a shorthand the differentiation is expressed as:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {f}}=\left[\left({\frac {\mathrm {d} }{\mathrm {d} t}}\right)_{\mathrm {r} }+{\boldsymbol {\Omega }}\times \right]{\boldsymbol {f}}\ .}</annotation>

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</math>

This result is also known as the <a href="Transport_theorem" title="Transport theorem">transport theorem</a> in analytical dynamics and is also sometimes referred to as the basic kinematic equation.<a href="#cite_note-16">[14]</a>

<div class="mw-heading mw-heading3"><h3 id="Relation_between_velocities_in_the_two_frames">Relation between velocities in the two frames</h3></div>

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

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} \ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\ .}</annotation>

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</math><img src="./b9048757e5d76acba69abe93dac60b50c74fec5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.286ex; height:5.509ex;" alt="{\displaystyle \mathbf {v} \ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\ .}" loading="lazy"></dd></dl>

The time derivative of a position <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {r}}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {r}}(t)}</annotation>

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</math><img src="./aa34186c096c01921320d71c58317ee7253c84fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.879ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {r}}(t)}" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {r}}(t),}">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {r}}(t),}</annotation>

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</math><img src="./57f833e71e136c58c752d858acf078427049b9d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.526ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {r}}(t),}" loading="lazy"> the <a href="Velocity" title="Velocity">velocities</a> in the two reference frames are related by the equation

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {v_{i}} \ {\stackrel {\mathrm {def} }{=}}\ \left({\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\right)_{\mathrm {i} }\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}=\left[\left({\frac {\mathrm {d} }{\mathrm {d} t}}\right)_{\mathrm {r} }+{\boldsymbol {\Omega }}\times \right]{\boldsymbol {r}}=\left({\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\right)_{\mathrm {r} }+{\boldsymbol {\Omega }}\times \mathbf {r} =\mathbf {v} _{\mathrm {r} }+{\boldsymbol {\Omega }}\times \mathbf {r} \ ,}</annotation>

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</math><img src="./7bcbdea46f9392f035de7365e98a4e5970c031f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:75.26ex; height:6.176ex;" alt="{\displaystyle \mathbf {v_{i}} \ {\stackrel {\mathrm {def} }{=}}\ \left({\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\right)_{\mathrm {i} }\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}=\left[\left({\frac {\mathrm {d} }{\mathrm {d} t}}\right)_{\mathrm {r} }+{\boldsymbol {\Omega }}\times \right]{\boldsymbol {r}}=\left({\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\right)_{\mathrm {r} }+{\boldsymbol {\Omega }}\times \mathbf {r} =\mathbf {v} _{\mathrm {r} }+{\boldsymbol {\Omega }}\times \mathbf {r} \ ,}" loading="lazy"></dd></dl>

where subscript <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {i} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {i} }</annotation>

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</math><img src="./18f0f09f6fc40e634d34aed6e205ac0f7a40e062.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:2.176ex;" alt="{\displaystyle \mathrm {i} }" loading="lazy"> means the inertial frame of reference, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {r} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {r} }</annotation>

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</math><img src="./e8c0f15bc8d24e551d7d0618c476a9b04448d695.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.912ex; height:1.676ex;" alt="{\displaystyle \mathrm {r} }" loading="lazy"> means the rotating frame of reference.

<div class="mw-heading mw-heading3"><h3 id="Relation_between_accelerations_in_the_two_frames">Relation between accelerations in the two frames</h3></div>

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

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</mfrac>

</mrow>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{\mathrm {i} }\ {\stackrel {\mathrm {def} }{=}}\ \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} }+{\boldsymbol {\Omega }}\times \right]\left[\left({\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\right)_{\mathrm {r} }+{\boldsymbol {\Omega }}\times \mathbf {r} \right]\ ,}</annotation>

</semantics>

</math><img src="./6c7e47b6eca1ceff62acfc705568a358cd648829.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:64.65ex; height:6.343ex;" alt="{\displaystyle \mathbf {a} _{\mathrm {i} }\ {\stackrel {\mathrm {def} }{=}}\ \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} }+{\boldsymbol {\Omega }}\times \right]\left[\left({\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\right)_{\mathrm {r} }+{\boldsymbol {\Omega }}\times \mathbf {r} \right]\ ,}" loading="lazy"></dd></dl>

where subscript <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {i} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {i} }</annotation>

</semantics>

</math><img src="./18f0f09f6fc40e634d34aed6e205ac0f7a40e062.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:2.176ex;" alt="{\displaystyle \mathrm {i} }" loading="lazy"> means the inertial frame of reference, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {r} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {r} }</annotation>

</semantics>

</math><img src="./e8c0f15bc8d24e551d7d0618c476a9b04448d695.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.912ex; height:1.676ex;" alt="{\displaystyle \mathrm {r} }" loading="lazy"> the rotating frame of reference, and where the expression, again, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\Omega }}\times }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\Omega }}\times }</annotation>

</semantics>

</math><img src="./c2177ad64bc944ff9a0dd1a7a42ad25a7a0ec35c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.739ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\Omega }}\times }" loading="lazy"> in the bracketed expression on the left is to be interpreted as an <a href="Operator_(mathematics)" title="Operator (mathematics)">operator</a> working onto the bracketed expression on the right.

As <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\Omega }}\times {\boldsymbol {\Omega }}={\boldsymbol {0}}}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\Omega }}\times {\boldsymbol {\Omega }}={\boldsymbol {0}}}</annotation>

</semantics>

</math><img src="./e40d07143fec074aaceef79dcbeb4ed3d6c10625.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.138ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\Omega }}\times {\boldsymbol {\Omega }}={\boldsymbol {0}}}" loading="lazy">, the first time derivatives of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\Omega }}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\Omega }}}</annotation>

</semantics>

</math><img src="./9c95a28793dac7413de04992d822822f7e2dba16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\Omega }}}" loading="lazy"> inside either frame, when expressed with respect to the basis of e.g. the inertial frame, coincide.

Carrying out the <a href="Derivative" title="Derivative">differentiations</a> and re-arranging some terms yields the acceleration relative to the rotating reference frame, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} _{\mathrm {r} }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{\mathrm {r} }}</annotation>

</semantics>

</math><img src="./9add9775278740fc942e56055ffe445f462c3668.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.176ex; height:2.009ex;" alt="{\displaystyle \mathbf {a} _{\mathrm {r} }}" loading="lazy">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{\mathrm {r} }=\mathbf {a} _{\mathrm {i} }-2{\boldsymbol {\Omega }}\times \mathbf {v} _{\mathrm {r} }-{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times \mathbf {r} )-{\frac {\mathrm {d} {\boldsymbol {\Omega }}}{\mathrm {d} t}}\times \mathbf {r} }</annotation>

</semantics>

</math><img src="./837d3c3ceb6786ba58c1fe8bfccc35421fb44fd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:44.464ex; height:5.509ex;" alt="{\displaystyle \mathbf {a} _{\mathrm {r} }=\mathbf {a} _{\mathrm {i} }-2{\boldsymbol {\Omega }}\times \mathbf {v} _{\mathrm {r} }-{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times \mathbf {r} )-{\frac {\mathrm {d} {\boldsymbol {\Omega }}}{\mathrm {d} t}}\times \mathbf {r} }" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} _{\mathrm {r} }\ {\stackrel {\mathrm {def} }{=}}\ \left({\tfrac {\mathrm {d} ^{2}\mathbf {r} }{\mathrm {d} t^{2}}}\right)_{\mathrm {r} }}">

<msub>

</mrow>

</mrow>

</msub>

<mover>

</mrow>

</mrow>

</mover>

</mrow>

<msub>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{\mathrm {r} }\ {\stackrel {\mathrm {def} }{=}}\ \left({\tfrac {\mathrm {d} ^{2}\mathbf {r} }{\mathrm {d} t^{2}}}\right)_{\mathrm {r} }}</annotation>

</semantics>

</math><img src="./2470ecfd8a458d8a17fbc8c7ffad450a64b7591f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.607ex; height:4.843ex;" alt="{\displaystyle \mathbf {a} _{\mathrm {r} }\ {\stackrel {\mathrm {def} }{=}}\ \left({\tfrac {\mathrm {d} ^{2}\mathbf {r} }{\mathrm {d} t^{2}}}\right)_{\mathrm {r} }}" loading="lazy"> is the apparent acceleration in the rotating reference frame, the term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times \mathbf {r} )}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times \mathbf {r} )}</annotation>

</semantics>

</math><img src="./d15c67e070622d658f4fc6de2d32059a93d878c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.263ex; height:2.843ex;" alt="{\displaystyle -{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times \mathbf {r} )}" loading="lazy"> represents <a href="Centrifugal_acceleration" class="mw-redirect" title="Centrifugal acceleration">centrifugal acceleration</a>, and the term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2{\boldsymbol {\Omega }}\times \mathbf {v} _{\mathrm {r} }}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -2{\boldsymbol {\Omega }}\times \mathbf {v} _{\mathrm {r} }}</annotation>

</semantics>

</math><img src="./df03842fe72be9bb50f9fc5f655c0bbccb16a61f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.03ex; height:2.509ex;" alt="{\displaystyle -2{\boldsymbol {\Omega }}\times \mathbf {v} _{\mathrm {r} }}" loading="lazy"> is the <a href="Coriolis_acceleration" class="mw-redirect" title="Coriolis acceleration">Coriolis acceleration</a>. The last term, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\tfrac {\mathrm {d} {\boldsymbol {\Omega }}}{\mathrm {d} t}}\times \mathbf {r} }">

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\tfrac {\mathrm {d} {\boldsymbol {\Omega }}}{\mathrm {d} t}}\times \mathbf {r} }</annotation>

</semantics>

</math><img src="./6aedb652cb192fc06117dcc16e9ec557c13f9114.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.866ex; height:3.843ex;" alt="{\displaystyle -{\tfrac {\mathrm {d} {\boldsymbol {\Omega }}}{\mathrm {d} t}}\times \mathbf {r} }" loading="lazy">, is the <a href="Euler_acceleration" class="mw-redirect" title="Euler acceleration">Euler acceleration</a> and is zero in uniformly rotating frames.

<div class="mw-heading mw-heading3"><h3 id="Newton's_second_law_in_the_two_frames">Newton's second law in the two frames</h3></div>

When the expression for acceleration is multiplied by the mass of the particle, the three extra terms on the right-hand side result in <a href="Fictitious_force" title="Fictitious force">fictitious forces</a> in the rotating reference frame, that is, apparent forces that result from being in a <a href="Non-inertial_reference_frame" title="Non-inertial reference frame">non-inertial reference frame</a>, rather than from any physical interaction between bodies.

Using <a href="Newton's_laws_of_motion" title="Newton's laws of motion">Newton's second law of motion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =m\mathbf {a} ,}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =m\mathbf {a} ,}</annotation>

</semantics>

</math><img src="./93aa54e6c7e8df66d85d06b6eb0b0a2d3ec4ce20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.768ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} =m\mathbf {a} ,}" loading="lazy"> we obtain:<a href="#cite_note-Arnold-1">[1]</a><a href="#cite_note-Lanczos-14">[12]</a><a href="#cite_note-Taylor-15">[13]</a><a href="#cite_note-Landau-17">[15]</a><a href="#cite_note-Hand-18">[16]</a>

<ul><li>the <a href="Coriolis_force" title="Coriolis force">Coriolis force</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{\mathrm {Coriolis} }=-2m{\boldsymbol {\Omega }}\times \mathbf {v} _{\mathrm {r} }}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{\mathrm {Coriolis} }=-2m{\boldsymbol {\Omega }}\times \mathbf {v} _{\mathrm {r} }}</annotation>

</semantics>

</math></li>

<li>the <a href="Centrifugal_force_(fictitious)" class="mw-redirect" title="Centrifugal force (fictitious)">centrifugal force</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{\mathrm {centrifugal} }=-m{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times \mathbf {r} )}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{\mathrm {centrifugal} }=-m{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times \mathbf {r} )}</annotation>

</semantics>

</math></li>

<li>and the <a href="Euler_force" title="Euler force">Euler force</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{\mathrm {Euler} }=-m{\frac {\mathrm {d} {\boldsymbol {\Omega }}}{\mathrm {d} t}}\times \mathbf {r} }">

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{\mathrm {Euler} }=-m{\frac {\mathrm {d} {\boldsymbol {\Omega }}}{\mathrm {d} t}}\times \mathbf {r} }</annotation>

</semantics>

</math></li></ul>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> is the mass of the object being acted upon by these <a href="Fictitious_force" title="Fictitious force">fictitious forces</a>. Notice that all three forces vanish when the frame is not rotating, that is, when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\Omega }}=0\ .}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\Omega }}=0\ .}</annotation>

</semantics>

</math><img src="./565018445fe417aa920fe15d1ef475f5e5a9a4b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.42ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\Omega }}=0\ .}" loading="lazy">

For completeness, the inertial acceleration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} _{\mathrm {i} }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{\mathrm {i} }}</annotation>

</semantics>

</math><img src="./4e5f0b3c39727a761e8685fe7aa73e1b0a70a696.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.989ex; height:2.009ex;" alt="{\displaystyle \mathbf {a} _{\mathrm {i} }}" loading="lazy"> due to impressed external forces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{\mathrm {imp} }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{\mathrm {imp} }}</annotation>

</semantics>

</math><img src="./20c26eb013b20287f4a65be81984b08e1c733e89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.655ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} _{\mathrm {imp} }}" loading="lazy"> can be determined from the total physical force in the inertial (non-rotating) frame (for example, force from physical interactions such as <a href="Electromagnetism" title="Electromagnetism">electromagnetic forces</a>) using <a href="Newton's_laws_of_motion" title="Newton's laws of motion">Newton's second law</a> in the inertial frame:

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{\mathrm {imp} }=m\mathbf {a} _{\mathrm {i} }}</annotation>

</semantics>

</math>

Newton's law in the rotating frame then becomes

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

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</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {F_{\mathrm {r} }} =\mathbf {F} _{\mathrm {imp} }+\mathbf {F} _{\mathrm {centrifugal} }+\mathbf {F} _{\mathrm {Coriolis} }+\mathbf {F} _{\mathrm {Euler} }=m\mathbf {a_{\mathrm {r} }} \ .}</annotation>

</semantics>

</math><img src="./d77b0d4a69ffe37acf7b598727c37ed78eb95994.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:50.225ex; height:2.843ex;" alt="{\displaystyle \mathbf {F_{\mathrm {r} }} =\mathbf {F} _{\mathrm {imp} }+\mathbf {F} _{\mathrm {centrifugal} }+\mathbf {F} _{\mathrm {Coriolis} }+\mathbf {F} _{\mathrm {Euler} }=m\mathbf {a_{\mathrm {r} }} \ .}" loading="lazy"></dd></dl></dd></dl>

In other words, to handle the laws of motion in a rotating reference frame:<a href="#cite_note-Hand-18">[16]</a><a href="#cite_note-Pui-19">[17]</a><a href="#cite_note-Taylor2-20">[18]</a>

/* start https://en.wikipedia.org/ */

.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}

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</style><blockquote class="templatequote">Treat the fictitious forces like real forces, and pretend you are in an inertial frame.</blockquote><div class="templatequotecite">— Louis N. Hand, Janet D. Finch Analytical Mechanics, p. 267</div>

<blockquote class="templatequote">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.</blockquote><div class="templatequotecite">— HS Hans & SP Pui: Mechanics; p. 341</div>

<blockquote class="templatequote">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. </blockquote><div class="templatequotecite">— John R. Taylor: Classical Mechanics; p. 328</div>

<div class="mw-heading mw-heading2"><h2 id="Use_in_magnetic_resonance">Use in magnetic resonance</h2></div>

It is convenient to consider <a href="Nuclear_magnetic_resonance" title="Nuclear magnetic resonance">magnetic resonance</a> in a frame that rotates at the <a href="Larmor_frequency" class="mw-redirect" title="Larmor frequency">Larmor frequency</a> of the spins. This is illustrated in the animation below. The <a href="Rotating_wave_approximation" class="mw-redirect" title="Rotating wave approximation">rotating wave approximation</a> may also be used.

<ul><li><a href="Absolute_rotation" title="Absolute rotation">Absolute rotation</a></li>

<li><a href="Centrifugal_force_(rotating_reference_frame)" class="mw-redirect" title="Centrifugal force (rotating reference frame)">Centrifugal force (rotating reference frame)</a> Centrifugal force as seen from systems rotating about a fixed axis</li>

<li><a href="Coriolis_force" title="Coriolis force">Coriolis force</a> The effect of the Coriolis force on the Earth and other rotating systems</li>

<li><a href="Inertial_frame_of_reference" title="Inertial frame of reference">Inertial frame of reference</a></li>

<li><a href="Non-inertial_frame" class="mw-redirect" title="Non-inertial frame">Non-inertial frame</a></li>

<li><a href="Fictitious_force" title="Fictitious force">Fictitious force</a> A more general treatment of the subject of this article</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFVladimir_Igorević_Arnolʹd1989" class="citation book cs1">Vladimir Igorević Arnolʹd (1989). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Pd8-s6rOt_cC&q=%22additional+terms+called+inertial+forces.+This+allows+us+to+detect+experimentally%22&pg=PT149">Mathematical Methods of Classical Mechanics</a> (2nd ed.). Springer. p. 130. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-96890-2</bdi>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFRobert_ResnickDavid_Halliday1966" class="citation book cs1">Robert Resnick & David Halliday (1966). <a rel="nofollow" class="external text" href="https://archive.org/details/physics00resn">Physics</a>. Wiley. p. <a rel="nofollow" class="external text" href="https://archive.org/details/physics00resn/page/121">121</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-34524-5</bdi>.</cite>

</li>

<li id="cite_note-Marsden-3"><a href="#cite_ref-Marsden_3-0">^</a> <cite id="CITEREFJerrold_E._MarsdenTudor_S._Ratiu1999" class="citation book cs1">Jerrold E. Marsden; Tudor S. Ratiu (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=I2gH9ZIs-3AC&pg=PA251">Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems</a>. Springer. p. 251. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-98643-X</bdi>.</cite>

</li>

<li id="cite_note-Taylor_A-4"><a href="#cite_ref-Taylor_A_4-0">^</a> <cite id="CITEREFJohn_Robert_Taylor2005" class="citation book cs1">John Robert Taylor (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=P1kCtNr-pJsC&pg=PP1">Classical Mechanics</a>. University Science Books. p. 343. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-891389-22-X</bdi>.</cite>

</li>

<li id="cite_note-Marion-5"><a href="#cite_ref-Marion_5-0">^</a> <cite id="CITEREFStephen_T._ThorntonJerry_B._Marion2004" class="citation book cs1">Stephen T. Thornton & Jerry B. Marion (2004). "Chapter 10". Classical Dynamics of Particles and Systems (5th ed.). Belmont CA: Brook/Cole. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-534-40896-6</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/52806908">52806908</a>.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFDavid_McNaughton" class="citation web cs1">David McNaughton. <a rel="nofollow" class="external text" href="http://dlmcn.com/circle.html">"Centrifugal and Coriolis Effects"</a>. Retrieved 2008-05-18.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFDavid_P._Stern" class="citation web cs1">David P. Stern. <a rel="nofollow" class="external text" href="http://www.phy6.org/stargaze/Lframes2.htm">"Frames of reference: The centrifugal force"</a>. Retrieved 2008-10-26.</cite>

</li>

<li id="cite_note-Morin-8"><a href="#cite_ref-Morin_8-0">^</a> <cite id="CITEREFDavid_Morin2008" class="citation book cs1">David Morin (2008). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontocl00mori">Introduction to classical mechanics: with problems and solutions</a>. Cambridge University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontocl00mori/page/469">469</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-87622-3</bdi>. <q>acceleration azimuthal Morin.</q></cite>

</li>

<li id="cite_note-Fowles-9"><a href="#cite_ref-Fowles_9-0">^</a> <cite id="CITEREFGrant_R._FowlesGeorge_L._Cassiday1999" class="citation book cs1">Grant R. Fowles & George L. Cassiday (1999). Analytical Mechanics (6th ed.). Harcourt College Publishers. p. 178.</cite>

</li>

<li id="cite_note-Battin-10"><a href="#cite_ref-Battin_10-0">^</a> <cite id="CITEREFRichard_H_Battin1999" class="citation book cs1">Richard H Battin (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OjH7aVhiGdcC&q=%22Euler+acceleration%22&pg=PA102">An introduction to the mathematics and methods of astrodynamics</a>. Reston, VA: <a href="American_Institute_of_Aeronautics_and_Astronautics" title="American Institute of Aeronautics and Astronautics">American Institute of Aeronautics and Astronautics</a>. p. 102. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-56347-342-9</bdi>.</cite>

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFJerrold_E._MarsdenTudor_S._Ratiu1999" class="citation book cs1">Jerrold E. Marsden; Tudor S. Ratiu (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=I2gH9ZIs-3AC&pg=PP1">Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems</a>. Springer. p. 251. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-98643-X</bdi>.</cite>

</li>

<li id="cite_note-Lanczos-14">^ <a href="#cite_ref-Lanczos_14-0">a</a> <a href="#cite_ref-Lanczos_14-1">b</a> <cite id="CITEREFCornelius_Lanczos1986" class="citation book cs1">Cornelius Lanczos (1986). <a rel="nofollow" class="external text" href="https://books.google.com/books?num=10&btnG=Google+Search">The Variational Principles of Mechanics</a> (Reprint of Fourth Edition of 1970 ed.). <a href="Dover_Publications" title="Dover Publications">Dover Publications</a>. Chapter 4, §5. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-65067-7</bdi>.</cite>

</li>

<li id="cite_note-Taylor-15">^ <a href="#cite_ref-Taylor_15-0">a</a> <a href="#cite_ref-Taylor_15-1">b</a> <cite id="CITEREFJohn_R_Taylor2005" class="citation book cs1">John R Taylor (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=P1kCtNr-pJsC&pg=PP1">Classical Mechanics</a>. University Science Books. p. 342. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-891389-22-X</bdi>.</cite>

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <cite id="CITEREFCorless" class="citation web cs1">Corless, Martin. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121024121222/https://engineering.purdue.edu/AAE/Academics/Courses/aae203/2003/fall/aae203F03supp.pdf">"Kinematics"</a> (PDF). Aeromechanics I Course Notes. <a href="Purdue_University" title="Purdue University">Purdue University</a>. p. 213. Archived from <a rel="nofollow" class="external text" href="https://engineering.purdue.edu/AAE/Academics/Courses/aae203/2003/fall/aae203F03supp.pdf">the original</a> (PDF) on 24 October 2012. Retrieved 18 July 2011.</cite>

</li>

<li id="cite_note-Landau-17"><a href="#cite_ref-Landau_17-0">^</a> <cite id="CITEREFLD_LandauLM_Lifshitz1976" class="citation book cs1">LD Landau & LM Lifshitz (1976). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=e-xASAehg1sC&pg=PA40">Mechanics</a> (Third ed.). Butterworth-Heinemann. p. 128. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7506-2896-9</bdi>.</cite>

</li>

<li id="cite_note-Hand-18">^ <a href="#cite_ref-Hand_18-0">a</a> <a href="#cite_ref-Hand_18-1">b</a> <cite id="CITEREFLouis_N._HandJanet_D._Finch1998" class="citation book cs1">Louis N. Hand; Janet D. Finch (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1J2hzvX2Xh8C&q=Hand+inauthor:Finch&pg=PA267">Analytical Mechanics</a>. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. 267. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-57572-9</bdi>.</cite>

</li>

<li id="cite_note-Pui-19"><a href="#cite_ref-Pui_19-0">^</a> <cite id="CITEREFHS_HansSP_Pui2003" class="citation book cs1">HS Hans & SP Pui (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mgVW00YV3zAC&q=inertial+force+%22rotating+frame%22&pg=PA341">Mechanics</a>. Tata McGraw-Hill. p. 341. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-047360-9</bdi>.</cite>

</li>

<li id="cite_note-Taylor2-20"><a href="#cite_ref-Taylor2_20-0">^</a> <cite id="CITEREFJohn_R_Taylor2005" class="citation book cs1">John R Taylor (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=P1kCtNr-pJsC&pg=PP1">Classical Mechanics</a>. University Science Books. p. 328. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-891389-22-X</bdi>.</cite>

</li>

</ol></div>

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<annotation encoding="application/x-tex">{\displaystyle x,y,z}</annotation>

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</math><img src="./bbeca34b28f569a407ef74a955d041df9f360268.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.641ex; height:2.009ex;" alt="{\displaystyle x,y,z}" loading="lazy"> are functions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x',y',z',}">

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</math><img src="./d553af3090132330a761f3c4059eda336448ab9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.349ex; height:2.843ex;" alt="{\displaystyle x',y',z',}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t.}">

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<annotation encoding="application/x-tex">{\displaystyle \left(x',y',z'\right)=(0,0,0)}</annotation>

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</math><img src="./294b99b8df8e9b87cd4751db400b9a5a354a1046.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.975ex; height:3.009ex;" alt="{\displaystyle \left(x',y',z'\right)=(0,0,0)}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)=(0,0,0).}">

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<annotation encoding="application/x-tex">{\displaystyle (x,y,z)=(0,0,0).}</annotation>

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</math><img src="./d22ffeba8b9f9087c2cfb4d3a88a0698ea4678db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.56ex; height:2.843ex;" alt="{\displaystyle (x,y,z)=(0,0,0).}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle f_{1},f_{2},f_{3}}</annotation>

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</math><img src="./8005d01ebb187d915834307d071c2ccb3aa4a852.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.648ex; height:2.509ex;" alt="{\displaystyle f_{1},f_{2},f_{3}}" loading="lazy"> are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {f}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {f}}}</annotation>

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</math><img src="./9637dfaeb3214b577efe019ad53b8269f042e306.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.45ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {f}}}" loading="lazy">'s coordinates with respect to the rotating basis vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\imath }}},\ {\hat {\boldsymbol {\jmath }}},\ {\hat {\boldsymbol {k}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\imath }}},\ {\hat {\boldsymbol {\jmath }}},\ {\hat {\boldsymbol {k}}}}</annotation>

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</math><img src="./227dda128fc49046fc65b7fec4331b9434a087a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.255ex; height:3.176ex;" alt="{\displaystyle {\hat {\boldsymbol {\imath }}},\ {\hat {\boldsymbol {\jmath }}},\ {\hat {\boldsymbol {k}}}}" loading="lazy"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {f}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {f}}}</annotation>

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</math><img src="./9637dfaeb3214b577efe019ad53b8269f042e306.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.45ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {f}}}" loading="lazy">'s coordinates with respect to the inertial frame are not used). Consequently, at any given instant, the rate of change of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {f}}}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {f}}}</annotation>

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</math><img src="./9637dfaeb3214b577efe019ad53b8269f042e306.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.45ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {f}}}" loading="lazy"> with respect to these rotating coordinates is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} f_{1}}{\mathrm {d} t}}{\hat {\boldsymbol {\imath }}}+{\frac {\mathrm {d} f_{2}}{\mathrm {d} t}}{\hat {\boldsymbol {\jmath }}}+{\frac {\mathrm {d} f_{3}}{\mathrm {d} t}}{\hat {\boldsymbol {k}}}.}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} f_{1}}{\mathrm {d} t}}{\hat {\boldsymbol {\imath }}}+{\frac {\mathrm {d} f_{2}}{\mathrm {d} t}}{\hat {\boldsymbol {\jmath }}}+{\frac {\mathrm {d} f_{3}}{\mathrm {d} t}}{\hat {\boldsymbol {k}}}.}</annotation>

</semantics>

</math><img src="./77a7c5c830529c3aab30b1894f984b45a763a76a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.32ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} f_{1}}{\mathrm {d} t}}{\hat {\boldsymbol {\imath }}}+{\frac {\mathrm {d} f_{2}}{\mathrm {d} t}}{\hat {\boldsymbol {\jmath }}}+{\frac {\mathrm {d} f_{3}}{\mathrm {d} t}}{\hat {\boldsymbol {k}}}.}" loading="lazy"> So for example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}\equiv 1}">

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<annotation encoding="application/x-tex">{\displaystyle f_{1}\equiv 1}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle f_{2}=f_{3}\equiv 0}</annotation>

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</math><img src="./db0b076b12d3047c571cecd0e8613fbd9f015ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.746ex; height:2.509ex;" alt="{\displaystyle f_{2}=f_{3}\equiv 0}" loading="lazy"> are constants, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {f}}\equiv {\hat {\boldsymbol {\imath }}}}">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {f}}\equiv {\hat {\boldsymbol {\imath }}}}</annotation>

</semantics>

</math><img src="./c8913953bb075d748d99be0093915de7b608c986.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.785ex; height:2.676ex;" alt="{\displaystyle {\boldsymbol {f}}\equiv {\hat {\boldsymbol {\imath }}}}" loading="lazy"> is just one of the rotating basis vectors and (as expected) its time rate of change with respect to these rotating coordinates is identically <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {0}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {0}}}</annotation>

</semantics>

</math><img src="./58fe04d1d35aac19861a0d7c9f7c374c88d42db0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {0}}}" loading="lazy"> (so the formula for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {f}}}">

<mfrac>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {f}}}</annotation>

</semantics>

</math><img src="./37b89eb663877434c11367fb10f9f2e222ff03d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.419ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {f}}}" loading="lazy"> given below implies that the derivative at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> of this rotating basis vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {f}}\equiv {\hat {\boldsymbol {\imath }}}}">

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {f}}\equiv {\hat {\boldsymbol {\imath }}}}</annotation>

</semantics>

</math><img src="./c8913953bb075d748d99be0093915de7b608c986.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.785ex; height:2.676ex;" alt="{\displaystyle {\boldsymbol {f}}\equiv {\hat {\boldsymbol {\imath }}}}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {i}}={\boldsymbol {\Omega }}(t)\times {\boldsymbol {i}}(t)}">

<mfrac>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {i}}={\boldsymbol {\Omega }}(t)\times {\boldsymbol {i}}(t)}</annotation>

</semantics>

</math><img src="./d357323cbd5a907c72fa6ffea9c9dcffbbde66e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.02ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {i}}={\boldsymbol {\Omega }}(t)\times {\boldsymbol {i}}(t)}" loading="lazy">); however, its rate of change with respect to the non-rotating inertial frame will not be constantly <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {0}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {0}}}</annotation>

</semantics>

</math><img src="./58fe04d1d35aac19861a0d7c9f7c374c88d42db0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {0}}}" loading="lazy"> except (of course) in the case where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\imath }}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\imath }}}}</annotation>

</semantics>

</math><img src="./d4bc189ac595a50b9975ac58220c7ff0295b37ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.237ex; height:2.343ex;" alt="{\displaystyle {\hat {\boldsymbol {\imath }}}}" loading="lazy"> is not moving in the inertial frame (this happens, for instance, when the axis of rotation is fixed as the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">-axis (assuming standard coordinates) in the inertial frame and also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\imath }}}\equiv (0,0,1)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\imath }}}\equiv (0,0,1)}</annotation>

</semantics>

</math><img src="./a9acb7084f5d27f31acde9bfbdfb4f8cf3ae2211.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.7ex; height:2.843ex;" alt="{\displaystyle {\hat {\boldsymbol {\imath }}}\equiv (0,0,1)}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\imath }}}\equiv (0,0,-1)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\imath }}}\equiv (0,0,-1)}</annotation>

</semantics>

</math><img src="./1c63bd2408f99b2c26b33c8be4078c570e175a91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.508ex; height:2.843ex;" alt="{\displaystyle {\hat {\boldsymbol {\imath }}}\equiv (0,0,-1)}" loading="lazy">).

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=49JwbrXcPjc">Animation clip</a> showing scenes as viewed from both an inertial frame and a rotating frame of reference, visualizing the Coriolis and centrifugal forces.</li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-04-17" href="https://en.wikipedia.org/wiki/?title=Rotating_reference_frame&oldid=1286094965">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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