Rotating Reference Frame

Rotating reference frame 1

Rotating reference frame

Classicalmechanics

Second law of motion

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A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertialreference frame. An everyday example of a rotating reference frame is the surface of the Earth. (This articleconsiders only frames rotating about a fixed axis. For more general rotations, see Euler angles.)

Fictitious forcesMain article: Fictitious forcesAll non-inertial reference frames exhibit fictitious forces. Rotating reference frames are characterized by threefictitious forces: the centrifugal force, the Coriolis force,and, for non-uniformly rotating reference frames, the Euler force.Scientists living in a rotating box can measure the speed and direction of their rotation by measuring these fictitiousforces. For example, Lon Foucault was able to show the Coriolis force that results from the Earth's rotation usingthe Foucault pendulum. If the Earth were to rotate many times faster, these fictitious forces could be felt by humans,as they are when on a spinning carousel.

Relating rotating frames to stationary framesThe following is a derivation of the formulas for accelerations as well as fictitious forces in a rotating frame. Itbegins 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 asseen in the two frames, and the acceleration relative to each frame. Using these accelerations, the fictitious forces areidentified by comparing Newton's second law as formulated in the two different frames.


Relation between positions in the two frames

To derive these fictitious forces, it's helpful to be able to convert between the coordinates of the rotatingreference frame and the coordinates of an inertial reference frame with the same origin. If the rotation isabout the axis with an angular velocity and the two reference frames coincide at time , thetransformation from rotating coordinates to inertial coordinates can be written

whereas the reverse transformation is

This result can be obtained from a rotation matrix.

Introduce the unit vectors representing standard unit basis vectors in the rotating frame. Thetime-derivatives of these unit vectors are found next. Suppose the frames are aligned at t = 0 and the z-axis is theaxis of rotation. Then for a counterclockwise rotation through angle t:

where the (x, y) components are expressed in the stationary frame. Likewise,

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

where . This result is the same as found using a vector cross product with the rotation vector

pointed along the z-axis of rotation , namely,

where is either or .

Time derivatives in the two frames

Introduce the unit vectors representing standard unit basis vectors in the rotating frame. As they rotatethey will remain normalized. If we let them rotate at the speed of about an axis then each unit vector of therotating coordinate system abides by the following equation:

Then if we have a vector function ,

and we want to examine its first dervative we have (using the product rule of differentiation):


where is the rate of change of as observed in the rotating coordinate system. As a shorthand the

differentiation is expressed as:

This result is also known as the Transport Theorem in analytical dynamics and is also sometimes referred to as theBasic Kinematic Equation.

Relation between velocities in the two framesA velocity of an object is the time-derivative of the object's position, or

The time derivative of a position in a rotating reference frame has two components, one from the explicit timedependence due to motion of the particle itself, and another from the frame's own rotation. Applying the result of theprevious subsection to the displacement , the velocities in the two reference frames are related by the equation

where subscript i means the inertial frame of reference, and r means the rotating frame of reference.

Relation between accelerations in the two framesAcceleration is the second time derivative of position, or the first time derivative of velocity

where subscript i means the inertial frame of reference. Carrying out the differentiations and re-arranging some termsyields the acceleration in the rotating reference frame

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

represents centrifugal acceleration, and the term is the coriolis acceleration.

Newton's second law in the two framesWhen the expression for acceleration is multiplied by the mass of the particle, the three extra terms on the right-handside result in fictitious forces in the rotating reference frame, that is, apparent forces that result from being in anon-inertial reference frame, rather than from any physical interaction between bodies.Using Newton's second law of motion , we obtain: the Coriolis force

the centrifugal force

and the Euler force


where is the mass of the object being acted upon by these fictitious forces. Notice that all three forces vanishwhen the frame is not rotating, that is, when

For completeness, the inertial acceleration due to impressed external forces can be determined from thetotal physical force in the inertial (non-rotating) frame (for example, force from physical interactions such aselectromagnetic forces) using Newton's second law in the inertial frame:

Newton's law in the rotating frame then becomes

In other words, to handle the laws of motion in a rotating reference frame:Treat the fictitious forces like real forces, and pretend you are in an inertial frame.

Louis N. Hand, Janet D. Finch Analytical Mechanics, p. 267Obviously, a rotating frame of reference is a case of a non-inertial frame. Thus the particle in addition to thereal force is acted upon by a fictitious force...The particle will move according to Newton's second law ofmotion if the total force acting on it is taken as the sum of the real and fictitious forces.

HS Hans & SP Pui: Mechanics; p. 341This equation has exactly the form of Newton's second law, except that in addition to F, the sum of all forcesidentified in the inertial frame, there is an extra term on the right...This means we can continue to use Newton'ssecond law in the noninertial frame provided we agree that in the noninertial frame we must add an extraforce-like term, often called the inertial force.

John R. Taylor: Classical Mechanics; p. 328

Centrifugal forceMain article: Centrifugal force (rotating reference frame)In classical mechanics, centrifugal force is an outward force associated with rotation. Centrifugal force is one ofseveral so-called pseudo-forces (also known as inertial forces), so named because, unlike real forces, they do notoriginate 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.


Coriolis effectMain article: Coriolis effect

Figure 1: In the inertial frame of reference (upperpart of the picture), the black object moves in a

straight line. However, the observer (red dot) whois standing in the rotating frame of reference(lower part of the picture) sees the object as

following a curved path.

The mathematical expression for the Coriolis force appeared in an1835 paper by a French scientist Gaspard-Gustave Coriolis inconnection with hydrodynamics, and also in the tidal equations ofPierre-Simon Laplace in 1778. Early in the 20th century, the termCoriolis force began to be used in connection with meteorology.

Perhaps the most commonly encountered rotating reference frame isthe Earth. Moving objects on the surface of the Earth experience aCoriolis force, and appear to veer to the right in the northernhemisphere, and to the left in the southern. Movements of air in theatmosphere and water in the ocean are notable examples of thisbehavior: rather than flowing directly from areas of high pressure tolow pressure, as they would on a non-rotating planet, winds andcurrents tend to flow to the right of this direction north of the equator,and to the left of this direction south of the equator. This effect isresponsible for the rotation of large cyclones (see Coriolis effects inmeteorology).

Euler force

Main article: Euler forceIn classical mechanics, the Euler acceleration (named for LeonhardEuler), also known as azimuthal acceleration or transverse acceleration is an acceleration that appears when anon-uniformly rotating reference frame is used for analysis of motion and there is variation in the angular velocity ofthe reference frame's axis. This article is restricted to a frame of reference that rotates about a fixed axis.

The Euler force is a fictitious force on a body that is related to the Euler acceleration by F =ma, where a is theEuler acceleration and m is the mass of the body.

Use in magnetic resonanceIt is convenient to consider magnetic resonance in a frame that rotates at the Larmor frequency of the spins. This isillustrated in the animation below. The rotating wave approximation may also be used.

References[1] http:/ / en. wikipedia. org/ w/ index. php?title=Template:Classical_mechanics& action=edit

External links Animation clip (http:/ / www. youtube. com/ watch?v=49JwbrXcPjc) showing scenes as viewed from both an

inertial frame and a rotating frame of reference, visualizing the Coriolis and centrifugal forces.

Article Sources and Contributors 6

Article Sources and ContributorsRotating reference frame Source: http://en.wikipedia.org/w/index.php?oldid=614551302 Contributors: Ancheta Wis, Bender235, Bh3u4m, Bobathon71, Boiler321, Brews ohare, Dario Gnani,Diptanshu.D, Dirac66, Eequor, ErkDemon, Frokor, Frdrick Lacasse, GavinMorley, Hairy Dude, Harald88, Iwfyita, J04n, Light current, Lunis, Matj Grabovsk, Montyv, Mor22, Mpatel,Netheril96, Paolo.dL, Pjacobi, Rjwilmsi, Robert Horning, Rossami, Saga City, Sanpaz, SebastianHelm, Skorkmaz, Spel-Punc-Gram, Ssiruuk25, Steuard, Svjo-2, Teapeat, The Anome,Thermochap, Udirock, WillowW, Wolfkeeper, Woodstone, 35 anonymous edits

Image Sources, Licenses and ContributorsImage:Corioliskraftanimation.gif Source: http://en.wikipedia.org/w/index.php?title=File:Corioliskraftanimation.gif License: GNU Free Documentation License Contributors: Hubi

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Rotating reference frameFictitious forcesRelating rotating frames to stationary framesRelation between positions in the two frames Time derivatives in the two frames Relation between velocities in the two frames Relation between accelerations in the two frames Newton's second law in the two frames

Centrifugal forceCoriolis effectEuler forceUse in magnetic resonanceReferencesExternal links

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Rotating Reference Frame