A rotating frame of reference is a special case of a
non-inertial reference frame
A non-inertial reference frame is a frame of reference that undergoes acceleration with respect to an inertial frame. An accelerometer at rest in a non-inertial frame will, in general, detect a non-zero acceleration. While the laws of motion are ...
that is
rotating
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
relative to an
inertial reference frame
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
. An everyday example of a rotating reference frame is the surface of the
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
. (This article considers only frames rotating about a fixed axis. For more general rotations, see
Euler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the Orientation (geometry), orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189†...
.)
Fictitious forces
All
non-inertial reference frame
A non-inertial reference frame is a frame of reference that undergoes acceleration with respect to an inertial frame. An accelerometer at rest in a non-inertial frame will, in general, detect a non-zero acceleration. While the laws of motion are ...
s exhibit
fictitious force
A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame.
It is related to Newton's second law of motion, which trea ...
s; rotating reference frames are characterized by three:
* the
centrifugal force
In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
,
* the
Coriolis force
In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...
,
and, for non-uniformly rotating reference frames,
* the
Euler force In classical mechanics, the Euler force is the fictitious tangential force
that appears when a non-uniformly rotating reference frame is used for analysis of motion and there is variation in the angular velocity of the reference frame's axes. The E ...
.
Scientists in a rotating box can measure the speed and direction of their rotation by measuring these fictitious forces. For example,
LĂ©on Foucault
Jean Bernard Léon Foucault (, ; ; 18 September 1819 – 11 February 1868) was a French physicist best known for his demonstration of the Foucault pendulum, a device demonstrating the effect of Earth's rotation. He also made an early measurement ...
was able to show the Coriolis force that results from Earth's rotation using the
Foucault pendulum
The Foucault pendulum or Foucault's pendulum is a simple device named after French physicist LĂ©on Foucault, conceived as an experiment to demonstrate the Earth's rotation. A long and heavy pendulum suspended from the high roof above a circular a ...
. If Earth were to rotate many times faster, these fictitious forces could be felt by humans, as they are when on a spinning
carousel
A carousel or carrousel (mainly North American English), merry-go-round (List of sovereign states, international), roundabout (British English), or hurdy-gurdy (an old term in Australian English, in South Australia, SA) is a type of amusement ...
.
Relating rotating frames to stationary frames
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.
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 rotating reference frame and the coordinates
of an
inertial reference frame
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
with the same origin.
[So are functions of and time Similarly are functions of and That these reference frames have the same origin means that for all if and only if ]
If the rotation is about the
axis with a constant
angular velocity
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
(so
and
which implies
for some constant
where
denotes the angle in the
-plane formed at time
by
and the
-axis),
and if the two reference frames coincide at time
(meaning
when
so take
or some other integer multiple of
), the transformation from rotating coordinates to inertial coordinates can be written
whereas the reverse transformation is
This result can be obtained from a
rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
:R = \begin
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end ...
.
Introduce the unit vectors
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
and the
-axis is the axis of rotation. Then for a counterclockwise rotation through angle
:
where the
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
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
with the rotation vector
pointed along the z-axis of rotation
namely,
where
is either
or
Time derivatives in the two frames
Introduce unit vectors
, 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
about an axis along the rotation vector
then each unit vector
of the rotating coordinate system (such as
or
) abides by the following equation:
So if
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
.
If
is a vector function that is written as
[So are 's coordinates with respect to the rotating basis vector ('s coordinates with respect to the inertial frame are not used). Consequently, at any given instant, the rate of change of with respect to these rotating coordinates is So for example, if and are constants, then is just one of the rotating basis vectors and (as expected) its time rate of change with respect to these rotating coordinates is identically (so the formula for given below implies that the derivative at time of this rotating basis vector is ); however, its rate of change with respect to the non-rotating inertial frame will not be constantly except (of course) in the case where is not moving in the inertial frame (this happens, for instance, when the axis of rotation is fixed as the -axis (assuming standard coordinates) in the inertial frame and also or ).]
and we want to examine its first derivative then (using the
product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
of differentiation):
where
denotes 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
The transport theorem (or transport equation, rate of change transport theorem or basic kinematic equation) is a vector equation that relates the time derivative of a Euclidean vector as evaluated in a non-rotating coordinate system to its time de ...
in analytical dynamics and is also sometimes referred to as the ''basic kinematic equation''.
Relation between velocities in the two frames
A velocity of an object is the time-derivative of the object's position, so
:
The time derivative of a position
in a rotating reference frame has two components, one from the explicit time dependence due to motion of the particle itself, and another from the frame's own rotation. Applying the result of the previous subsection to the displacement
the
velocities
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
in the two reference frames are related by the equation
:
where subscript
means the inertial frame of reference, and
means the rotating frame of reference.
Relation between accelerations in the two frames
Acceleration is the second time derivative of position, or the first time derivative of velocity
:
where subscript
means the inertial frame of reference,
the rotating frame of reference, and where the expression, again,
in the bracketed expression on the left is to be interpreted as an
operator working onto the bracketed expression on the right.
As
, the first time derivatives of
inside either frame, when expressed with respect to the basis of e.g. the inertial frame, coincide.
Carrying out the
differentiations and re-arranging some terms yields the acceleration ''relative to the rotating'' reference frame,
:
where
is the apparent acceleration in the rotating reference frame, the term
represents
centrifugal acceleration
In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is parallel ...
, and the term
is the
Coriolis acceleration
In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...
. The last term,
, is the
Euler acceleration In classical mechanics, the Euler force is the fictitious tangential force
that appears when a non-uniformly rotating reference frame is used for analysis of motion and there is variation in the angular velocity of the reference frame's axes. The E ...
and is zero in uniformly rotating frames.
Newton's second law in the two frames
When the expression for acceleration is multiplied by the mass of the particle, the three extra terms on the right-hand side result in
fictitious force
A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame.
It is related to Newton's second law of motion, which trea ...
s in the rotating reference frame, that is, apparent forces that result from being in a
non-inertial reference frame
A non-inertial reference frame is a frame of reference that undergoes acceleration with respect to an inertial frame. An accelerometer at rest in a non-inertial frame will, in general, detect a non-zero acceleration. While the laws of motion are ...
, rather than from any physical interaction between bodies.
Using
Newton's second law of motion
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motion ...
we obtain:
[
* the ]Coriolis force
In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...
* the centrifugal force
In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
* and the Euler force In classical mechanics, the Euler force is the fictitious tangential force
that appears when a non-uniformly rotating reference frame is used for analysis of motion and there is variation in the angular velocity of the reference frame's axes. The E ...
where is the mass of the object being acted upon by these fictitious force
A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame.
It is related to Newton's second law of motion, which trea ...
s. Notice that all three forces vanish when the frame is not rotating, that is, when
For completeness, the inertial acceleration due to impressed external forces can be determined from the total physical force in the inertial (non-rotating) frame (for example, force from physical interactions such as electromagnetic forces) using Newton's second law
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motion ...
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:
Centrifugal force
In classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
, centrifugal force is an outward force associated with rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
. Centrifugal force is one of several so-called pseudo-force
A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame.
It is related to Newton's second law of motion, which trea ...
s (also known as inertial force
A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial reference frame, non-inertial frame of reference, such as a linearly accelerating or rotating reference frame.
It is related to Newton's la ...
s), so named because, unlike real forces, they do not originate in interactions with other bodies situated in the environment of the particle upon which they act. Instead, centrifugal force originates in the rotation of the frame of reference within which observations are made.
Coriolis effect
The mathematical expression for the Coriolis force appeared in an 1835 paper by a French scientist Gaspard-Gustave Coriolis
Gaspard-Gustave de Coriolis (; 21 May 1792 – 19 September 1843) was a French mathematician, mechanical engineer and scientist. He is best known for his work on the supplementary forces that are detected in a rotating frame of reference, le ...
in connection with hydrodynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...
, and also in the tidal equations of Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
in 1778. Early in the 20th century, the term Coriolis force began to be used in connection with meteorology
Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did not ...
.
Perhaps the most commonly encountered rotating reference frame is the Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
. Moving objects on the surface of the Earth experience a Coriolis force, and appear to veer to the right in the northern hemisphere
The Northern Hemisphere is the half of Earth that is north of the Equator. For other planets in the Solar System, north is defined as being in the same celestial hemisphere relative to the invariable plane of the solar system as Earth's Nort ...
, and to the left in the southern
Southern may refer to:
Businesses
* China Southern Airlines, airline based in Guangzhou, China
* Southern Airways, defunct US airline
* Southern Air, air cargo transportation company based in Norwalk, Connecticut, US
* Southern Airways Express, M ...
. Movements of air in the atmosphere and water in the ocean are notable examples of this behavior: rather than flowing directly from areas of high pressure to low pressure, as they would on a non-rotating planet, winds and currents tend to flow to the right of this direction north of the equator
The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can als ...
, and to the left of this direction south of the equator. This effect is responsible for the rotation of large cyclones
In meteorology, a cyclone () is a large air mass that rotates around a strong center of low atmospheric pressure, counterclockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere as viewed from above (opposite to an anti ...
(see Coriolis effects in meteorology).
Euler force
In classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
, the Euler acceleration (named for Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
), also known as azimuthal acceleration or transverse acceleration is an acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
that appears when a non-uniformly rotating reference frame is used for analysis of motion and there is variation in the angular velocity
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
of the reference frame
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin (mathematics), origin, orientation (geometry), orientation, and scale (geometry), scale are specified by a set of reference point ...
's axis. This article is restricted to a frame of reference that rotates about a fixed axis.
The Euler force is a fictitious force
A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame.
It is related to Newton's second law of motion, which trea ...
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.
Use in magnetic resonance
It is convenient to consider magnetic resonance
Magnetic resonance is a process by which a physical excitation (resonance) is set up via magnetism.
This process was used to develop magnetic resonance imaging and Nuclear magnetic resonance spectroscopy technology.
It is also being used to ...
in a frame that rotates at the Larmor frequency
In physics, Larmor precession (named after Joseph Larmor) is the precession of the magnetic moment of an object about an external magnetic field. The phenomenon is conceptually similar to the precession of a tilted classical gyroscope in an extern ...
of the spins. This is illustrated in the animation below. The rotating wave approximation
The rotating-wave approximation is an approximation used in atom optics and magnetic resonance. In this approximation, terms in a Hamiltonian that oscillate rapidly are neglected. This is a valid approximation when the applied electromagnetic radi ...
may also be used.
See also
* Absolute rotation
In physics, the concept of absolute rotation— rotation independent of any external reference—is a topic of debate about relativity, cosmology, and the nature of physical laws.
For the concept of absolute rotation to be scientifically meaning ...
* Centrifugal force (rotating reference frame)
In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is parallel ...
Centrifugal force as seen from systems rotating about a fixed axis
* Mechanics of planar particle motion
This article describes a particle in planar motionSee for example, , when observed from non-inertial reference frames.''Fictitious forces'' (also known as a ''pseudo forces'', ''inertial forces'' or ''d'Alembert forces''), exist for observers i ...
Fictitious forces exhibited by a particle in planar motion as seen by the particle itself and by observers in a co-rotating frame of reference
* Coriolis force
In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...
The effect of the Coriolis force on the Earth and other rotating systems
* Inertial frame of reference
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
* Non-inertial frame
A non-inertial reference frame is a frame of reference that undergoes acceleration with respect to an inertial frame. An accelerometer at rest in a non-inertial frame will, in general, detect a non-zero acceleration. While the laws of motion ar ...
* Fictitious force
A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame.
It is related to Newton's second law of motion, which trea ...
A more general treatment of the subject of this article
References
{{reflist, group=note
External links
Animation clip
showing scenes as viewed from both an inertial frame and a rotating frame of reference, visualizing the Coriolis and centrifugal forces.
Frames of reference
Classical mechanics
Astronomical coordinate systems
Surveying
Rotation