text:
Phosphorus cycle
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Phosphorus Cycle on land
The phosphorus cycle is the biogeochemical cycle that describes the movement of phosphorus through the lithosphere, hydrosphere, and biosphere. Unlike many other biogeochemical cycles, the atmosphere does not play a significant role in the movement of phosphorus, because phosphorus and phosphorus-based compounds are usually solids at the typical ranges of temperature and pressure found on Earth. The production of phosphine gas occurs in only specialized, local conditions.
On the land, phosphorus (chemical symbol, P) gradually becomes less available to plants over thousands of years, because it is slowly lost in runoff. Low concentration of P in soils reduces plant growth, and slows soil microbial growth - as shown in studies of soil microbial biomass. Soil microorganisms act as both sinks and sources of available P in the biogeochemical cycle.[1] Locally, transformations of P are chemical, biological and microbiological: the major long-term transfers in the global cycle, however, are driven by tectonic movements in geologic time.[2]
Humans have caused major changes to the global P cycle through shipping of P minerals, and use of P fertilizer, and also the shipping of food from farms to cities, where it is lost as effluent.
Contents
[hide]
1 Phosphorus in the environment
1.1 Ecological function
1.2 Biological function
1.3 Process of the cycle
1.4 Phosphatic minerals
2 Human influences
3 See also
4 References
5 External links
Phosphorus in the environment[edit]
The aquatic phosphorus cycle
Ecological function[edit]
Phosphorus is an essential nutrient for plants and animals. Phosphorus is a limiting nutrient for aquatic organisms. Phosphorus forms parts of important life-sustaining molecules that are very common in the biosphere. Phosphorus does not enter the atmosphere, remaining mostly on land and in rock and soil minerals. Eighty percent of the mined phosphorus is used to make fertilizers. Phosphates from fertilizers, sewage and detergents can cause pollution in lakes and streams. Overenrichment of phosphate in both fresh and inshore marine waters can lead to massive algae blooms which, when they die and decay, leads to eutrophication of fresh waters only. An example of this is the Canadian Experimental Lakes Area. These freshwater algal blooms should not be confused with those in saltwater environments. Recent research suggests that the predominant pollutant responsible for algal blooms in salt water estuaries and coastal marine habitats is Nitrogen.[3]
Phosphorus occurs most abundantly in nature as part of the orthophosphate ion (PO4)3−, consisting of a P atom and 4 oxygen atoms. On land most phosphorus is found in rocks and minerals. Phosphorus rich deposits have generally formed in the ocean or from guano, and over time, geologic processes bring ocean sediments to land. Weathering of rocks and minerals release phosphorus in a soluble form where it is taken up by plants, and it is transformed into organic compounds. The plants may then be consumed by herbivores and the phosphorus is either incorporated into their tissues or excreted. After death, the animal or plant decays, and phosphorus is returned to the soil where a large part of the phosphorus is transformed into insoluble compounds. Runoff may carry a small part of the phosphorus back to the ocean. Generally with time (thousands of years) soils become deficient in phosphorus leading to ecosystem retrogression.[4]
Biological function[edit]
The primary biological importance of phosphates is as a component of nucleotides, which serve as energy storage within cells (ATP) or when linked together, form the nucleic acids DNA and RNA. The double helix of our DNA is only possible because of the phosphate ester bridge that binds the helix. Besides making biomolecules, phosphorus is also found in bone and the enamel of mammalian teeth, whose strength is derived from calcium phosphate in the form of Hydroxylapatite. It is also found in the exoskeleton of insects, and phospholipids (found in all biological membranes).[5] It also functions as a buffering agent in maintaining acid base homeostasis in the human body.[6]
Process of the cycle[edit]
Phosphates move quickly through plants and animals; however, the processes that move them through the soil or ocean are very slow, making the phosphorus cycle overall one of the slowest biogeochemical cycles.[2][7]
Initially, phosphate weathers from rocks and minerals, the most common mineral being apatite. Overall small losses occur in terrestrial environments by leaching and erosion, through the action of rain. In soil, phosphate is adsorbed on iron oxides, aluminium hydroxides, clay surfaces, and organic matter particles, and becomes incorporated (immobilized or fixed). Plants and fungi can also be active in making P soluble.[citation needed]
Unlike other cycles, P is not normally found in the air as a gas; it only occurs under highly reducing conditions as the gas phosphine PH3.[citation needed]
Phosphatic minerals[edit]
The availability of phosphorus in an ecosystem is restricted by the rate of release of this element during weathering. The release of phosphorus from apatite dissolution is a key control on ecosystem productivity. The primary mineral with significant phosphorus content, apatite [Ca5(PO4)3OH] undergoes carbonation.[2][8]
Little of this released phosphorus is taken up by biota (organic form), whereas a larger proportion reacts with other soil minerals. This leads to precipitation into unavailable forms in the later stage of weathering and soil development. Available phosphorus is found in a biogeochemical cycle in the upper soil profile, while phosphorus found at lower depths is primarily involved in geochemical reactions with secondary minerals. Plant growth depends on the rapid root uptake of phosphorus released from dead organic matter in the biochemical cycle. Phosphorus is limited in supply for plant growth. Phosphates move quickly through plants and animals; however, the processes that move them through the soil or ocean are very slow, making the phosphorus cycle overall one of the slowest biogeochemical cycles.[2][7]
Low-molecular-weight (LMW) organic acids are found in soils. They originate from the activities of various microorganisms in soils or may be exuded from the roots of living plants. Several of those organic acids are capable of forming stable organo-metal complexes with various metal ions found in soil solutions. As a result, these processes may lead to the release of inorganic phosphorus associated with aluminium, iron, and calcium in soil minerals. The production and release of oxalic acid by mycorrhizal fungi explain their importance in maintaining and supplying phosphorus to plants.[2][9]
The availability of organic phosphorus to support microbial, plant and animal growth depends on the rate of their degradation to generate free phosphate. There are various enzymes such as phosphatases, nucleases and phytase involved for the degradation. Some of the abiotic pathways in the environment studied are hydrolytic reactions and photolytic reactions. Enzymatic hydrolysis of organic phosphorus is an essential step in the biogeochemical phosphorus cycle, including the phosphorus nutrition of plants and microorganisms and the transfer of organic phosphorus from soil to bodies of water.[1] Many organisms rely on the soil derived phosphorus for their phosphorus nutrition.[citation needed]
Human influences[edit]
Nutrients are important to the growth and survival of living organisms, and hence, are essential for development and maintenance of healthy ecosystems. Humans have greatly influenced the phosphorus cycle by mining phosphorus, converting it to fertilizer, and by shipping fertilizer and products around the globe. Transporting phosphorus in food from farms to cities has made a major change in the global Phosphorus cycle. However, excessive amounts of nutrients, particularly phosphorus and nitrogen, are detrimental to aquatic ecosystems. Waters are enriched in phosphorus from farms' run-off, and from effluent that is inadequately treated before it is discharged to waters. Natural eutrophication is a process by which lakes gradually age and become more productive and may take thousands of years to progress. Cultural or anthropogenic eutrophication, however, is water pollution caused by excessive plant nutrients; this results in excessive growth in the algal population; when this algae dies its putrefaction depletes the water of oxygen.Such eutrophication may also give rise to toxic algal bloom.Both these effects cause animal and plant death rates to increase as the plants take in poisonous water while the animals drink the poisoned water. Surface and subsurface runoff and erosion from high-phosphorus soils may be major contributing factors to this fresh water eutrophication. The processes controlling soil Phosphorus release to surface runoff and to subsurface flow are a complex interaction between the type of phosphorus input, soil type and management, and transport processes depending on hydrological conditions.[10][11]
Repeated application of liquid hog manure in excess to crop needs can have detrimental effects on soil phosphorus status. Also, application of biosolids may increase available phosphorus in soil.[12] In poorly drained soils or in areas where snowmelt can cause periodic waterlogging, dereducing conditions can be attained in 7–10 days. This causes a sharp increase in phosphorus concentration in solution and phosphorus can be leached. In addition, reduction of the soil causes a shift in phosphorus from resilient to more labile forms. This could eventually increase the potential for phosphorus loss. This is of particular concern for the environmentally sound management of such areas, where disposal of agricultural wastes has already become a problem. It is suggested that the water regime of soils that are to be used for organic wastes disposal is taken into account in the preparation of waste management regulations.[13]
Human interference in the phosphorus cycle occurs by overuse or careless use of phosphorus fertilizers. This results in increased amounts of phosphorus as pollutants in bodies of water resulting in eutrophication. Eutrophication devastates water ecosystems by inducing anoxic conditions.[14]
See also[edit]
Peak phosphorus
Planetary boundaries
References[edit]
^ Jump up to: a b Turner B.L et al. (2003) Organic phosphorus in the environment. CABI publishing
^ Jump up to: a b c d e Schlesinger W.H. Biogeochemistry: An analysis of global change. (1991)
Jump up ^ https://www.soils.org/discover-soils/soils-in-the-city/green-infrastructure/important-terms/eutrophication
Jump up ^ Peltzer, D.A. et al. 2010. Understanding ecosystem retrogression. Ecological Monographs 80; 509-529.
Jump up ^ http://www.enviroliteracy.org/article.php/480.html
Jump up ^ Voet D and Voet J.G. (2003) Biochemistry pg 607-608.
^ Jump up to: a b Oelkers E.H (2008) Phosphate mineral reactivity: from global cycles to sustainable development Mineralogical Magazine 72: 337 - 340.
Jump up ^ Filippelli G.M (2002) The Global Phosphorus Cycle. Reviews in Mineralogy and geochemistry 48: 391 – 425
Jump up ^ Harrold S.A. and Tabatabai M.A (2006) Release of Inorganic Phosphorus from Soils by Low-Molecular-Weight Organic Acids. Communications in soil science and plant analysis, volume, issue 9 & 10
Jump up ^ Branom J.R and Sarkar D (2004) Phosphorus bioavailability in sediments of a sludge-disposal lake. Environmental Geoscience 11: 42 - 52
Jump up ^ Schelde K et al. (2006) Effects of Manure Application and Plowing on Transport of Colloids and Phosphorus to Tile Drains. 5: 445 - 458.
Jump up ^ "The effects of incubation on phosphorus desorption properties, phosphorus availability, and salinity of biosolids-amended soils". Environmental Earth Sciences. 69: 899–908. doi:10.1007/s12665-012-1975-6.
Jump up ^ Ajmone-Marsan. F et al. (2005) Phosphorus transformations under reduction in long-term manured soils.
Jump up ^ Conley, Daniel J.; et al. (2009). "Controlling Eutrophication: Nitrogen and Phosphorus". Science. 323: 1014–1015. CS1 maint: Explicit use of et al. (link)
External links[edit]
Part III of "Matter cycles": The phosphorus cycle, Lenntech Water treatment & air purification, Holding B.V. 2006
Environmental Literacy Council - Phosphorus Cycle
Monitoring and assessing water quality, section 5.6 Phosphorus - EPA
Prentice Hall Biology, Kenneth R. Miller and Joseph Levine 2001
Katie Corbin-Soil Microbiology
Environmental Literacy Council: Phosphorus Cycle
[show]
v
t
e
Biogeochemical cycles
Cycles
Carbon cycle
Hydrogen cycle
Iron cycle
Mercury cycle
Mineral cycle
Nitrogen cycle
Oxygen cycle
ozone
Phosphorus cycle
Rock cycle
Selenium cycle
Sulfur cycle
Water cycle
Research groups
DAAC
GEOTRACES
IMBER
NOBM
SOLAS
Related topics
Biogeochemistry
geochemical cycle
chemical cycling
environmental chemistry
Biosequestration
carbon sequestration
carbon sink
soil carbon
biological pump
mycorrhizal fungi
Ocean acidification
acid rain
Methane clathrate
clathrate gun hypothesis
Arctic methane emissions
Human impact on the nitrogen cycle
Nitrification
Nitrogen fixation
Nitrogen assimilation
Phosphorus assimilation
Sulfur assimilation
Planetary boundaries
[show]
v
t
e
Human impact on the environment
General
Anthropocene
Environmental issues
Human impact
Impact assessment
List of issues
Planetary boundaries
Causes
Agriculture
fishing
irrigation
meat production
palm oil
Energy industry
biofuels
biodiesel
coal
electricity generation
nuclear power
oil shale
petroleum
reservoirs
wind power
Manufacturing
cleaning agents
concrete
plastics
nanotechnology
paint
paper
pesticides
pharmaceuticals and personal care
Tourism
Transport
aviation
roads
shipping
reservoirs
Other
land use
mining
genetic pollution
human overpopulation
overexploitation
particulates
pollution
war
Effects
Biodiversity threats
Biodiversity loss
Climate change
Deforestation
Desertification
Ecocide
Erosion
Coral reefs
Freshwater cycle
Global warming
Habitat destruction
Holocene extinction
Nitrogen cycle
Land degradation
Land surface effects on climate
Phosphorus cycle
Ocean acidification
Ozone depletion
Runaway climate change
Mitigation
Birth control
Cleaner production
Climate change mitigation
Climate engineering
Ecological engineering
Environmental engineering
Environmental mitigation
Industrial ecology
Mitigation banking
Organic farming
Reforestation
urban
Restoration ecology
Sustainable consumption
Waste minimization
Commons
Category
by country
assessment
mitigation
Retrieved from "https://en.wikipedia.org/w/index.php?title=Phosphorus_cycle&oldid=806939757"
Categories:
Biogeochemical cycle
Soil biology
Soil chemistry
Phosphorus
Hidden categories:
CS1 maint: Explicit use of et al.
All articles with unsourced statements
Articles with unsourced statements from December 2014
Navigation menu
Personal tools
Not logged in
Talk
Contributions
Create account
Log in
Namespaces
Article
Talk
Variants
Views
Read
Edit
View history
More
Search
Navigation
Main page
Contents
Featured content
Current events
Random article
Donate to Wikipedia
Wikipedia store
Interaction
Help
About Wikipedia
Community portal
Recent changes
Contact page
Tools
What links here
Related changes
Upload file
Special pages
Permanent link
Page information
Wikidata item
Cite this page
Print/export
Create a book
Download as PDF
Printable version
Languages
العربية
Azərbaycanca
Dansk
Deutsch
Eesti
Español
Euskara
فارسی
Français
Galego
한국어
Italiano
ქართული
Polski
Simple English
Slovenščina
ไทย
Türkçe
Українська
中文
Edit links
This page was last edited on 25 October 2017, at 01:28.
Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
Privacy policy
About Wikipedia
Disclaimers
Contact Wikipedia
Developers
Cookie statement
Mobile view
Enable previews

key: html

type: local

value: div

attrs: id class

values: mw-page-base noprint

values: mw-head-base noprint

value: a

attrs: id class lang

values: firstHeading firstHeading en

values: siteSub noprint

values: jump-to-nav mw-jump

value: img

attrs: alt src width height class srcset data-file-width data-file-height

attrs: href class

attrs: href class title

attrs: class style

value: b

attrs: href title

values:
/wiki/Earth%27s_atmosphere
mw-redirect
Earth's atmosphere

value: p

values: /wiki/Microbial mw-redirect Microbial

values:
/wiki/Tectonic_plate
mw-redirect
Tectonic plate

values:
/wiki/Geologic_time
mw-redirect
Geologic time

values: Edit section: Phosphorus in the environment

values:
/wiki/Algae_bloom
mw-redirect
Algae bloom

values:
/wiki/Runoff_(water)
mw-redirect
Runoff (water)

values: This claim needs references to reliable sources. (December 2014)

values:
noprint Inline-Template Template-Fact
white-space:nowrap;

values: Category:CS1 maint: Explicit use of et al.

values: -moz-column-width: 32em; -webkit-column-width: 32em; column-width: 32em; list-style-type: decimal;

values: Mycorrhizal fungi and soil carbon storage

values: nowraplinks collapsible autocollapse navbox-inner

values: Template:Human impact on the environment

values: Template talk:Human impact on the environment

attrs: alt src width height srcset data-file-width data-file-height

values:
/wiki/File:Planetary_boundaries.svg
image
Planetary boundaries

values:
navbox-image
4
width:1px;padding:0px 0px 0px 2px

values: Social and environmental impact of palm oil

values: Environmental impact of the energy industry

values: Indirect land use change impacts of biofuels

values: Environmental impact of the coal industry

values: Environmental impact of electricity generation

values: Environmental impact of the oil shale industry

values: Environmental impact of the petroleum industry

values:
/wiki/Environmental_impact_of_plastics
mw-redirect
Environmental impact of plastics

values: Environmental impact of pharmaceuticals and personal care products

values:
/wiki/Environmental_impact_of_tourism
mw-redirect
Environmental impact of tourism

values:
/wiki/Environmental_impact_of_roads
mw-redirect
Environmental impact of roads

values:
/wiki/Biodiversity_threats
mw-redirect
Biodiversity threats

values:
/wiki/Freshwater_cycle
mw-redirect
Freshwater cycle

attrs: alt src title width height srcset data-file-width data-file-height

values: Category:Environmental impact by country

values: nowraplinks hlist collapsible autocollapse navbox-inner

values: Category:All articles with unsourced statements

values: Category:Articles with unsourced statements from December 2014

values: Discussion about edits from this IP address [ctrl-alt-n]

values: A list of edits made from this IP address [ctrl-alt-y]

values:
/wiki/Phosphorus_cycle
View the content page [ctrl-alt-c]
c

values: p-views navigation vectorTabs p-views-label

values:
p-cactions
navigation
vectorMenu emptyPortlet
p-cactions-label

values: portal navigation p-navigation p-navigation-label

values: Guidance on how to use and edit Wikipedia

values: About the project, what you can do, where to find things

values: A list of recent changes in the wiki [ctrl-alt-r]

values: portal navigation p-interaction p-interaction-label

values: List of all English Wikipedia pages containing links to this page [ctrl-alt-j]

values: Recent changes in pages linked from this page [ctrl-alt-k]

values:
/wiki/Wikipedia:File_Upload_Wizard
Upload files [ctrl-alt-u]
u

values: A list of all special pages [ctrl-alt-q]

values: Permanent link to this revision of the page

values: Link to connected data repository item [ctrl-alt-g]

values: Printable version of this page [ctrl-alt-p]

values:
uls-settings-trigger
Language settings
0
button
true

values: دورة الفوسفور – Arabic

values: چرخه فسفر – Persian

values: ფოსფორის ციკლი – Georgian

values: วัฏจักรฟอสฟอรัส – Thai

values: Обіг фосфору в природі – Ukrainian

values: mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Phosphorus_cycle rootpage-Phosphorus_cycle vector-experimental-print-styles vector-nav-directionality skin-vector action-view


text:
Centrifugal force
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Not to be confused with Centripetal force.
Part of a series of articles about
Classical mechanics
F → = m a → {\displaystyle {\vec {F}}=m{\vec {a}}}
Second law of motion
History
Timeline
Branches[show]
Applied
Celestial
Continuum
Dynamics
Kinematics
Kinetics
Statics
Statistical
Fundamentals[show]
Acceleration
Angular momentum
Couple
D'Alembert's principle
Energy
kinetic
potential
Force
Frame of reference
Inertial frame of reference
Impulse
Inertia / Moment of inertia
Mass

Mechanical power
Mechanical work

Moment
Momentum
Space
Speed
Time
Torque
Velocity
Virtual work
Formulations[show]
Newton's laws of motion
Analytical mechanics
Lagrangian mechanics
Hamiltonian mechanics
Routhian mechanics
Hamilton–Jacobi equation
Appell's equation of motion
Udwadia–Kalaba equation
Koopman–von Neumann mechanics
Core topics[show]
Damping (ratio)
Displacement
Equations of motion
Euler's laws of motion
Fictitious force
Friction
Harmonic oscillator
Inertial / Non-inertial reference frame
Mechanics of planar particle motion
Motion (linear)
Newton's law of universal gravitation
Newton's laws of motion
Relative velocity
Rigid body
dynamics
Euler's equations
Simple harmonic motion
Vibration
Rotation[show]
Circular motion
Rotating reference frame
Centripetal force
Centrifugal force
reactive
Coriolis force
Pendulum
Tangential speed
Rotational speed
Angular acceleration / displacement / frequency / velocity
Scientists[show]
Galileo
Huygens
Newton
Kepler
Horrocks
Halley
Euler
d'Alembert
Clairaut
Lagrange
Laplace
Hamilton
Poisson
Daniel Bernoulli
Johann Bernoulli
Cauchy
v
t
e
In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) directed away from the axis of rotation that appears to act on all objects when viewed in a rotating frame of reference.
The concept of centrifugal force can be applied in rotating devices, such as centrifuges, centrifugal pumps, centrifugal governors, and centrifugal clutches, and in centrifugal railways, planetary orbits and banked curves, when they are analyzed in a rotating coordinate system. The term has sometimes also been used for the reactive centrifugal force that is a reaction to a centripetal force.
Contents
[hide]
1 Introduction
2 Examples
2.1 Vehicle driving round a curve
2.2 Stone on a string
2.3 Earth
2.3.1 Weight of an object at the poles and on the equator
2.3.2 Equatorial railway
3 Derivation
3.1 Time derivatives in a rotating frame
3.2 Acceleration
3.3 Force
4 Absolute rotation
5 Applications
6 History of conceptions of centrifugal and centripetal forces
7 Other uses of the term
8 See also
9 References
Introduction[edit]
Centrifugal force is an outward force apparent in a rotating reference frame.[1][2][3] It does not exist when a system is described relative to an inertial frame of reference. All measurements of position and velocity must be made relative to some frame of reference. For example, an analysis of the motion of an object in an airliner in flight could be made relative to the airliner, to the surface of the Earth, or even to the Sun.[4] A reference frame that is at rest (or one that moves with no rotation and at constant velocity) relative to the "fixed stars" is generally taken to be an inertial frame. Any system can be analyzed in an inertial frame (and so with no centrifugal force). However, it is often more convenient to describe a rotating system by using a rotating frame—the calculations are simpler, and descriptions more intuitive. When this choice is made, fictitious forces, including the centrifugal force, arise.
In a rotating reference frame, all objects, regardless of their state of motion, appear to be under the influence of a radially (from the axis of rotation) outward force that is proportional to their mass, to the distance from the axis of rotation of the frame, and to the square of the angular velocity of the frame.[5][6] This is the centrifugal force. As humans usually experience centrifugal force from within the rotating reference frame, e.g. on a merry-go-round or vehicle, this is much more well-known than centripetal force.
Motion relative to a rotating frame results in another fictitious force: the Coriolis force. If the rate of rotation of the frame changes, a third fictitious force (the Euler force) is required. These fictitious forces are necessary for the formulation of correct equations of motion in a rotating reference frame[7][8] and allow Newton's laws to be used in their normal form in such a frame (with one exception: the fictitious forces do not obey Newton's third law: they have no equal and opposite counterparts).[7]
Examples[edit]
Vehicle driving round a curve[edit]
A common experience that gives rise to the idea of a centrifugal force is encountered by passengers riding in a vehicle, such as a car, that is changing direction. If a car is traveling at a constant speed along a straight road, then a passenger inside is not accelerating and, according to Newton's second law of motion, the net force acting on him is therefore zero (all forces acting on him cancel each other out). If the car enters a curve that bends to the left, the passenger experiences an apparent force that seems to be pulling him towards the right. This is the fictitious centrifugal force. It is needed within the passenger's local frame of reference to explain his sudden tendency to start accelerating to the right relative to the car—a tendency which he must resist by applying a rightward force to the car (for instance, a frictional force against the seat) in order to remain in a fixed position inside. Since he pushes the seat toward the right, Newton's third law says that the seat pushes him toward the left. The centrifugal force must be included in the passenger's reference frame (in which the passenger remains at rest): it counteracts the leftward force applied to the passenger by the seat, and explains why this otherwise unbalanced force does not cause him to accelerate.[9] However, it would be apparent to a stationary observer watching from an overpass above that the frictional force exerted on the passenger by the seat is not being balanced; it constitutes a net force to the left, causing the passenger to accelerate toward the inside of the curve, as he must in order to keep moving with the car rather than proceeding in a straight line as he otherwise would. Thus the "centrifugal force" he feels is the result of a "centrifugal tendency" caused by inertia.[10] Similar effects are encountered in aeroplanes and roller coasters where the magnitude of the apparent force is often reported in "G's".
Stone on a string[edit]
If a stone is whirled round on a string, in a horizontal plane, the only real force acting on the stone in the horizontal plane is applied by the string (gravity acts vertically). There is a net force on the stone in the horizontal plane which acts toward the center.
In an inertial frame of reference, were it not for this net force acting on the stone, the stone would travel in a straight line, according to Newton's first law of motion. In order to keep the stone moving in a circular path, a centripetal force, in this case provided by the string, must be continuously applied to the stone. As soon as it is removed (for example if the string breaks) the stone moves in a straight line. In this inertial frame, the concept of centrifugal force is not required as all motion can be properly described using only real forces and Newton's laws of motion.
In a frame of reference rotating with the stone around the same axis as the stone, the stone is stationary. However, the force applied by the string is still acting on the stone. If one were to apply Newton's laws in their usual (inertial frame) form, one would conclude that the stone should accelerate in the direction of the net applied force—towards the axis of rotation—which it does not do. The centrifugal force and other fictitious forces must be included along with the real forces in order to apply Newton's laws of motion in the rotating frame.
Earth[edit]
The Earth constitutes a rotating reference frame because it rotates once a day on its axis. Because the rotation is slow, the fictitious forces it produces are small, and in everyday situations can generally be neglected. Even in calculations requiring high precision, the centrifugal force is generally not explicitly included, but rather lumped in with the gravitational force: the strength and direction of the local "gravity" at any point on the Earth's surface is actually a combination of gravitational and centrifugal forces.
Weight of an object at the poles and on the equator[edit]
If an object is weighed with a simple spring balance at one of the Earth's poles, there are two forces acting on the object: the Earth's gravity, which acts in a downward direction, and the equal and opposite restoring force in the spring, acting upward. Since the object is stationary and not accelerating, there is no net force acting on the object and the force from the spring is equal in magnitude to the force of gravity on the object. In this case, the balance shows the value of the force of gravity on the object.
When the same object is weighed on the equator, the same two real forces act upon the object. However, the object is moving in a circular path as the Earth rotates and therefore experiencing a centripetal acceleration. When considered in an inertial frame (that is to say, one that is not rotating with the Earth), the non-zero acceleration means that force of gravity will not balance with the force from the spring. In order to have a net centripetal force, the magnitude of the restoring force of the spring must be less than the magnitude of force of gravity. Less restoring force in the spring is reflected on the scale as less weight — about 0.3% less at the equator than at the poles.[11] In the Earth reference frame (in which the object being weighed is at rest), the object does not appear to be accelerating, however the two real forces, gravity and the force from the spring, are the same magnitude and do not balance. The centrifugal force must be included to make the sum of the forces be zero to match the apparent lack of acceleration.
Note: In fact, the observed weight difference is more — about 0.53%. Earth's gravity is a bit stronger at the poles than at the equator, because the Earth is not a perfect sphere, so an object at the poles is slightly closer to the center of the Earth than one at the equator; this effect combines with the centrifugal force to produce the observed weight difference.[12]
Equatorial railway[edit]
This thought experiment is more complicated than the previous examples in that it requires the use of the Coriolis force as well as the centrifugal force.
If there were a railway line running round the Earth's equator, a train moving westward along it fast enough would remain stationary in a frame moving (but not rotating) with the Earth; it would stand still as the Earth spun beneath it. In this inertial frame the situation is easy to analyze. The only forces acting on the train (assuming no wind resistance or other horizontal forces) are its gravity (downward) and the equal and opposite (upward) force from the track. There is no net force on the train and it therefore remains stationary.
In a frame rotating with the Earth the train moves in a circular orbit as it travels round the Earth. In this frame, the upward reaction force from the track and the force of gravity on the train remain the same, as they are real forces. However, in the Earth's (rotating) frame, the train is traveling in a circular path and therefore requires a centripetal (downward) force to keep it on this path. Because this uses a rotating frame, the (fictitious) centrifugal force must be applied to the train. This is equal in value to the required centripetal force but acts in an upward direction — the opposite direction to that required. It would seem that there is a net upward force on the train and it should therefore accelerate upward.
The resolution to this paradox lies in the fact that the train is in motion with respect to the rotating frame and is subject to (in addition to the centrifugal force) the Coriolis force, which, in this example, acts downward direction and is twice as strong as centrifugal force.
Derivation[edit]
Main article: Rotating reference frame
See also: Fictitious force and Mechanics of planar particle motion
For the following formalism, the rotating frame of reference is regarded as a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame denoted the stationary frame.
Time derivatives in a rotating frame[edit]
In a rotating frame of reference, the time derivatives of any vector function P of time—such as the velocity and acceleration vectors of an object—will differ from its time derivatives in the stationary frame. If P1 P2, P3 are the components of P with respect to unit vectors i, j, k directed along the axes of the rotating frame,[13] then the first time derivative [dP/dt] of P with respect to the rotating frame is, by definition, dP1/dt i + dP2/dt j + dP3/dt k. If the absolute angular velocity of the rotating frame is ω then the derivative dP/dt of P with respect to the stationary frame is related to [dP/dt] by the equation:[14]
d  P d  t = [ d  P d  t ] + ω × P , {\displaystyle {\frac {\operatorname {d} {\boldsymbol {P}}}{\operatorname {d} t}}=\left[{\frac {\operatorname {d} {\boldsymbol {P}}}{\operatorname {d} t}}\right]+{\boldsymbol {\omega }}\times {\boldsymbol {P}}\ ,}
where × {\displaystyle \times } denotes the vector cross product. In other words, the rate of change of P in the stationary frame is the sum of its apparent rate of change in the rotating frame and a rate of rotation ω × P {\displaystyle {\boldsymbol {\omega }}\times {\boldsymbol {P}}} attributable to the motion of the rotating frame. The vector ω has magnitude ω equal to the rate of rotation and is directed along the axis of rotation according to the right-hand rule.
Acceleration[edit]
Newton's law of motion for a particle of mass m written in vector form is:
F = m a , {\displaystyle {\boldsymbol {F}}=m{\boldsymbol {a}}\ ,}
where F is the vector sum of the physical forces applied to the particle and a is the absolute acceleration (that is, acceleration in an inertial frame) of the particle, given by:
a = d 2  r d  t 2 , {\displaystyle {\boldsymbol {a}}={\frac {\operatorname {d} ^{2}{\boldsymbol {r}}}{\operatorname {d} t^{2}}}\ ,}
where r is the position vector of the particle.
By applying the transformation above from the stationary to the rotating frame three times,[15] the absolute acceleration of the particle can be written as:
a = d 2  r d  t 2 = d d  t d  r d  t = d d  t ( [ d  r d  t ] + ω × r ) = [ d 2  r d  t 2 ] + ω × [ d  r d  t ] + d  ω d  t × r + ω × d  r d  t = [ d 2  r d  t 2 ] + ω × [ d  r d  t ] + d  ω d  t × r + ω × ( [ d  r d  t ] + ω × r ) = [ d 2  r d  t 2 ] + d  ω d  t × r + 2 ω × [ d  r d  t ] + ω × ( ω × r ) . {\displaystyle {\begin{aligned}{\boldsymbol {a}}&={\frac {\operatorname {d} ^{2}{\boldsymbol {r}}}{\operatorname {d} t^{2}}}={\frac {\operatorname {d} }{\operatorname {d} t}}{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}={\frac {\operatorname {d} }{\operatorname {d} t}}\left(\left[{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\right]+{\boldsymbol {\omega }}\times {\boldsymbol {r}}\ \right)\\&=\left[{\frac {\operatorname {d} ^{2}{\boldsymbol {r}}}{\operatorname {d} t^{2}}}\right]+{\boldsymbol {\omega }}\times \left[{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\right]+{\frac {\operatorname {d} {\boldsymbol {\omega }}}{\operatorname {d} t}}\times {\boldsymbol {r}}+{\boldsymbol {\omega }}\times {\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\\&=\left[{\frac {\operatorname {d} ^{2}{\boldsymbol {r}}}{\operatorname {d} t^{2}}}\right]+{\boldsymbol {\omega }}\times \left[{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\right]+{\frac {\operatorname {d} {\boldsymbol {\omega }}}{\operatorname {d} t}}\times {\boldsymbol {r}}+{\boldsymbol {\omega }}\times \left(\left[{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\right]+{\boldsymbol {\omega }}\times {\boldsymbol {r}}\ \right)\\&=\left[{\frac {\operatorname {d} ^{2}{\boldsymbol {r}}}{\operatorname {d} t^{2}}}\right]+{\frac {\operatorname {d} {\boldsymbol {\omega }}}{\operatorname {d} t}}\times {\boldsymbol {r}}+2{\boldsymbol {\omega }}\times \left[{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\right]+{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {r}})\ .\end{aligned}}}
Force[edit]
The apparent acceleration in the rotating frame is [d2r/dt2]. An observer unaware of the rotation would expect this to be zero in the absence of outside forces. However, Newton's laws of motion apply only in the inertial frame and describe dynamics in terms of the absolute acceleration d2r/dt2. Therefore, the observer perceives the extra terms as contributions due to fictitious forces. These terms in the apparent acceleration are independent of mass; so it appears that each of these fictitious forces, like gravity, pulls on an object in proportion to its mass. When these forces are added, the equation of motion has the form:[16][17][18]
F − m d  ω d  t × r − 2 m ω × [ d  r d  t ] − m ω × ( ω × r ) {\displaystyle {\boldsymbol {F}}-m{\frac {\operatorname {d} {\boldsymbol {\omega }}}{\operatorname {d} t}}\times {\boldsymbol {r}}-2m{\boldsymbol {\omega }}\times \left[{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\right]-m{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {r}})} = m [ d 2  r d  t 2 ] . {\displaystyle =m\left[{\frac {\operatorname {d} ^{2}{\boldsymbol {r}}}{\operatorname {d} t^{2}}}\right]\ .}
From the perspective of the rotating frame, the additional force terms are experienced just like the real external forces and contribute to the apparent acceleration.[19][20] The additional terms on the force side of the equation can be recognized as, reading from left to right, the Euler force − m d  ω / d  t × r {\displaystyle -m\operatorname {d} {\boldsymbol {\omega }}/\operatorname {d} t\times {\boldsymbol {r}}} , the Coriolis force − 2 m ω × [ d  r / d  t ] {\displaystyle -2m{\boldsymbol {\omega }}\times \left[\operatorname {d} {\boldsymbol {r}}/\operatorname {d} t\right]} , and the centrifugal force − m ω × ( ω × r ) {\displaystyle -m{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {r}})} , respectively.[21] Unlike the other two fictitious forces, the centrifugal force always points radially outward from the axis of rotation of the rotating frame, with magnitude mω2r, and unlike the Coriolis force in particular, it is independent of the motion of the particle in the rotating frame. As expected, for a non-rotating inertial frame of reference ( ω = 0 ) {\displaystyle ({\boldsymbol {\omega }}=0)} the centrifugal force and all other fictitious forces disappear.[22] Similarly, as the centrifugal force is proportional to the distance from object to the axis of rotation of the frame, the centrifugal force vanishes for objects that lie upon the axis.
Absolute rotation[edit]
Main article: Absolute rotation
The interface of two immiscible liquids rotating around a vertical axis is an upward-opening circular paraboloid.
When analysed in a rotating reference frame of the planet, centrifugal force causes rotating planets to assume the shape of an oblate spheroid
Three scenarios were suggested by Newton to answer the question of whether the absolute rotation of a local frame can be detected; that is, if an observer can decide whether an observed object is rotating or if the observer is rotating.[23][24]
The shape of the surface of water rotating in a bucket. The shape of the surface becomes concave to balance the centrifugal force against the other forces upon the liquid.
The tension in a string joining two spheres rotating about their center of mass. The tension in the string will be proportional to the centrifugal force on each sphere as it rotates around the common center of mass.
In these scenarios, the effects attributed to centrifugal force are only observed in the local frame (the frame in which the object is stationary) if the object is undergoing absolute rotation relative to an inertial frame. By contrast, in an inertial frame, the observed effects arise as a consequence of the inertia and the known forces without the need to introduce a centrifugal force. Based on this argument, the privileged frame, wherein the laws of physics take on the simplest form, is a stationary frame in which no fictitious forces need to be invoked.
Within this view of physics, any other phenomenon that is usually attributed to centrifugal force can be used to identify absolute rotation. For example, the oblateness of a sphere of freely flowing material is often explained in terms of centrifugal force. The oblate spheroid shape reflects, following Clairaut's theorem, the balance between containment by gravitational attraction and dispersal by centrifugal force. That the Earth is itself an oblate spheroid, bulging at the equator where the radial distance and hence the centrifugal force is larger, is taken as one of the evidences for its absolute rotation.[25]
Applications[edit]
The operations of numerous common rotating mechanical systems are most easily conceptualized in terms of centrifugal force. For example:
A centrifugal governor regulates the speed of an engine by using spinning masses that move radially, adjusting the throttle, as the engine changes speed. In the reference frame of the spinning masses, centrifugal force causes the radial movement.
A centrifugal clutch is used in small engine-powered devices such as chain saws, go-karts and model helicopters. It allows the engine to start and idle without driving the device but automatically and smoothly engages the drive as the engine speed rises. Inertial drum brake ascenders used in rock climbing and the inertia reels used in many automobile seat belts operate on the same principle.
Centrifugal forces can be used to generate artificial gravity, as in proposed designs for rotating space stations. The Mars Gravity Biosatellite would have studied the effects of Mars-level gravity on mice with gravity simulated in this way.
Spin casting and centrifugal casting are production methods that uses centrifugal force to disperse liquid metal or plastic throughout the negative space of a mold.
Centrifuges are used in science and industry to separate substances. In the reference frame spinning with the centrifuge, the centrifugal force induces a hydrostatic pressure gradient in fluid-filled tubes oriented perpendicular to the axis of rotation, giving rise to large buoyant forces which push low-density particles inward. Elements or particles denser than the fluid move outward under the influence of the centrifugal force. This is effectively Archimedes' principle as generated by centrifugal force as opposed to being generated by gravity.
Some amusement rides make use of centrifugal forces. For instance, a Gravitron's spin forces riders against a wall and allows riders to be elevated above the machine's floor in defiance of Earth's gravity.[26]
Nevertheless, all of these systems can also be described without requiring the concept of centrifugal force, in terms of motions and forces in a stationary frame, at the cost of taking somewhat more care in the consideration of forces and motions within the system.
History of conceptions of centrifugal and centripetal forces[edit]
Main article: History of centrifugal and centripetal forces
The conception of centrifugal force has evolved since the time of Huygens, Newton, Leibniz, and Hooke who expressed early conceptions of it. Its modern conception as a fictitious force arising in a rotating reference frame evolved in the eighteenth and nineteenth centuries.[citation needed]
Centrifugal force has also played a role in debates in classical mechanics about detection of absolute motion. Newton suggested two arguments to answer the question of whether absolute rotation can be detected: the rotating bucket argument, and the rotating spheres argument.[27] According to Newton, in each scenario the centrifugal force would be observed in the object's local frame (the frame where the object is stationary) only if the frame were rotating with respect to absolute space. Nearly two centuries later, Mach's principle was proposed where, instead of absolute rotation, the motion of the distant stars relative to the local inertial frame gives rise through some (hypothetical) physical law to the centrifugal force and other inertia effects. Today's view is based upon the idea of an inertial frame of reference, which privileges observers for which the laws of physics take on their simplest form, and in particular, frames that do not use centrifugal forces in their equations of motion in order to describe motions correctly.
The analogy between centrifugal force (sometimes used to create artificial gravity) and gravitational forces led to the equivalence principle of general relativity.[28][29]
Other uses of the term[edit]
While the majority of the scientific literature uses the term centrifugal force to refer to the particular fictitious force that arises in rotating frames, there are a few limited instances in the literature of the term applied to other distinct physical concepts. One of these instances occurs in Lagrangian mechanics. Lagrangian mechanics formulates mechanics in terms of generalized coordinates {qk}, which can be as simple as the usual polar coordinates ( r , θ ) {\displaystyle (r,\ \theta )} or a much more extensive list of variables.[30][31] Within this formulation the motion is described in terms of generalized forces, using in place of Newton's laws the Euler–Lagrange equations. Among the generalized forces, those involving the square of the time derivatives {(dqk ⁄ dt )2} are sometimes called centrifugal forces.[32][33][34][35] In the case of motion in a central potential the Lagrangian centrifugal force has the same form as the fictitious centrifugal force derived in a co-rotating frame.[36] However, the Lagrangian use of "centrifugal force" in other, more general cases has only a limited connection to the Newtonian definition.
In another instance the term refers to the reaction force to a centripetal force, or reactive centrifugal force. A body undergoing curved motion, such as circular motion, is accelerating toward a center at any particular point in time. This centripetal acceleration is provided by a centripetal force, which is exerted on the body in curved motion by some other body. In accordance with Newton's third law of motion, the body in curved motion exerts an equal and opposite force on the other body. This reactive force is exerted by the body in curved motion on the other body that provides the centripetal force and its direction is from that other body toward the body in curved motion.[37][38] [39][40]
This reaction force is sometimes described as a centrifugal inertial reaction,[41][42] that is, a force that is centrifugally directed, which is a reactive force equal and opposite to the centripetal force that is curving the path of the mass.
The concept of the reactive centrifugal force is sometimes used in mechanics and engineering. It is sometimes referred to as just centrifugal force rather than as reactive centrifugal force[43][44] although this usage is deprecated in elementary mechanics.[45]
See also[edit]
Physics portal
Wikimedia Commons has media related to Centrifugal force.
Centrifugal mechanism of acceleration
Equivalence principle
Folk physics
Lamm equation
Lagrangian point
Balancing of rotating masses
References[edit]
Jump up ^ Richard T. Weidner and Robert L. Sells (1973). Mechanics, mechanical waves, kinetic theory, thermodynamics (2 ed.). Allyn and Bacon. p. 123.
Jump up ^ John Robert Taylor (2004). Classical Mechanics. Sausalito CA: University Science Books. Chapter 9, pp. 344 ff. ISBN 1-891389-22-X.
Jump up ^ Kobayashi, Yukio (2008). "Remarks on viewing situation in a rotating frame". European Journal of Physics. 29 (3): 599–606. doi:10.1088/0143-0807/29/3/019.
Jump up ^ David P. Stern (2006). "Frames of Reference: The Basics". From Stargazers to Starships. Goddard Space Flight Center Space Physics Data Facility. Retrieved 20 April 2017.
Jump up ^ Encyclopædia Britannica, article on Centrifuge
Jump up ^ Feynman Lectures on Physics, Book 1, 12-11.
^ Jump up to: a b Alexander L. Fetter; John Dirk Walecka (2003). Theoretical Mechanics of Particles and Continua. Courier Dover Publications. pp. 38–39. ISBN 0-486-43261-0.
Jump up ^ Jerrold E. Marsden; Tudor S. Ratiu (1999). Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems. Springer. p. 251. ISBN 0-387-98643-X.
Jump up ^ "Centrifugal force". Encyclopædia Britannica. 17 August 2016. Retrieved 20 April 2017.
Jump up ^ Knight, Judson (2016). Schlager, Neil, ed. Centripetal Force. Science of Everyday Things, Volume 2: Real-Life Physics. Thomson Learning. p. 47. Retrieved 19 April 2017.
Jump up ^ "Curious About Astronomy?" Archived January 17, 2015, at the Wayback Machine., Cornell University, retrieved June 2007
Jump up ^ Boynton, Richard (2001). "Precise Measurement of Mass" (PDF). Sawe Paper No. 3147. Arlington, Texas: S.A.W.E., Inc. Retrieved 2007-01-21.
Jump up ^ i.e. P = P1 i + P2 j +P3 k
Jump up ^ John L. Synge; Byron A. Griffith (2007). Principles of Mechanics (Reprint of Second Edition of 1942 ed.). Read Books. p. 347. ISBN 1-4067-4670-3.
Jump up ^ Twice to d  r d  t {\displaystyle {\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}} and once to d d  t [ d  r d  t ] {\displaystyle {\frac {\operatorname {d} }{\operatorname {d} t}}\left[{\frac {\operatorname {d} {\boldsymbol {r}}}{\operatorname {d} t}}\right]} .
Jump up ^ Taylor (2005). p. 342.
Jump up ^ LD Landau; LM Lifshitz (1976). Mechanics (Third ed.). Oxford: Butterworth-Heinemann. p. 128. ISBN 978-0-7506-2896-9.
Jump up ^ Louis N. Hand; Janet D. Finch (1998). Analytical Mechanics. Cambridge University Press. p. 267. ISBN 0-521-57572-9.
Jump up ^ Mark P Silverman (2002). A universe of atoms, an atom in the universe (2 ed.). Springer. p. 249. ISBN 0-387-95437-6.
Jump up ^ Taylor (2005). p. 329.
Jump up ^ Cornelius Lanczos (1986). The Variational Principles of Mechanics (Reprint of Fourth Edition of 1970 ed.). Dover Publications. Chapter 4, §5. ISBN 0-486-65067-7.
Jump up ^ Morton Tavel (2002). Contemporary Physics and the Limits of Knowledge. Rutgers University Press. p. 93. ISBN 0-8135-3077-6. Noninertial forces, like centrifugal and Coriolis forces, can be eliminated by jumping into a reference frame that moves with constant velocity, the frame that Newton called inertial.
Jump up ^ Louis N. Hand; Janet D. Finch (1998). Analytical Mechanics. Cambridge University Press. p. 324. ISBN 0-521-57572-9.
Jump up ^ I. Bernard Cohen; George Edwin Smith (2002). The Cambridge companion to Newton. Cambridge University Press. p. 43. ISBN 0-521-65696-6.
Jump up ^ Simon Newcomb (1878). Popular astronomy. Harper & Brothers. pp. 86–88.
Jump up ^ Myers, Rusty L. (2006). The basics of physics. Greenwood Publishing Group. p. 57. ISBN 0-313-32857-9.
Jump up ^ An English translation is found at Isaac Newton (1934). Philosophiae naturalis principia mathematica (Andrew Motte translation of 1729, revised by Florian Cajori ed.). University of California Press. pp. 10–12.
Jump up ^ Barbour, Julian B. and Herbert Pfister (1995). Mach's principle: from Newton's bucket to quantum gravity. Birkhäuser. ISBN 0-8176-3823-7, p. 69.
Jump up ^ Eriksson, Ingrid V. (2008). Science education in the 21st century. Nova Books. ISBN 1-60021-951-9, p. 194.
Jump up ^ For an introduction, see for example Cornelius Lanczos (1986). The variational principles of mechanics (Reprint of 1970 University of Toronto ed.). Dover. p. 1. ISBN 0-486-65067-7.
Jump up ^ For a description of generalized coordinates, see Ahmed A. Shabana (2003). "Generalized coordinates and kinematic constraints". Dynamics of Multibody Systems (2 ed.). Cambridge University Press. p. 90 ff. ISBN 0-521-54411-4.
Jump up ^ Christian Ott (2008). Cartesian Impedance Control of Redundant and Flexible-Joint Robots. Springer. p. 23. ISBN 3-540-69253-3.
Jump up ^ Shuzhi S. Ge; Tong Heng Lee; Christopher John Harris (1998). Adaptive Neural Network Control of Robotic Manipulators. World Scientific. pp. 47–48. ISBN 981-02-3452-X. In the above Euler–Lagrange equations, there are three types of terms. The first involves the second derivative of the generalized co-ordinates. The second is quadratic in q ˙ {\displaystyle {\boldsymbol {\dot {q}}}} where the coefficients may depend on q {\displaystyle {\boldsymbol {q}}} . These are further classified into two types. Terms involving a product of the type q ˙ i 2 {\displaystyle {{\dot {q}}_{i}}^{2}} are called centrifugal forces while those involving a product of the type q ˙ i q ˙ j {\displaystyle {\dot {q}}_{i}{\dot {q}}_{j}} for i ≠ j are called Coriolis forces. The third type is functions of q {\displaystyle {\boldsymbol {q}}} only and are called gravitational forces.
Jump up ^ R. K. Mittal; I. J. Nagrath (2003). Robotics and Control. Tata McGraw-Hill. p. 202. ISBN 0-07-048293-4.
Jump up ^ T Yanao; K Takatsuka (2005). "Effects of an intrinsic metric of molecular internal space". In Mikito Toda; Tamiki Komatsuzaki; Stuart A. Rice; Tetsuro Konishi; R. Stephen Berry. Geometrical Structures Of Phase Space In Multi-dimensional Chaos: Applications to chemical reaction dynamics in complex systems. Wiley. p. 98. ISBN 0-471-71157-8. As is evident from the first terms ..., which are proportional to the square of ϕ ˙ {\displaystyle {\dot {\phi }}} , a kind of "centrifugal force" arises ... We call this force "democratic centrifugal force". Of course, DCF is different from the ordinary centrifugal force, and it arises even in a system of zero angular momentum.
Jump up ^ See p. 5 in Donato Bini; Paolo Carini; Robert T Jantzen (1997). "The intrinsic derivative and centrifugal forces in general relativity: I. Theoretical foundations". International Journal of Modern Physics D. 6 (1). . The companion paper is Donato Bini; Paolo Carini; Robert T Jantzen (1997). "The intrinsic derivative and centrifugal forces in general relativity: II. Applications to circular orbits in some stationary axisymmetric spacetimes". International Journal of Modern Physics D. 6 (1).
Jump up ^ Mook, Delo E. & Thomas Vargish (1987). Inside relativity. Princeton NJ: Princeton University Press. ISBN 0-691-02520-7, p. 47.
Jump up ^ G. David Scott (1957). "Centrifugal Forces and Newton's Laws of Motion". 25. American Journal of Physics. p. 325.
Jump up ^ Signell, Peter (2002). "Acceleration and force in circular motion" Physnet. Michigan State University, "Acceleration and force in circular motion", §5b, p. 7.
Jump up ^ Mohanty, A. K. (2004). Fluid Mechanics. PHI Learning Pvt. Ltd. ISBN 81-203-0894-8, p. 121.
Jump up ^ Roche, John (September 2001). "Introducing motion in a circle". Physics Education 43 (5), pp. 399-405, "Introducing motion in a circle". Retrieved 2009-05-07.
Jump up ^ Lloyd William Taylor (1959). Physics, the pioneer science. 1. Dover Publications. p. 173.
Jump up ^ Edward Albert Bowser (1920). An elementary treatise on analytic mechanics: with numerous examples (25th ed.). D. Van Nostrand Company. p. 357.
Jump up ^ Joseph A. Angelo (2007). Robotics: a reference guide to the new technology. Greenwood Press. p. 267. ISBN 1-57356-337-4.
Jump up ^ Eric M Rogers (1960). Physics for the Inquiring Mind. Princeton University Press. p. 302.
Authority control
LCCN: sh85021931
GND: 4257608-8
Retrieved from "https://en.wikipedia.org/w/index.php?title=Centrifugal_force&oldid=831927579"
Categories:
Fictitious forces
Force
Mechanics
Rotation
Acceleration
Hidden categories:
Webarchive template wayback links
All articles with unsourced statements
Articles with unsourced statements from December 2012
Wikipedia articles with LCCN identifiers
Wikipedia articles with GND identifiers
Navigation menu
Personal tools
Not logged in
Talk
Contributions
Create account
Log in
Namespaces
Article
Talk
Variants
Views
Read
Edit
View history
More
Search
Navigation
Main page
Contents
Featured content
Current events
Random article
Donate to Wikipedia
Wikipedia store
Interaction
Help
About Wikipedia
Community portal
Recent changes
Contact page
Tools
What links here
Related changes
Upload file
Special pages
Permanent link
Page information
Wikidata item
Cite this page
Print/export
Create a book
Download as PDF
Printable version
In other projects
Wikimedia Commons
Languages
Afrikaans
العربية
Asturianu
Azərbaycanca
Беларуская
Български
Català
Čeština
ChiShona
Dansk
Deutsch
Eesti
Ελληνικά
Español
Esperanto
Euskara
فارسی
Français
Galego
한국어
Հայերեն
हिन्दी
Hrvatski
Bahasa Indonesia
Íslenska
Italiano
עברית
Қазақша
Kreyòl ayisyen
Magyar
Македонски
മലയാളം
Bahasa Melayu
မြန်မာဘာသာ
Nederlands
日本語
Norsk
Norsk nynorsk
Occitan
Polski
Português
Русский
Sicilianu
Simple English
Slovenčina
Slovenščina
کوردی
Српски / srpski
Srpskohrvatski / српскохрватски
Suomi
Svenska
தமிழ்
Татарча/tatarça
ไทย
Türkçe
Українська
Tiếng Việt
粵語
中文
50 more
Edit links
This page was last edited on 22 March 2018, at 21:41.
Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
Privacy policy
About Wikipedia
Disclaimers
Contact Wikipedia
Developers
Cookie statement
Mobile view
Enable previews

values: mw-page-base noprint

values: mw-head-base noprint

values: firstHeading firstHeading en

values: siteSub noprint

values: jump-to-nav mw-jump

attrs: role class

values:
note
hatnote navigation-not-searchable

value: td

value: tr

value: th

value: mi

value: mo

value: mover

value: mrow

values: vertical-align: -0.338ex; width:8.14ex; height:2.843ex;

values:
/wiki/Second_law_of_motion
mw-redirect
Second law of motion

values: font-size: 105%; padding: 0.35em 0px 0.4em; text-align: center; display: none;

values: float:right;clear:right;width:22.0em;margin:0 0 1.0em 1.0em;background:#f9f9f9;border:1px solid #aaa;padding:0.2em;border-spacing:0.4em 0;text-align:center;line-height:1.4em;font-size:88%

values:
/wiki/Newtonian_mechanics
mw-redirect
Newtonian mechanics

values:
/wiki/Inertial_force
mw-redirect
Inertial force

values:
/wiki/Planetary_orbit
mw-redirect
Planetary orbit

values:
/wiki/Banked_curve
mw-redirect
Banked curve

values:
/wiki/Rotating_coordinate_system
mw-redirect
Rotating coordinate system

values: Edit section: Vehicle driving round a curve

values:
/wiki/Gravitational_force
mw-redirect
Gravitational force

values: Edit section: Weight of an object at the poles and on the equator

values:
/wiki/Spring_balance
mw-redirect
Spring balance

values:
/wiki/Oblate_spheroid
mw-redirect
Oblate spheroid

values:
/wiki/Rotating_frame_of_reference
mw-redirect
Rotating frame of reference

values:
/wiki/Inertial_reference_frame
mw-redirect
Inertial reference frame

values: Edit section: Time derivatives in a rotating frame

values: vertical-align: -2.505ex; width:25.067ex; height:6.176ex;

values: vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;

values:
/wiki/Vector_cross_product
mw-redirect
Vector cross product

values: vertical-align: -0.338ex; width:6.478ex; height:2.176ex;

values: {\displaystyle {\boldsymbol {F}}=m{\boldsymbol {a}}\ ,}

values: vertical-align: -0.671ex; width:9.718ex; height:2.509ex;

values: vertical-align: -2.171ex; width:10.597ex; height:6.009ex;

values: {\boldsymbol  {a}}={\frac  {\operatorname {d}^{2}{\boldsymbol  {r}}}{\operatorname {d}t^{2}}}\ ,

attrs: columnalign rowspacing columnspacing displaystyle

values: right left right left right left right left right left right left

values: 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em

values: vertical-align: -12.171ex; width:63.353ex; height:25.509ex;

values: vertical-align: -2.505ex; width:48.707ex; height:6.176ex;

values: {\boldsymbol  {F}}-m{\frac  {\operatorname {d}{\boldsymbol  {\omega }}}{\operatorname {d}t}}\times {\boldsymbol  {r}}-2m{\boldsymbol  {\omega }}\times \left[{\frac  {\operatorname {d}{\boldsymbol  {r}}}{\operatorname {d}t}}\right]-m{\boldsymbol  {\omega }}\times ({\boldsymbol  {\omega }}\times {\boldsymbol  {r}})

values: vertical-align: -2.505ex; width:13.75ex; height:6.343ex;

values: =m\left[{\frac  {\operatorname {d}^{2}{\boldsymbol  {r}}}{\operatorname {d}t^{2}}}\right]\ .

values: vertical-align: -0.838ex; width:15.723ex; height:2.843ex;

values: vertical-align: -0.838ex; width:17.792ex; height:2.843ex;

values: vertical-align: -0.838ex; width:15.906ex; height:2.843ex;

values: {\displaystyle ({\boldsymbol {\omega }}=0)}

values: vertical-align: -0.838ex; width:7.739ex; height:2.843ex;

values:
/wiki/Rotating_bucket
mw-redirect
Rotating bucket

values:
/wiki/Buoyant_force
mw-redirect
Buoyant force

values:
/wiki/Amusement_ride
mw-redirect
Amusement ride

values: Edit section: History of conceptions of centrifugal and centripetal forces

values: History of centrifugal and centripetal forces

values: This claim needs references to reliable sources. (December 2012)

values:
noprint Inline-Template Template-Fact
white-space:nowrap;

values: vertical-align: -0.838ex; width:5.563ex; height:2.843ex;

values:
/wiki/Newton%27s_laws
mw-redirect
Newton's laws

values:
/wiki/Euler%E2%80%93Lagrange_equations
mw-redirect
Euler–Lagrange equations

values:
/wiki/Centripetal_acceleration
mw-redirect
Centripetal acceleration

values: display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle

values: margin:0.5em 0 0.5em 1em;border:solid #aaa 1px

values:
/wiki/Folk_physics
mw-redirect
Folk physics

values: vertical-align: -2.005ex; width:3.746ex; height:5.509ex;

values: vertical-align: -2.505ex; width:9.943ex; height:6.176ex;

values: {\displaystyle {\boldsymbol {\dot {q}}}}

values: vertical-align: -0.671ex; width:1.571ex; height:2.676ex;

values: vertical-align: -0.671ex; width:1.279ex; height:2.009ex;

values: vertical-align: -0.838ex; width:3.231ex; height:3.176ex;

values: {\displaystyle {\dot {q}}_{i}{\dot {q}}_{j}}

values: vertical-align: -1.171ex; width:4.464ex; height:3.009ex;

values: vertical-align: -0.671ex; width:1.467ex; height:3.009ex;

values:
nofollow
external text
https://arxiv.org/abs/gr-qc/0106014v1

values: -moz-column-width: 30em; -webkit-column-width: 30em; column-width: 30em; list-style-type: decimal;

values:
nofollow
external text
https://d-nb.info/gnd/4257608-8

values: Category:Webarchive template wayback links

values: Category:All articles with unsourced statements

values: Category:Articles with unsourced statements from December 2012

values: Category:Wikipedia articles with LCCN identifiers

values: Category:Wikipedia articles with GND identifiers

values: Discussion about edits from this IP address [ctrl-alt-n]

values: A list of edits made from this IP address [ctrl-alt-y]

values: You are encouraged to create an account and log in; however, it is not mandatory

values: You're encouraged to log in; however, it's not mandatory. [ctrl-alt-o]

values:
/wiki/Centrifugal_force
View the content page [ctrl-alt-c]
c

values: Discussion about the content page [ctrl-alt-t]

values: p-namespaces navigation vectorTabs p-namespaces-label

values:
p-variants
navigation
vectorMenu emptyPortlet
p-variants-label

values: Past revisions of this page [ctrl-alt-h]

values:
mw-wiki-logo
/wiki/Main_Page
Visit the main page

values:
/wiki/Main_Page
Visit the main page [ctrl-alt-z]
z

values: Featured content – the best of Wikipedia

values: Find background information on current events

values:
/wiki/Special:Random
Load a random article [ctrl-alt-x]
x

values: portal navigation p-wikibase-otherprojects p-wikibase-otherprojects-label

values:
interlanguage-link interwiki-af
display: none;

values: قوة الطرد المركزي – Arabic

values:
interlanguage-link interwiki-ast
display: none;

values:
interlanguage-link interwiki-az
display: none;

values: Цэнтрабежная сіла – Belarusian

values:
interlanguage-link interwiki-be
display: none;

values: Центробежна сила – Bulgarian

values:
interlanguage-link interwiki-bg
display: none;

values:
interlanguage-link interwiki-ca
display: none;

values:
interlanguage-link interwiki-cs
display: none;

values:
interlanguage-link interwiki-sn
display: none;

values:
interlanguage-link interwiki-da
display: none;

values:
interlanguage-link interwiki-de
display: none;

values:
interlanguage-link interwiki-et
display: none;

values: Φυγόκεντρος δύναμη – Greek

values:
interlanguage-link interwiki-el
display: none;

values:
interlanguage-link interwiki-eo
display: none;

values:
interlanguage-link interwiki-eu
display: none;

values: نیروی گریز از مرکز – Persian

values:
interlanguage-link interwiki-fa
display: none;

values:
interlanguage-link interwiki-gl
display: none;

values:
interlanguage-link interwiki-ko
display: none;

values: Կենտրոնախույս ուժ – Armenian

values:
interlanguage-link interwiki-hy
display: none;

values: अपकेन्द्रिय बल – Hindi

values: Centrifugalna i centripetalna sila – Croatian

values:
interlanguage-link interwiki-hr
display: none;

values:
interlanguage-link interwiki-is
display: none;

values:
interlanguage-link interwiki-it
display: none;

values: כוח צנטריפוגלי – Hebrew

values:
interlanguage-link interwiki-he
display: none;

values: Центрден тепкіш күш – Kazakh

values:
interlanguage-link interwiki-kk
display: none;

values:
interlanguage-link interwiki-ht
display: none;

values:
interlanguage-link interwiki-hu
display: none;

values: Центрифугална сила – Macedonian

values:
interlanguage-link interwiki-mk
display: none;

values: അപകേന്ദ്രബലം – Malayalam

values:
interlanguage-link interwiki-ml
display: none;

values: ဗဟိုခွာအား – Burmese

values:
interlanguage-link interwiki-my
display: none;

values:
interlanguage-link interwiki-nl
display: none;

values:
interlanguage-link interwiki-ja
display: none;

values:
interlanguage-link interwiki-no
display: none;

values:
interlanguage-link interwiki-nn
display: none;

values:
interlanguage-link interwiki-oc
display: none;

values:
interlanguage-link interwiki-pl
display: none;

values: Центробежная сила – Russian

values:
interlanguage-link interwiki-scn
display: none;

values:
interlanguage-link interwiki-simple
display: none;

values:
interlanguage-link interwiki-sk
display: none;

values:
interlanguage-link interwiki-sl
display: none;

values: ھێزی دژەناوەند – Central Kurdish

values:
interlanguage-link interwiki-ckb
display: none;

values: Центрифугална сила – Serbian

values:
interlanguage-link interwiki-sr
display: none;

values:
interlanguage-link interwiki-sh
display: none;

values:
interlanguage-link interwiki-fi
display: none;

values:
interlanguage-link interwiki-sv
display: none;

values: மையவிலக்கு விசை – Tamil

values:
interlanguage-link interwiki-ta
display: none;

values: Үзәктән куу көче – Tatar

values:
interlanguage-link interwiki-tt
display: none;

values: แรงหนีศูนย์กลาง – Thai

values:
interlanguage-link interwiki-th
display: none;

values:
interlanguage-link interwiki-tr
display: none;

values: Відцентрова сила – Ukrainian

values:
interlanguage-link interwiki-uk
display: none;

values:
interlanguage-link interwiki-vi
display: none;

values:
interlanguage-link interwiki-zh-yue
display: none;

values: All languages (initial selection from common choices by you and others)

values: /static/images/wikimedia-button-1.5x.png 1.5x, /static/images/wikimedia-button-2x.png 2x

values: /static/images/poweredby_mediawiki_132x47.png 1.5x, /static/images/poweredby_mediawiki_176x62.png 2x

values:
suggestions
display: none; font-size: 13px;

values: mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Centrifugal_force rootpage-Centrifugal_force skin-vector action-view