In physics, power is the amount of

^{2} and is ^{3}/s in SI units.

energy
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

transferred or converted per unit time. In the International System of Units
The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms_and_initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wi ...

, the unit of power is the watt
The watt (symbol: W) is a unit of power
Power typically refers to:
* Power (physics)
In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equa ...

, equal to one joule
The joule ( ; symbol: J) is a SI derived unit, derived unit of energy in the International System of Units. It is equal to the energy transferred to (or work (physics), work done on) an object when a force of one Newton (unit), newton acts on th ...

per second. In older works, power is sometimes called ''activity''. Power is a scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
*Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...

quantity.
Power is related to other quantities; for example, the power involved in moving a ground vehicle is the product of the traction force on the wheels and the velocity
The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

of the vehicle. The output power of a motor
An engine or motor is a machine
A machine is a man-made device that uses power to apply forces and control movement to perform an action. Machines can be driven by animals and people
A people is a plurality of person
A person ( ...

is the product of the torque
In physics and mechanics, torque is the rotational equivalent of linear force
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the na ...

that the motor generates and the angular velocity
In physics, angular velocity (\boldsymbol or \boldsymbol), also known as angular frequency vector,(UP1) is a vector measure of rotation rate, that refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angu ...

of its output shaft. Likewise, the power dissipated in an electrical element
Electrical elements are conceptual abstractions representing idealized electrical components, such as resistors, capacitor
A capacitor is a device that stores electric charge in an electric field. It is a passivity (engineering), passive el ...

of a circuitCircuit may refer to:
Science and technology
Electrical engineering
* Electrical circuit, a complete electrical network with a closed-loop giving a return path for current
** Analog circuit, uses continuous signal levels
** Balanced circuit, p ...

is the product of the current
Currents or The Current may refer to:
Science and technology
* Current (fluid)
A current in a fluid
In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied shear stress, or external force. ...

flowing through the element and of the voltage
Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is the ...

across the element.
Definition

Power is the rate with respect to time at which work is done; it is the timederivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

of work:
:$P\; =\backslash frac$
where ''P'' is power, ''W'' is work, and ''t'' is time.
If a constant force F is applied throughout a distance
Distance is a numerical measurement
'
Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

x, the work done is defined as $W\; =\; \backslash mathbf\; \backslash cdot\; \backslash mathbf$. In this case, power can be written as:
$P\; =\backslash frac=\; \backslash frac\backslash left(\backslash mathbf\backslash cdot\backslash mathbf\backslash right)=\; \backslash mathbf\backslash cdot\; \backslash frac\; =\; \backslash mathbf\; \backslash cdot\; \backslash mathbf$
If instead the force is variable over a three-dimensional curve C, then the work is expressed in terms of the line integral:
$W\; =\; \backslash int\_C\; \backslash mathbf\; \backslash cdot\; d\backslash mathbf\; =\; \backslash int\_\; \backslash mathbf\; \backslash cdot\; \backslash frac\; \backslash \; dt\; =\; \backslash int\_\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; \backslash \; dt$
From the fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of derivative, differentiating a function (mathematics), function (calculating the gradient) with the concept of integral, integrating a function (calculating the area under t ...

, we know that $P\; =\backslash frac=\; \backslash frac\; \backslash int\_\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; \backslash \; dt\; =\; \backslash mathbf\; \backslash cdot\; \backslash mathbf$. Hence the formula is valid for any general situation.
Units

The dimension of power is energy divided by time. In theInternational System of Units
The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms_and_initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wi ...

(SI), the unit of power is the watt
The watt (symbol: W) is a unit of power
Power typically refers to:
* Power (physics)
In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equa ...

(W), which is equal to one joule
The joule ( ; symbol: J) is a SI derived unit, derived unit of energy in the International System of Units. It is equal to the energy transferred to (or work (physics), work done on) an object when a force of one Newton (unit), newton acts on th ...

per second. Other common and traditional measures are horsepower
Horsepower (hp) is a unit of measurement
A unit of measurement is a definite magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the ...

(hp), comparing to the power of a horse; one ''mechanical horsepower'' equals about 745.7 watts. Other units of power include erg
The erg is a unit of energy equal to 10−7joule
The joule ( ; symbol: J) is a SI derived unit, derived unit of energy in the International System of Units. It is equal to the energy transferred to (or work (physics), work done on) an objec ...

s per second (erg/s), foot-pounds
The foot-pound force (symbol: ft⋅lbf, ft⋅lbf, or ft⋅lb ) is a unit of Mechanical work, work or energy in the English Engineering Units, engineering and Foot–pound–second_system#force, gravitational systems in United States customary u ...

per minute, dBm
dBm or dBmW (decibel-milliwatts) is a unit of level
Level or levels may refer to:
Engineering
* Level (instrument), a device used to measure true horizontal or relative heights
*Canal pound or level
*Regrading or levelling, the process of rais ...

, a logarithmic measure relative to a reference of 1 milliwatt, calorie
The calorie is a unit of energy
Unit may refer to:
Arts and entertainment
* UNIT
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a di ...

s per hour, BTU
The British thermal unit (BTU or Btu) is a unit of heat
In thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical ...

per hour (BTU/h), and tons of refrigeration.
Average power

As a simple example, burning one kilogram ofcoal
Coal is a combustible
, Germany
)
, image_map =
, map_caption =
, map_width = 250px
, capital = Berlin
, coordinates =
, largest_city = capital
, languages_type = Official language
, languages = German language, German
, ...

releases much more energy than detonating a kilogram of TNT
Trinitrotoluene (; TNT), or more specifically 2,4,6-trinitrotoluene, is a chemical compound
A chemical compound is a chemical substance composed of many identical molecules (or molecular entity, molecular entities) composed of atoms from more ...

,Burning coal produces around 15-30 megajoule
The joule ( ; symbol: J) is a derived unit of energy
In physics, energy is the physical quantity, quantitative physical property, property that must be #Energy transfer, transferred to a physical body, body or physical system to perform W ...

s per kilogram, while detonating TNT produces about 4.7 megajoules per kilogram. For the coal value, see For the TNT value, see the article TNT equivalent
TNT equivalent is a convention for expressing energy, typically used to describe the energy released in an explosion. The is a unit of energy
As energy
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ ...

. Neither value includes the weight of oxygen from the air used during combustion. but because the TNT reaction releases energy much more quickly, it delivers far more power than the coal.
If is the amount of work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
* Work (physics), the product of ...

performed during a period of time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

of duration , the average power over that period is given by the formula:
:$P\_\backslash mathrm\; =\; \backslash frac$
It is the average amount of work done or energy converted per unit of time. The average power is often simply called "power" when the context makes it clear.
The instantaneous power is then the limiting value of the average power as the time interval approaches zero.
:$P\; =\; \backslash lim\; \_\; P\_\backslash mathrm\; =\; \backslash lim\; \_\; \backslash frac\; =\; \backslash frac$
In the case of constant power , the amount of work performed during a period of duration is given by:
:$W\; =\; Pt$
In the context of energy conversion, it is more customary to use the symbol rather than .
Mechanical power

Power in mechanical systems is the combination of forces and movement. In particular, power is the product of a force on an object and the object's velocity, or the product of a torque on a shaft and the shaft's angular velocity. Mechanical power is also described as the time derivative of work. Inmechanics
Mechanics (Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million ...

, the work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
* Work (physics), the product of ...

done by a force on an object that travels along a curve is given by the line integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

:
: $W\_C\; =\; \backslash int\_\backslash mathbf\backslash cdot\; \backslash mathbf\backslash ,\backslash mathrmt\; =\backslash int\_\; \backslash mathbf\; \backslash cdot\; \backslash mathrm\backslash mathbf$
where defines the path and is the velocity along this path.
If the force is derivable from a potential (conservative
Conservatism is an aesthetic
Aesthetics, or esthetics (), is a branch of philosophy that deals with the nature of beauty and taste (sociology), taste, as well as the philosophy of art (its own area of philosophy that comes out of aest ...

), then applying the gradient theorem
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a Conservative vector field, gradient field can be evaluated by evaluating the original scalar field at the endpoints of ...

(and remembering that force is the negative of the gradient
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Prod ...

of the potential energy) yields:
:$W\_C\; =\; U(A)-U(B)$
where and are the beginning and end of the path along which the work was done.
The power at any point along the curve is the time derivative:
:$P(t)\; =\; \backslash frac=\backslash mathbf\backslash cdot\; \backslash mathbf=-\backslash frac$
In one dimension, this can be simplified to:
:$P(t)\; =\; F\backslash cdot\; v$
In rotational systems, power is the product of the torque
In physics and mechanics, torque is the rotational equivalent of linear force
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the na ...

and angular velocity
In physics, angular velocity (\boldsymbol or \boldsymbol), also known as angular frequency vector,(UP1) is a vector measure of rotation rate, that refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angu ...

,
:$P(t)\; =\; \backslash boldsymbol\; \backslash cdot\; \backslash boldsymbol$
where measured in radian
The radian, denoted by the symbol \text, is the SI unit for measuring angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''verte ...

s per second. The $\backslash cdot$ represents scalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

.
In fluid power systems such as hydraulic actuators, power is given by
:$P(t)\; =\; pQ$
where is pressure
Pressure (symbol: ''p'' or ''P'') is the force
In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

in pascals, or N/mvolumetric flow rate
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

in mMechanical advantage

If a mechanical system has no losses, then the input power must equal the output power. This provides a simple formula for themechanical advantage
Mechanical advantage is a measure of the force
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and ti ...

of the system.
Let the input power to a device be a force acting on a point that moves with velocity and the output power be a force acts on a point that moves with velocity . If there are no losses in the system, then
:$P\; =\; F\_\backslash text\; v\_\backslash text\; =\; F\_\backslash text\; v\_\backslash text$
and the mechanical advantage
Mechanical advantage is a measure of the force
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and ti ...

of the system (output force per input force) is given by
: $\backslash mathrm\; =\; \backslash frac\; =\; \backslash frac$
The similar relationship is obtained for rotating systems, where and are the torque and angular velocity of the input and and are the torque and angular velocity of the output. If there are no losses in the system, then
:$P\; =\; T\_\backslash text\; \backslash omega\_\backslash text\; =\; T\_\backslash text\; \backslash omega\_\backslash text$
which yields the mechanical advantage
Mechanical advantage is a measure of the force
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and ti ...

:$\backslash mathrm\; =\; \backslash frac\; =\; \backslash frac$
These relations are important because they define the maximum performance of a device in terms of velocity ratio
The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction o ...

s determined by its physical dimensions. See for example gear ratio
A gear train is a mechanical system
A machine is any physical system with ordered structural and functional properties. It may represent human-made or naturally occurring device molecular machine
A molecular machine, nanite, or nanomachin ...

s.
Electrical power

The instantaneous electrical power ''P'' delivered to a component is given by :$P(t)\; =\; I(t)\; \backslash cdot\; V(t)$ where :$P(t)$ is the instantaneous power, measured inwatt
The watt (symbol: W) is a unit of power
Power typically refers to:
* Power (physics)
In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equa ...

s (joule
The joule ( ; symbol: J) is a SI derived unit, derived unit of energy in the International System of Units. It is equal to the energy transferred to (or work (physics), work done on) an object when a force of one Newton (unit), newton acts on th ...

s per second
The second (symbol: s, also abbreviated: sec) is the base unit of time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, th ...

)
:$V(t)$ is the potential difference
Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is def ...

(or voltage drop) across the component, measured in volt
The volt is the derived unit for electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work (physics), work energy needed to move a ...

s
:$I(t)$ is the current
Currents or The Current may refer to:
Science and technology
* Current (fluid)
A current in a fluid
In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied shear stress, or external force. ...

through it, measured in ampere
The ampere (, ; symbol: A), often shortened to "amp",SI supports only the use of symbols and deprecates the use of abbreviations for units. is the base unit of electric current
An electric current is a stream of charged particles, such as ele ...

s
If the component is a resistor
A resistor is a passive
Passive may refer to:
* Passive voice, a grammatical voice common in many languages, see also Pseudopassive (disambiguation), Pseudopassive
* Passive language, a language from which an interpreter works
* Passivity (b ...

with time-invariant voltage
Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is the ...

to current
Currents or The Current may refer to:
Science and technology
* Current (fluid)
A current in a fluid
In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied shear stress, or external force. ...

ratio, then:
:$P=I\; \backslash cdot\; V\; =\; I^2\; \backslash cdot\; R\; =\; \backslash frac$
where
:$R\; =\; \backslash frac$
is the resistance
Resistance may refer to:
Arts, entertainment, and media Comics
* Either of two similarly named but otherwise unrelated comic book series, both published by Wildstorm:
** ''Resistance'' (comics), based on the video game of the same title
** ''Th ...

, measured in ohm
The ohm (symbol: Ω) is the SI derived unit
SI derived units are units of measurement
'
Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, ...

s.
Peak power and duty cycle

In the case of a periodic signal $s(t)$ of period $T$, like a train of identical pulses, the instantaneous power $p(t)\; =\; ,\; s(t),\; ^2$ is also a periodic function of period $T$. The ''peak power'' is simply defined by: :$P\_0\; =\; \backslash max;\; href="/html/ALL/s/(t).html"\; ;"title="(t)">(t)$Radiant power

Power is related to intensity at a radius $r$; the power emitted by a source can be written as: :$P(r)\; =\; I(4\backslash pi\; r^2)$See also

*Simple machines
A simple machine is a mechanical device that changes the direction or magnitude of a force
In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with m ...

* Orders of magnitude (power)
This page lists examples of the power
Power typically refers to:
* Power (physics)
In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to ...

* Pulsed powerPulsed power is the science and technology of accumulating energy over a relatively long period of time and releasing it instantly, thus increasing the instantaneous power. They can be used in some applications such as food processing, water treatmen ...

* Intensity – in the radiative sense, power per area
* Power gainThe power gain of an electrical network
An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductor
An inductor, also called a coil, choke, or reactor, is a incremental passivity, passive two-t ...

– for linear, two-port networks
* Power density
Power density is the amount of power
Power typically refers to:
* Power (physics)
In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to ...

* Signal strength
In telecommunications
Telecommunication is the transmission of information
Information can be thought of as the resolution of uncertainty; it answers the question of "What an entity is" and thus defines both its essence and the nature of ...

* Sound power
Sound power or acoustic power is the rate at which sound energy is emitted, reflected, transmitted or received, per unit time. It is defined as "through a surface, the product of the sound pressure, and the component of the particle velocity, at ...

References

{{Authority control Force Temporal rates Physical quantities