View source for Power (physics)

{{Short description|Amount of energy transferred or converted per unit time}}
{{Use dmy dates|date=July 2021}}
{{Other uses|Power (disambiguation)}}{{Infobox physical quantity
| name = Power
| unit = [[watt]] (W)
| symbols = {{mvar|P}}
| baseunits = [[kilogram|kg]]⋅[[metre|m]]{{sup|2}}⋅[[second|s]]{{sup|−3}}
| dimension = wikidata
| derivations =
{{ublist
  | {{math|1=''P'' = [[Energy|''E'']]/[[Time|''t'']]}}
  | {{math|1=''P'' = [[Force|''F'']]·[[Velocity|''v'']]}}
  | {{math|1=''P'' = [[Voltage|''V'']]·[[Electric current|''I'']] }}
  | {{math|1=''P'' = [[Torque|''τ'']]·[[Angular velocity|''ω'']] }}
}}
}}
{{Classical mechanics}}
'''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 one [[joule]] per second. Power is a [[Scalar (physics)|scalar]] quantity.

The output power of a [[Engine|motor]] is the product of the [[torque]] that the motor generates and the [[angular velocity]] of its output shaft.  Likewise, the power dissipated in an [[electrical element]] of a [[electrical circuit|circuit]] is the product of the [[electric current|current]] flowing through the element and of the [[voltage]] across the element.<ref>{{Cite book |chapter= 6. Power |author=David Halliday |author2=Robert Resnick |title=Fundamentals of Physics |year=1974}}</ref><ref>Chapter 13, § 3, pp 13-2,3 ''[[The Feynman Lectures on Physics]]'' Volume I, 1963</ref>

==Definition==
Power is the [[Rate (mathematics)|rate]] with respect to time at which work is done or, more generally, the rate of change of total mechanical energy. It is given by:
<math display="block">P = \frac{dE}{dt},</math>
where {{mvar|P}} is power, {{mvar|E}} is the total mechanical energy (sum of kinetic and potential energy), and {{mvar|t}} is time.

For cases where only work is considered, power is also expressed as:
<math display="block">P = \frac{dW}{dt},</math>
where {{mvar|W}} is the work done on the system. However, in systems where potential energy changes without explicit work being done (e.g., changing fields or conservative forces), the total energy definition is more general.

We will now show that the mechanical power generated by a force F on a body moving at the velocity v can be expressed as the product: <math display="block">P = \frac{dW}{dt} = \mathbf{F} \cdot \mathbf {v}</math>

If a ''constant'' force '''F''' is applied throughout a [[distance]] '''x''', the work done is defined as <math>W = \mathbf{F} \cdot \mathbf{x}</math>. In this case, power can be written as:
<math display="block">P = \frac{dW}{dt} = \frac{d}{dt} \left(\mathbf{F} \cdot \mathbf{x}\right) = \mathbf{F}\cdot \frac{d\mathbf{x}}{dt} = \mathbf{F} \cdot \mathbf {v}.</math>

If instead the force is ''variable over a three-dimensional curve C'', then the work is expressed in terms of the line integral:
<math display="block">W = \int_C \mathbf{F} \cdot d\mathbf {r}
  = \int_{\Delta t} \mathbf{F} \cdot \frac{d\mathbf {r}}{dt} \  dt
  = \int_{\Delta t} \mathbf{F} \cdot \mathbf {v} \, dt.</math>

From the [[fundamental theorem of calculus]], we know that <math display="block">P = \frac{dW}{dt} = \frac{d}{dt} \int_{\Delta t} \mathbf{F} \cdot \mathbf {v} \, dt = \mathbf{F} \cdot \mathbf {v}.</math> Hence the formula is valid for any general situation.

In older works, power is sometimes called ''activity''.<ref name="Smithsonian Tables">{{cite book|title=Smithsonian Physical Tables|publisher=[[Smithsonian Institution]]|date=1921|editor-first1=Frederick E.|editor-last1=Fowle |edition=7th revised |location=Washington, D.C.|url=https://books.google.com/books?id=tCoJAQAAIAAJ&q=%22Power%20or%20Activity%20is%20the%20time%20rate%20of%20doing%20work%22|oclc=1142734534|archive-url=https://web.archive.org/web/20200423151426/https://www.google.com/books/edition/Smithsonian_Physical_Tables/tCoJAQAAIAAJ?hl=en&gbpv=1&bsq=%22Power%20or%20Activity%20is%20the%20time%20rate%20of%20doing%20work%22 |archive-date=23 April 2020|url-status=live |quote='''Power or Activity''' is the time rate of doing work, or if {{math|''W''}} represents work and {{math|''P''}} power, {{math|1=''P'' = ''dw''/''dt''}}. (p. xxviii) ... ACTIVITY. Power or rate of doing work; unit, the watt. (p. 435)}}</ref><ref name="Heron Motors">{{cite journal |last1=Heron|first1=C. A. |date=1906 |title=Electrical Calculations for Railway Motors |url=https://books.google.com/books?id=b5A4AQAAMAAJ&dq=%22The+activity+of+a+motor+is+the+work+done+per+second%22+%22Where+the+joule+is+employed+as+the+unit+of+work,+the+international+unit+of+activity+is+the+joule-per-second,+or,+as+it+is+commonly+called,+the+watt.%22&pg=PA78 |journal=Purdue Eng. Rev.|issue=2 |pages=77–93 |access-date=23 April 2020 |archive-url=https://web.archive.org/web/20200423142933/https://www.google.com/books/edition/The_Purdue_Engineering_Review/b5A4AQAAMAAJ?hl=en&gbpv=1&dq=%22The+activity+of+a+motor+is+the+work+done+per+second%22+%22Where+the+joule+is+employed+as+the+unit+of+work,+the+international+unit+of+activity+is+the+joule-per-second,+or,+as+it+is+commonly+called,+the+watt.%22&pg=PA78&printsec=frontcover |archive-date=23 April 2020 |url-status=live |quote=The activity of a motor is the work done per second, ... Where the joule is employed as the unit of work, the international unit of activity is the joule-per-second, or, as it is commonly called, the watt. (p. 78)}}</ref><ref name="Nature 1902">{{cite journal|date=1902 |title=Societies and Academies |journal=Nature |volume=66|issue=1700 |pages=118–120 |doi=10.1038/066118b0 |bibcode=1902Natur..66R.118. |quote=If the watt is assumed as unit of activity... |doi-access=free}}</ref>

==Units==
The dimension of power is energy divided by time. In the [[International System of Units]] (SI), the unit of power is the [[watt]] (W), which is equal to one [[joule]] per second. Other common and traditional measures are [[horsepower]] (hp), comparing to the power of a horse; one [[horsepower#Mechanical horsepower|''mechanical horsepower'']] equals about 745.7 watts. Other units of power include [[erg]]s per second (erg/s), [[foot-pound force|foot-pounds]] per minute, [[dBm]], a logarithmic measure relative to a reference of 1 milliwatt, [[calorie]]s per hour, [[BTU]] per hour (BTU/h), and [[ton of refrigeration|tons of refrigeration]].

==Average power and instantaneous power==
As a simple example, burning one kilogram of [[coal]] releases more energy than detonating a kilogram of [[TNT]],<ref>Burning coal produces around 15-30 [[megajoule]]s per kilogram, while detonating TNT produces about 4.7 megajoules per kilogram. For the coal value, see {{cite web | last = Fisher | first = Juliya | title = Energy Density of Coal | work = The Physics Factbook | url = http://hypertextbook.com/facts/2003/JuliyaFisher.shtml|year=2003|access-date =30 May 2011}} For the TNT value, see the article [[TNT equivalent]]. Neither value includes the weight of oxygen from the air used during combustion.</ref> but because the TNT reaction releases energy more quickly, it delivers more power than the coal.
If {{math|Δ''W''}}  is the amount of [[mechanical work|work]] performed during a period of [[time]] of duration {{math|Δ''t''}}, the average power {{math|''P''<sub>avg</sub>}} over that period is given by the formula
<math display="block">P_\mathrm{avg} = \frac{\Delta W}{\Delta t}.</math>
It is the average amount of work done or energy converted per unit of time. Average power is often called "power" when the context makes it clear.

Instantaneous power is the limiting value of the average power as the time interval {{math|Δ''t''}} approaches zero.
<math display="block">P = \lim_{\Delta t \to 0} P_\mathrm{avg} = \lim_{\Delta t \to 0} \frac{\Delta W}{\Delta t} = \frac{dW}{dt}.</math>

When power {{math|''P''}} is constant, the amount of work performed in time period {{mvar|t}} can be calculated as
<math display="block">W = Pt.</math>

In the context of energy conversion, it is more customary to use the symbol {{mvar|E}} rather than {{mvar|W}}.

==Mechanical power==
[[File:Horsepower plain.svg|thumb|One ''metric horsepower'' is needed to lift 75&nbsp;[[kilogram]]s by 1&nbsp;[[metre]] in 1&nbsp;[[second]].]]
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.  In [[mechanics]], the [[mechanical work|work]] done by a force {{math|'''F'''}} on an object that travels along a curve {{mvar|C}} is given by the [[line integral]]:
<math display="block">W_C = \int_C \mathbf{F} \cdot \mathbf{v} \, dt = \int_C \mathbf{F} \cdot d\mathbf{x},</math>
where {{math|'''x'''}} defines the path {{mvar|C}} and {{math|'''v'''}} is the velocity along this path.

If the force {{math|'''F'''}} is derivable from a potential ([[Conservative force|conservative]]), then applying the [[gradient theorem]] (and remembering that force is the negative of the [[gradient]] of the potential energy) yields:
<math display="block">W_C = U(A) - U(B),</math>
where {{mvar|A}} and {{mvar|B}} are the beginning and end of the path along which the work was done.

The power at any point along the curve {{mvar|C}} is the time derivative:
<math display="block">P(t) = \frac{dW}{dt} = \mathbf{F} \cdot \mathbf{v} = -\frac{dU}{dt}.</math>

In one dimension, this can be simplified to:
<math display="block">P(t) = F \cdot v.</math>

In rotational systems, power is the product of the [[torque]] {{math|'''τ'''}} and [[angular velocity]] {{math|'''ω'''}},
<math display="block">P(t) = \boldsymbol{\tau} \cdot \boldsymbol{\omega},</math>
where {{math|'''''ω'''''}} is [[angular frequency]], measured in [[radians per second]].  The <math> \cdot </math> represents [[scalar product]].

In fluid power systems such as [[hydraulic]] actuators, power is given by <math display="block"> P(t) = pQ,</math> where {{mvar|p}} is [[pressure]] in [[pascal (unit)|pascals]] or N/m<sup>2</sup>, and {{mvar|Q}} is [[volumetric flow rate]] in m<sup>3</sup>/s in SI units.

===Mechanical advantage===
If a mechanical system has no losses, then the input power must equal the output power. This provides a simple formula for the [[mechanical advantage]] of the system.

Let the input power to a device be a force {{math|''F''<sub>A</sub>}} acting on a point that moves with velocity {{math|''v''<sub>A</sub>}} and the output power be a force {{math|''F''<sub>B</sub>}} acts on a point that moves with velocity {{math|''v''<sub>B</sub>}}.  If there are no losses in the system, then
<math display="block">P = F_\text{B} v_\text{B} = F_\text{A} v_\text{A},</math>
and the [[mechanical advantage]] of the system (output force per input force) is given by
<math display="block"> \mathrm{MA} = \frac{F_\text{B}}{F_\text{A}} = \frac{v_\text{A}}{v_\text{B}}.</math>

The similar relationship is obtained for rotating systems, where {{math|''T''<sub>A</sub>}} and {{math|''ω''<sub>A</sub>}} are the torque and angular velocity of the input and {{math|''T''<sub>B</sub>}} and {{math|''ω''<sub>B</sub>}} are the torque and angular velocity of the output.  If there are no losses in the system, then
<math display="block">P = T_\text{A} \omega_\text{A} = T_\text{B} \omega_\text{B},</math>
which yields the [[mechanical advantage]]
<math display="block"> \mathrm{MA} = \frac{T_\text{B}}{T_\text{A}} = \frac{\omega_\text{A}}{\omega_\text{B}}.</math>

These relations are important because they define the maximum performance of a device in terms of [[velocity ratio]]s determined by its physical dimensions.  See for example [[gear ratio]]s.

==Electrical power==
{{main|Electric power}}
[[File:Ansel Adams - National Archives 79-AAB-02.jpg|right|thumb|alt=Ansel Adams photograph of electrical wires of the Boulder Dam Power Units|[[Ansel Adams]] photograph of electrical wires of the Boulder Dam Power Units, 1941–1942]]
The instantaneous electrical power ''P'' delivered to a component is given by
<math display="block">P(t) = I(t) \cdot V(t),</math>
where
*<math>P(t)</math> is the instantaneous power, measured in [[watt]]s ([[joule]]s per [[second]]),
*<math>V(t)</math> is the [[voltage|potential difference]] (or voltage drop) across the component, measured in [[volt]]s, and
*<math>I(t)</math> is the [[Electric current|current]] through it, measured in [[ampere]]s.

If the component is a [[resistor]] with time-invariant [[voltage]] to [[electric current|current]] ratio, then:
<math display="block">P = I \cdot V = I^2 \cdot R = \frac{V^2}{R}, </math>
where
<math display="block">R = \frac{V}{I}</math>
is the [[electrical resistance]], measured in [[ohm]]s.

==Peak power and duty cycle==
[[File:peak-power-average-power-tau-T.png|thumb|right|In a train of identical pulses, the instantaneous power is a periodic function of time. The ratio of the pulse duration to the period is equal to the ratio of the average power to the peak power. It is also called the duty cycle (see text for definitions).]]

In the case of a periodic signal <math>s(t)</math> of period <math>T</math>, like a train of identical pulses, the instantaneous power <math display="inline">p(t) = |s(t)|^2</math> is also a periodic function of period <math>T</math>.  The ''peak power'' is simply defined by:
<math display="block">P_0 = \max [p(t)].</math>

The peak power is not always readily measurable, however, and the measurement of the average power <math>P_\mathrm{avg}</math> is more commonly performed by an instrument.  If one defines the energy per pulse as
<math display="block">\varepsilon_\mathrm{pulse} = \int_0^T p(t) \, dt </math>
then the average power is
<math display="block">P_\mathrm{avg} = \frac{1}{T} \int_0^T p(t) \, dt = \frac{\varepsilon_\mathrm{pulse}}{T}. </math>

One may define the pulse length <math>\tau</math> such that <math>P_0\tau = \varepsilon_\mathrm{pulse}</math> so that the ratios
<math display="block">\frac{P_\mathrm{avg}}{P_0} = \frac{\tau}{T} </math>
are equal. These ratios are called the ''duty cycle'' of the pulse train.

==Radiant power==
Power is related to intensity at a radius <math>r</math>; the power emitted by a source can be written as:{{citation needed|date=August 2017}}
<math display="block">P(r) = I(4\pi r^2). </math>

==See also==
* [[Simple machines]]
* [[Orders of magnitude (power)]]
* [[Pulsed power]]
* [[Intensity (physics)|Intensity]] – in the radiative sense, power per area
* [[Power gain]] – for linear, two-port networks
* [[Power density]]
* [[Signal strength]]
* [[Sound power]]

==References==
{{Commons category}}
{{Wikiquote}}
{{Reflist}}

{{Classical mechanics derived SI units}}
{{Authority control}}

[[Category:Power (physics)| ]]
[[Category:Force]]
[[Category:Temporal rates]]
[[Category:Physical quantities]]