View source for Power (physics)

{{Use dmy dates|date=July 2012}}
{{ Infobox physical quantity
| name = Power
| unit = [[watt]] (W)
| symbols = {{mvar|P}}
| baseunits = [[kilogram|kg]]&sdot;[[meter|m]]{{sup|2}}&sdot;[[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|''T'']]·[[Angular velocity|''ω'']] }}
}}
  }}
{{Classical mechanics}}
In physics, '''power''' is the rate of doing [[Work (physics)|work]] or of [[transferring heat]], i.e. the amount of [[energy (physics)|energy]] transferred or converted per unit time.  Having no direction, it is a [[Scalar (physics)|scalar]] quantity.  In the [[International System of Units]], the unit of power is the [[joule]] per [[second]] (J/s), known as the [[watt]] (W) in honour of [[James Watt]], the eighteenth-century developer of the [[Watt steam engine|condenser steam engine]].  Another common and traditional measure is [[horsepower]] (comparing to the power of a horse); 1 horsepower equals about 745.7 watts. Being the rate of work, the equation for power can be written as:
:<math>\text{power}=\frac{\text{work}}{\text{time}}</math>

As a physical concept, power requires both a change in the physical system and a specified time in which the change occurs. This is distinct from the concept of work, which is measured only in terms of a net change in the state of the physical system. The same amount of work is done when carrying a load up a flight of stairs whether the person carrying it walks or runs, but more power is needed for running because the work is done in a shorter amount of time. 

The output power of an electric motor is the product of the [[torque]] that the motor generates and the [[angular velocity]] of its output shaft. The power involved in moving a ground vehicle is the product of the [[traction (engineering)|traction]] force on the wheels and the velocity of the vehicle. The power of a [[jet propulsion|jet-propelled]] vehicle is the product of the engine [[thrust]] and the [[velocity]] of the vehicle. The rate at which a light bulb converts electrical energy into light and heat is measured in watts&mdash;the higher the wattage, the more power, or equivalently the more electrical energy is used per unit time.<ref>{{Cite book|chapter= 6. Power|authors= Halliday and Resnick|title=Fundamentals of Physics|year= 1974}}</ref><ref>Chapter 13, &sect; 3, pp 13-2,3 ''[[The Feynman Lectures on Physics]]'' Volume I, 1963</ref>

==Units==
The dimension of power is energy divided by time. The [[International System of Units|SI]] unit of power is the [[watt]] (W), which is equal to one [[joule]] per second. Other units of power include [[erg]]s per second (erg/s), [[horsepower]] (hp), [[horsepower#PS|metric horsepower]] (Pferdestärke (PS) or cheval vapeur (CV)), and [[foot-pound force|foot-pounds]] per minute. Other units include [[dBm]], a logarithmic measure relative to a reference of 1 milliwatt, [[calorie]]s per hour, [[BTU]] per hour (BTU/h), and [[refrigeration ton|tons of refrigeration]].

== Equations for power==
Power, as a function of time, is the rate (i.e. ''[[derivative]]'') at which work is done, so can be expressed by this equation:

:<math>P =\frac{dW}{dt}</math>

where ''P'' is power, ''W'' is work, and ''t'' is time. Because work is a [[force]] '''F''' applied over a [[distance]] '''x''',
:<math>W = \mathbf{F} \cdot \mathbf{x}</math>

for a constant force, power can be rewritten as:
:<math>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>

In fact, this is valid for ''any'' force, as a consequence of applying the [[fundamental theorem of calculus]].

==Average power==
As a simple example, burning one kilogram of [[coal]] releases much more energy than does detonating a kilogram of [[Trinitrotoluene|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|accessdate =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 much more quickly, it delivers far more power than the coal.
If Δ''W''  is the amount of [[mechanical work|work]] performed during a period of [[time]] of duration Δ''t'', the '''average power''' ''P''<sub>avg</sub> over that period is given by the formula
:<math>
P_\mathrm{avg} = \frac{\Delta W}{\Delta t}\,.
</math>
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 Δ''t'' approaches zero.
:<math>
P = \lim _{\Delta t\rightarrow 0} P_\mathrm{avg} = \lim _{\Delta t\rightarrow 0} \frac{\Delta W}{\Delta t} =  \frac{\mathrm{d}W}{\mathrm{d}t}\,.
</math>

In the case of constant power ''P'', the amount of work performed during a period of duration ''t'' is given by:
:<math>
W = Pt\,.
</math>

In the context of energy conversion, it is more customary to use the symbol ''E'' rather than ''W''.

==Mechanical power==
[[File:Horsepower plain.svg|thumb|One ''metric horsepower'' is needed to lift 75&nbsp;[[:en:Kilogram|kilograms]] by 1&nbsp;[[meter]] 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 '''F''' on an object that travels along a curve ''C'' is given by the [[line integral]]:
: <math>W_C = \int_{C}\mathbf{F}\cdot \mathbf{v}\,\mathrm{d}t =\int_{C} \mathbf{F} \cdot \mathrm{d}\mathbf{x},</math>
where '''x''' defines the path ''C'' and '''v''' is the velocity along this path.

If the force '''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>W_C = U(B)-U(A),</math>

where ''A'' and ''B'' are the beginning and end of the path along which the work was done.

The power at any point along the curve ''C'' is the time derivative

:<math>P(t) = \frac{\mathrm{d}W}{\mathrm{d}t}=\mathbf{F}\cdot \mathbf{v}=-\frac{\mathrm{d}U}{\mathrm{d}t}.</math>

In one dimension, this can be simplified to:

:<math>P(t) = F\cdot v.</math>

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

In fluid power systems such as hydraulic actuators, power is given by
:<math> P(t) = pQ, \!</math>
where ''p'' is [[pressure]] in [[pascal (unit)|pascals]], or N/m<sup>2</sup> and ''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 ''F<sub>A</sub>'' acting on a point that moves with velocity ''v<sub>A</sub>'' and the output power be a force ''F<sub>B</sub>'' acts on a point that moves with velocity ''v<sub>B</sub>''.  If there are no losses in the system, then
:<math>P = F_B v_B = F_A v_A, \!</math>
and the [[mechanical advantage]] of the system (output force per input force) is given by
: <math> \mathrm{MA} = \frac{F_B}{F_A} = \frac{v_A}{v_B}. </math>

The similar relationship is obtained for rotating systems, where ''T<sub>A</sub>'' and ''ω<sub>A</sub>'' are the torque and angular velocity of the input and  ''T<sub>B</sub>'' and ''ω<sub>B</sub>'' are the torque and angular velocity of the output.  If there are no losses in the system, then
:<math>P = T_A \omega_A = T_B \omega_B, \!</math>
which yields the [[mechanical advantage]]
:<math> \mathrm{MA} = \frac{T_B}{T_A} = \frac{\omega_A}{\omega_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==
[[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]]
{{main|Electric power}}
The instantaneous electrical power ''P'' delivered to a component is given by
:<math>
P(t) = I(t) \cdot V(t) \,
</math>

where 
:''P''(''t'') is the instantaneous power, measured in [[watt]]s ([[joule]]s per [[second]])
:''V''(''t'') is the [[potential difference]] (or voltage drop) across the component, measured in [[volt]]s
:''I''(''t'') 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>
P=I \cdot V = I^2 \cdot R = \frac{V^2}{R} \,
</math>

where
:<math>
R = \frac{V}{I} \,
</math>
is the [[electrical resistance|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>p(t) = |s(t)|^2</math> is also a periodic function of period <math>T</math>.  The ''peak power'' is simply defined by:
:<math>
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>
\epsilon_\mathrm{pulse} = \int_{0}^{T}p(t) \mathrm{d}t \,
</math>
then the average power is:
:<math>
P_\mathrm{avg} = \frac{1}{T} \int_{0}^{T}p(t) \mathrm{d}t = \frac{\epsilon_\mathrm{pulse}}{T} \,
</math>.

One may define the pulse length <math>\tau</math> such that <math>P_0\tau = \epsilon_\mathrm{pulse}</math> so that the ratios
:<math>
\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>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==
{{Reflist}}

==External links==
*{{Commonscat-inline}}
{{Wikiquote}}

{{Classical mechanics derived SI units}}

[[Category:Concepts in physics]]
[[Category:Power (physics)| ]]