power (physics)

Power is the 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:

$$P\; =\; \backslash frac\{dE\}\{dt\},$$

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:

$$P\; =\; \backslash frac\{dW\}\{dt\},$$

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: $$P\; =\; \backslash frac\{dW\}\{dt\}\; =\; \backslash mathbf\{F\}\; \backslash cdot\; \backslash mathbf\; \{v\}$$

If a constant force F is applied throughout a distance x, the work done is defined as $W\; =\; \backslash mathbf\{F\}\; \backslash cdot\; \backslash mathbf\{x\}$. In this case, power can be written as:

$$P\; =\; \backslash frac\{dW\}\{dt\}\; =\; \backslash frac\{d\}\{dt\}\; \backslash left(\backslash mathbf\{F\}\; \backslash cdot\; \backslash mathbf\{x\}\backslash right)\; =\; \backslash mathbf\{F\}\backslash cdot\; \backslash frac\{d\backslash mathbf\{x\}\}\{dt\}\; =\; \backslash mathbf\{F\}\; \backslash cdot\; \backslash mathbf\; \{v\}.$$

If instead the force is variable over a three-dimensional curve C, then the work is expressed in terms of the line integral:

$$W\; =\; \backslash int\_C\; \backslash mathbf\{F\}\; \backslash cdot\; d\backslash mathbf\; \{r\}$$

= \int_{\Delta t} \mathbf{F} \cdot \frac{d\mathbf {r}}{dt} \ dt

= \int_{\Delta t} \mathbf{F} \cdot \mathbf {v} \, dt.

From the fundamental theorem of calculus, we know that $$P\; =\; \backslash frac\{dW\}\{dt\}\; =\; \backslash frac\{d\}\{dt\}\; \backslash int\_\{\backslash Delta\; t\}\; \backslash mathbf\{F\}\; \backslash cdot\; \backslash mathbf\; \{v\}\; \backslash ,\; dt\; =\; \backslash mathbf\{F\}\; \backslash cdot\; \backslash mathbf\; \{v\}.$$ Hence the formula is valid for any general situation.

In older works, power is sometimes called activity.{{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)}}{{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)}}{{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}}

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 mechanical horsepower equals about 745.7 watts. Other units of power include ergs per second (erg/s), foot-pounds per minute, dBm, a logarithmic measure relative to a reference of 1 milliwatt, calories per hour, BTU per hour (BTU/h), and tons of refrigeration.

As a simple example, burning one kilogram of coal releases more energy than detonating a kilogram of TNT,Burning coal produces around 15-30 megajoules 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. but because the TNT reaction releases energy more quickly, it delivers more power than the coal.

If {{math|ΔW}} is the amount of work performed during a period of time of duration {{math|Δt}}, the average power {{math|P_avg}} over that period is given by the formula

$$P\_\backslash mathrm\{avg\}\; =\; \backslash frac\{\backslash Delta\; W\}\{\backslash Delta\; t\}.$$

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|Δt}} approaches zero.

$$P\; =\; \backslash lim\_\{\backslash Delta\; t\; \backslash to\; 0\}\; P\_\backslash mathrm\{avg\}\; =\; \backslash lim\_\{\backslash Delta\; t\; \backslash to\; 0\}\; \backslash frac\{\backslash Delta\; W\}\{\backslash Delta\; t\}\; =\; \backslash frac\{dW\}\{dt\}.$$

When power {{math|P}} is constant, the amount of work performed in time period {{mvar|t}} can be calculated as

$$W\; =\; Pt.$$

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

File:Horsepower plain.svgs by 1 metre in 1 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 work done by a force {{math|F}} on an object that travels along a curve {{mvar|C}} is given by the line integral:

$$W\_C\; =\; \backslash int\_C\; \backslash mathbf\{F\}\; \backslash cdot\; \backslash mathbf\{v\}\; \backslash ,\; dt\; =\; \backslash int\_C\; \backslash mathbf\{F\}\; \backslash cdot\; d\backslash mathbf\{x\},$$

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), then applying the gradient theorem (and remembering that force is the negative of the gradient of the potential energy) yields:

$$W\_C\; =\; U(A)\; -\; U(B),$$

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:

$$P(t)\; =\; \backslash frac\{dW\}\{dt\}\; =\; \backslash mathbf\{F\}\; \backslash cdot\; \backslash mathbf\{v\}\; =\; -\backslash frac\{dU\}\{dt\}.$$

In one dimension, this can be simplified to:

$$P(t)\; =\; F\; \backslash cdot\; v.$$

In rotational systems, power is the product of the torque {{math|τ}} and angular velocity {{math|ω}},

$$P(t)\; =\; \backslash boldsymbol\{\backslash tau\}\; \backslash cdot\; \backslash boldsymbol\{\backslash omega\},$$

where {{math|ω}} is angular frequency, measured in radians per second. The $\backslash cdot$ represents scalar product.

In fluid power systems such as hydraulic actuators, power is given by $$P(t)\; =\; pQ,$$ where {{mvar|p}} is pressure in pascals or N/m², and {{mvar|Q}} is volumetric flow rate in m³/s in SI units.

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_A}} acting on a point that moves with velocity {{math|v_A}} and the output power be a force {{math|F_B}} acts on a point that moves with velocity {{math|v_B}}. If there are no losses in the system, then

$$P\; =\; F\_\backslash text\{B\}\; v\_\backslash text\{B\}\; =\; F\_\backslash text\{A\}\; v\_\backslash text\{A\},$$

and the mechanical advantage of the system (output force per input force) is given by

$$\backslash mathrm\{MA\}\; =\; \backslash frac\{F\_\backslash text\{B\}\}\{F\_\backslash text\{A\}\}\; =\; \backslash frac\{v\_\backslash text\{A\}\}\{v\_\backslash text\{B\}\}.$$

The similar relationship is obtained for rotating systems, where {{math|T_A}} and {{math|ω_A}} are the torque and angular velocity of the input and {{math|T_B}} and {{math|ω_B}} are the torque and angular velocity of the output. If there are no losses in the system, then

$$P\; =\; T\_\backslash text\{A\}\; \backslash omega\_\backslash text\{A\}\; =\; T\_\backslash text\{B\}\; \backslash omega\_\backslash text\{B\},$$

which yields the mechanical advantage

$$\backslash mathrm\{MA\}\; =\; \backslash frac\{T\_\backslash text\{B\}\}\{T\_\backslash text\{A\}\}\; =\; \backslash frac\{\backslash omega\_\backslash text\{A\}\}\{\backslash omega\_\backslash text\{B\}\}.$$

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

{{main|Electric power}}

File:Ansel Adams - National Archives 79-AAB-02.jpg photograph of electrical wires of the Boulder Dam Power Units, 1941–1942]]

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 in watts (joules per second),
• $V(t)$ is the potential difference (or voltage drop) across the component, measured in volts, and
• $I(t)$ is the current through it, measured in amperes.

If the component is a resistor with time-invariant voltage to current ratio, then:

$$P\; =\; I\; \backslash cdot\; V\; =\; I^2\; \backslash cdot\; R\; =\; \backslash frac\{V^2\}\{R\},$$

where

$$R\; =\; \backslash frac\{V\}\{I\}$$

is the electrical resistance, measured in ohms.

File:peak-power-average-power-tau-T.png

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:

$$P\_0\; =\; \backslash max\; [p(t)].$$

The peak power is not always readily measurable, however, and the measurement of the average power $P\_\backslash mathrm\{avg\}$ is more commonly performed by an instrument. If one defines the energy per pulse as

$$\backslash varepsilon\_\backslash mathrm\{pulse\}\; =\; \backslash int\_0^T\; p(t)\; \backslash ,\; dt$$

then the average power is

$$P\_\backslash mathrm\{avg\}\; =\; \backslash frac\{1\}\{T\}\; \backslash int\_0^T\; p(t)\; \backslash ,\; dt\; =\; \backslash frac\{\backslash varepsilon\_\backslash mathrm\{pulse\}\}\{T\}.$$

One may define the pulse length $\backslash tau$ such that $P\_0\backslash tau\; =\; \backslash varepsilon\_\backslash mathrm\{pulse\}$ so that the ratios

$$\backslash frac\{P\_\backslash mathrm\{avg\}\}\{P\_0\}\; =\; \backslash frac\{\backslash tau\}\{T\}$$

are equal. These ratios are called the duty cycle of the pulse train.

Power is related to intensity at a radius $r$; the power emitted by a source can be written as:{{citation needed|date=August 2017}}

$$P(r)\; =\; I(4\backslash pi\; r^2).$$

• Simple machines
• Orders of magnitude (power)
• Pulsed power
• Intensity – in the radiative sense, power per area
• Power gain – for linear, two-port networks
• Power density
• Signal strength
• Sound power

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