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 one joule per second. In older works, power is sometimes called ''activity''. Power is a scalar quantity.
The output power of a 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 force on the wheels and the velocity of the vehicle. In classical mechanics, as quantified from a stationary frame of reference, the motive power of a jet-propelled vehicle is the product of the engine thrust and the velocity of the vehicle (note that by this definition, a propelled vehicle hovering at stationary elevation over a gravitational body, where the upward thrust exactly cancels the downward acceleration of gravity, the motive power is zero). The rate at which a light bulb converts electrical energy into light and heat is measured in watts – the electrical energy used per unit of time.

Definition

Power is the rate with respect to time at which work is done; it is the time derivative 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 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, 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 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.

Average power

As a simple example, burning one kilogram of coal releases much more energy than does 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 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 much more quickly, it delivers far more power than the coal. If Δ''W'' is the amount of work performed during a period of time of duration Δ''t'', the average power ''P''_{avg} 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 Δ''t'' approaches zero.
:$P\; =\; \backslash lim\; \_\; P\_\backslash mathrm\; =\; \backslash lim\; \_\; \backslash frac\; =\; \backslash frac$
In the case of constant power ''P'', the amount of work performed during a period of duration ''t'' is given by:
:$W\; =\; Pt$
In the context of energy conversion, it is more customary to use the symbol ''E'' rather than ''W''.

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. In mechanics, the work done by a force F on an object that travels along a curve ''C'' is given by the line integral: : $W\_C\; =\; \backslash int\_\backslash mathbf\backslash cdot\; \backslash mathbf\backslash ,\backslash mathrmt\; =\backslash int\_\; \backslash mathbf\; \backslash cdot\; \backslash mathrm\backslash mathbf$ where x defines the path ''C'' and v is the velocity along this path. If the force 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 ''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: :$P(t)\; =\; \backslash frac=\backslash mathbf\backslash cdot\; \backslash mathbf=-\backslash frac$ In one dimension, this can be simplified to: :$P(t)\; =\; F\backslash cdot\; v$ In rotational systems, power is the product of the torque`τ` and angular velocity `ω`,
:$P(t)\; =\; \backslash boldsymbol\; \backslash cdot\; \backslash boldsymbol$
where ω 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 ''p'' is pressure in pascals, or N/m^{2} and ''Q'' is volumetric flow rate in m^{3}/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''_{A} acting on a point that moves with velocity ''v''_{A} and the output power be a force ''F''_{B} acts on a point that moves with velocity ''v''_{B}. 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 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 ''T''_{A} and ''ω''_{A} are the torque and angular velocity of the input and ''T''_{B} and ''ω''_{B} 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
:$\backslash mathrm\; =\; \backslash frac\; =\; \backslash frac$
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.

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 in watts (joules per second) :$V(t)$ is the potential difference (or voltage drop) across the component, measured in volts :$I(t)$ is the current through it, measured in amperes If the component is a resistor with time-invariant voltage to current ratio, then: :$P=I\; \backslash cdot\; V\; =\; I^2\; \backslash cdot\; R\; =\; \backslash frac$ where :$R\; =\; \backslash frac$ is the resistance, measured in ohms.

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: :$P\_0\; =\; \backslash max(t)/math>\; The\; peak\; power\; is\; not\; always\; readily\; measurable,\; however,\; and\; the\; measurement\; of\; the\; average\; power$ P\_\backslash mathrm$is\; more\; commonly\; performed\; by\; an\; instrument.\; If\; one\; defines\; the\; energy\; per\; pulse\; as:\; :$ \backslash epsilon\_\backslash mathrm\; =\; \backslash int\_^p(t)\; \backslash mathrmt$then\; the\; average\; power\; is:\; :$ P\_\backslash mathrm\; =\; \backslash frac\; \backslash int\_^p(t)\; \backslash mathrmt\; =\; \backslash frac$One\; may\; define\; the\; pulse\; length$ \backslash tau$such\; that$ P\_0\backslash tau\; =\; \backslash epsilon\_\backslash mathrm$so\; that\; the\; ratios\; :$ \backslash frac\; =\; \backslash frac$are\; equal.\; These\; ratios\; are\; called\; the\; \text{'}\text{'}duty\; cycle\text{'}\text{'}\; of\; the\; pulse\; train.$

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 * 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

References

{{Authority control Category:Force Category:Temporal rates Category:Physical quantities

Definition

Power is the rate with respect to time at which work is done; it is the time derivative 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 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, 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 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.

Average power

As a simple example, burning one kilogram of coal releases much more energy than does 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 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 much more quickly, it delivers far more power than the coal. If Δ''W'' is the amount of work performed during a period of time of duration Δ''t'', the average power ''P''

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. In mechanics, the work done by a force F on an object that travels along a curve ''C'' is given by the line integral: : $W\_C\; =\; \backslash int\_\backslash mathbf\backslash cdot\; \backslash mathbf\backslash ,\backslash mathrmt\; =\backslash int\_\; \backslash mathbf\; \backslash cdot\; \backslash mathrm\backslash mathbf$ where x defines the path ''C'' and v is the velocity along this path. If the force 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 ''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: :$P(t)\; =\; \backslash frac=\backslash mathbf\backslash cdot\; \backslash mathbf=-\backslash frac$ In one dimension, this can be simplified to: :$P(t)\; =\; F\backslash cdot\; v$ In rotational systems, power is the product of the torque

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''

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 in watts (joules per second) :$V(t)$ is the potential difference (or voltage drop) across the component, measured in volts :$I(t)$ is the current through it, measured in amperes If the component is a resistor with time-invariant voltage to current ratio, then: :$P=I\; \backslash cdot\; V\; =\; I^2\; \backslash cdot\; R\; =\; \backslash frac$ where :$R\; =\; \backslash frac$ is the resistance, measured in ohms.

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: :$P\_0\; =\; \backslash max(t)/math>\; The\; peak\; power\; is\; not\; always\; readily\; measurable,\; however,\; and\; the\; measurement\; of\; the\; average\; power$ P\_\backslash mathrm$is\; more\; commonly\; performed\; by\; an\; instrument.\; If\; one\; defines\; the\; energy\; per\; pulse\; as:\; :$ \backslash epsilon\_\backslash mathrm\; =\; \backslash int\_^p(t)\; \backslash mathrmt$then\; the\; average\; power\; is:\; :$ P\_\backslash mathrm\; =\; \backslash frac\; \backslash int\_^p(t)\; \backslash mathrmt\; =\; \backslash frac$One\; may\; define\; the\; pulse\; length$ \backslash tau$such\; that$ P\_0\backslash tau\; =\; \backslash epsilon\_\backslash mathrm$so\; that\; the\; ratios\; :$ \backslash frac\; =\; \backslash frac$are\; equal.\; These\; ratios\; are\; called\; the\; \text{'}\text{'}duty\; cycle\text{'}\text{'}\; of\; the\; pulse\; train.$

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 * 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

References

{{Authority control Category:Force Category:Temporal rates Category:Physical quantities