In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, power is the amount of
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
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 pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
, the unit of power is the
watt
The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after James Wa ...
, equal to one
joule
The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force applie ...
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 ...
quantity.
Power is related to other quantities; for example, the power involved in moving a ground vehicle is the product of the
aerodynamic drag
In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding fl ...
plus
traction force on the wheels, and the
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
of the vehicle. The output power of a
motor
An engine or motor is a machine designed to convert one or more forms of energy into mechanical energy.
Available energy sources include potential energy (e.g. energy of the Earth's gravitational field as exploited in hydroelectric power g ...
is the product of the
torque
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
that the motor generates and the
angular velocity of its output shaft. Likewise, the power dissipated in an
electrical element
Electrical elements are conceptual abstractions representing idealized electrical components, such as resistors, capacitors, and inductors, used in the circuit analysis, analysis of electrical networks. All electrical networks can be analyzed as ...
of a
circuit is the product of the
current
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (stre ...
flowing through the element and of the
voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
across the element.
Definition
Power is the rate with respect to time at which work is done; it is the time
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
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
** Working animal, an animal t ...
:
where is power, is work, and is time.
If a constant force F is applied throughout a
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
x, the work done is defined as
. In this case, power can be written as:
If instead the force is variable over a three-dimensional curve C, then the work is expressed in terms of the line integral:
From the
fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
, we know that
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
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
(SI), the unit of power is the
watt
The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after James Wa ...
(W), which is equal to one
joule
The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force applie ...
per second. Other common and traditional measures are
horsepower
Horsepower (hp) is a unit of measurement of power, or the rate at which work is done, usually in reference to the output of engines or motors. There are many different standards and types of horsepower. Two common definitions used today are t ...
(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 10−7joules (100 nJ). It originated in the Centimetre–gram–second system of units (CGS). It has the symbol ''erg''. The erg is not an SI unit. Its name is derived from (), a Greek word meaning 'work' o ...
s per second (erg/s),
foot-pounds
The foot-pound force (symbol: ft⋅lbf, ft⋅lbf, or ft⋅lb ) is a unit of work or energy in the engineering and gravitational systems in United States customary and imperial units of measure. It is the energy transferred upon applying a fo ...
per minute,
dBm
DBM or dbm may refer to:
Science and technology
* dBm, a unit for power measurement
* DBM (computing), family of key-value database engines including dbm, ndbm, gdbm, and Berkeley DB
* Database Manager (DBM), a component of 1987's ''Extended Edi ...
, a logarithmic measure relative to a reference of 1 milliwatt,
calories per hour,
BTU
The British thermal unit (BTU or Btu) is a unit of heat; it is defined as the amount of heat required to raise the temperature of one pound of water by one degree Fahrenheit. It is also part of the United States customary units. The modern SI u ...
per hour (BTU/h), and
tons of refrigeration.
Average power
As a simple example, burning one kilogram of
coal
Coal is a combustible black or brownish-black sedimentary rock, formed as rock strata called coal seams. Coal is mostly carbon with variable amounts of other elements, chiefly hydrogen, sulfur, oxygen, and nitrogen.
Coal is formed when dea ...
releases much more energy than detonating a kilogram of
TNT
Trinitrotoluene (), more commonly known as TNT, more specifically 2,4,6-trinitrotoluene, and by its preferred IUPAC name 2-methyl-1,3,5-trinitrobenzene, is a chemical compound with the formula C6H2(NO2)3CH3. TNT is occasionally used as a reagen ...
,
[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
TNT equivalent is a convention for expressing energy, typically used to describe the energy released in an explosion. The is a unit of energy defined by that convention to be , which is the approximate energy released in the detonation of a ...
. 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
** Working animal, an animal t ...
performed during a period of
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
of duration , the average power over that period is given by the formula:
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.
In the case of constant power , the amount of work performed during a period of duration is given by:
In the context of energy conversion, it is more customary to use the symbol rather than .
Mechanical power
![Horsepower plain](https://upload.wikimedia.org/wikipedia/commons/c/c5/Horsepower_plain.svg)
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
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
, 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
** Working animal, an animal t ...
done by a force on an object that travels along a curve is given by the
line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, al ...
:
where defines the path and is the velocity along this path.
If the force is derivable from a potential (
conservative
Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization i ...
), 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 gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is ...
(and remembering that force is the negative of the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of the potential energy) yields:
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:
In one dimension, this can be simplified to:
In rotational systems, power is the product of the
torque
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
and
angular velocity ,
where measured in
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before tha ...
s per second. The
represents
scalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
.
In fluid power systems such as
hydraulic
Hydraulics (from Greek: Υδραυλική) is a technology and applied science using engineering, chemistry, and other sciences involving the mechanical properties and use of liquids. At a very basic level, hydraulics is the liquid counte ...
actuators, power is given by
where is
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
in
pascals
The pascal (symbol: Pa) is the unit of pressure in the International System of Units (SI), and is also used to quantify internal pressure, stress, Young's modulus, and ultimate tensile strength. The unit, named after Blaise Pascal, is defined ...
, or N/m
2 and 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
Mechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. The device trades off input forces against movement to obtain a desired amplification in the output force. The model for ...
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
and the
mechanical advantage
Mechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. The device trades off input forces against movement to obtain a desired amplification in the output force. The model for ...
of the system (output force per input force) is given by
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
which yields the
mechanical advantage
Mechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. The device trades off input forces against movement to obtain a desired amplification in the output force. The model for ...
These relations are important because they define the maximum performance of a device in terms of
velocity ratio
A gear train is a mechanical system formed by mounting gears on a frame so the teeth of the gears engage.
Gear teeth are designed to ensure the pitch circles of engaging gears roll on each other without slipping, providing a smooth transmission ...
s determined by its physical dimensions. See for example
gear ratio
A gear train is a mechanical system formed by mounting gears on a frame so the teeth of the gears engage.
Gear teeth are designed to ensure the Pitch circle diameter (gears), pitch circles of engaging gears roll on each other without slipping, pr ...
s.
Electrical power
![Ansel Adams - National Archives 79-AAB-02](https://upload.wikimedia.org/wikipedia/commons/3/39/Ansel_Adams_-_National_Archives_79-AAB-02.jpg)
The instantaneous electrical power ''P'' delivered to a component is given by
where
*
is the instantaneous power, measured in
watt
The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after James Wa ...
s (
joule
The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force applie ...
s per
second
The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds ...
)
*
is the
potential difference
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
(or voltage drop) across the component, measured in
volt
The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI). It is named after the Italian physicist Alessandro Volta (1745–1827).
Defi ...
s
*
is the
current
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (stre ...
through it, measured in
amperes
If the component is a
resistor with time-invariant
voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
to
current
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (stre ...
ratio, then:
where
is the
electrical resistance
The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares some conceptual parallels ...
, measured in
ohm
Ohm (symbol Ω) is a unit of electrical resistance named after Georg Ohm.
Ohm or OHM may also refer to:
People
* Georg Ohm (1789–1854), German physicist and namesake of the term ''ohm''
* Germán Ohm (born 1936), Mexican boxer
* Jörg Ohm (b ...
s.
Peak power and duty cycle
![peak-power-average-power-tau-T](https://upload.wikimedia.org/wikipedia/commons/4/4c/Peak-power-average-power-tau-T.png)
In the case of a periodic signal
of period
, like a train of identical pulses, the instantaneous power
is also a periodic function of period
. The ''peak power'' is simply defined by: