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 ...
, 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 ...
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. ...
, 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 ...
, 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 appli ...
per second. In older works, power is sometimes called ''activity''.
Power is a
scalar quantity.
Power is related to other quantities; for example, the power involved in moving a ground vehicle is the product of the
aerodynamic drag 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 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 of a
circuit is the product of the
current 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 ...
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:
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, ...
, 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. ...
(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 ...
(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 appli ...
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 per minute,
dBm, a logarithmic measure relative to a reference of 1 milliwatt,
calorie
The calorie is a unit of energy. For historical reasons, two main definitions of "calorie" are in wide use. The large calorie, food calorie, or kilogram calorie was originally defined as the amount of heat needed to raise the temperature of o ...
s per hour,
BTU 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 ...
releases much 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 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 is the amount of
work performed during a period of
time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
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
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 objec ...
, the
work 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, ...
:
where defines the path and is the velocity along this path.
If the force is derivable from a potential (
conservative), then applying the
gradient theorem (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 that ...
s per second. The
represents
scalar product.
In fluid power systems such as
hydraulic 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 a ...
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 defin ...
, 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 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 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
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
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 ...
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 appli ...
s per
second)
*
is the
potential difference (or voltage drop) across the component, measured in
volts
*
is the
current 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 ...
to
current 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 parallel ...
, 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 (bor ...
s.
Peak power and duty cycle
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: