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
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.
Power is the rate with respect to time at which work is done; it is the time derivative
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
. 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
, we know that
. Hence the formula is valid for any general situation.
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 erg
s per second (erg/s), foot-pounds
per minute, dBm
, a logarithmic measure relative to a reference of 1 milliwatt, calorie
s per hour, BTU
per hour (BTU/h), and tons of refrigeration
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:
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.
In the case of constant power ''P'', the amount of work performed during a period of duration ''t'' is given by:
In the context of energy conversion, it is more customary to use the symbol ''E'' rather than ''W''.
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
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:
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:
In one dimension, this can be simplified to:
In rotational systems, power is the product of the torque τ
and angular velocity ω
where ω measured in radian
s per second. The
represents scalar product
In fluid power systems such as hydraulic actuators, power is given by
where ''p'' is pressure
, or N/m2
and ''Q'' is volumetric flow rate
/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 ''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
and the mechanical advantage
of the system (output force per input force) is given by
The similar relationship is obtained for rotating systems, where ''T''A
are the torque and angular velocity of the input and ''T''B
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 ratio
s determined by its physical dimensions. See for example gear ratio
The instantaneous electrical power ''P'' delivered to a component is given by
is the instantaneous power, measured in watt
s per second
is the potential difference
(or voltage drop) across the component, measured in volt
is the current
through it, measured in ampere
If the component is a resistor
with time-invariant voltage
is the resistance
, measured in ohm
Peak power and duty cycle
In the case of a periodic signal
, like a train of identical pulses, the instantaneous power
is also a periodic function of period
. The ''peak power'' is simply defined by: