In physics, power is the rate of doing work.
It is equivalent to an amount of energy consumed per unit time.
In the MKS system, the unit of power is the joule per second, known as the watt in honor
of James Watt, the eighteenth-century developer of the steam engine.
The integral of power over time defines the work performed.
Because this integral depends on the trajectory of the point of application of the force and
torque, this calculation of work is said to be path dependent.
The same amount of work is done when carrying a load up a flight of stairs whether the person
carrying it walks or runs, but more power is needed for running because the work is
done in a shorter amount of time.
The output power of an electric 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 vehicle is the product of the traction force of the wheels
and the velocity of the vehicle.
The rate at which a light bulb converts electrical energy into light and heat is measured in
watts—the higher the wattage, the more power, or equivalently the more electrical energy
is used per unit time.
Units
The dimension of power is energy divided by time.
The SI unit of power is the watt, which is equal to one joule per second.
Other units of power include ergs per second, horsepower, metric horsepower or cheval vapeur,
CV), and foot-pounds per minute.
One horsepower is equivalent to 33,000 foot-pounds per minute, or the power required to lift
550 pounds by one foot in one second, and is equivalent to about 746 watts.
Other units include dBm, a relative logarithmic measure with 1 milliwatt as reference; calories
per hour; Btu per hour; and tons of refrigeration.
Average power As a simple example, burning a kilogram of
coal releases much more energy than does detonating a kilogram of TNT, but because the TNT reaction
releases energy much more quickly, it delivers far more power than the coal.
If ΔW is the amount of work performed during a period of time of duration Δt, the average
power Pavg 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
Δt 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. 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:
where x defines the path C and v is the velocity along this path.
If the force F is derivable from a potential, then applying the gradient theorem 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 radians per second.
The represents scalar product.
In fluid power systems such as hydraulic actuators, power is given by
where p is pressure in pascals, or N/m2 and Q is volumetric flow rate in m3/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 FA acting on a point that moves with velocity
vA and the output power be a force FB acts on a point that moves with velocity vB.
If there are no losses in the system, then
and the mechanical advantage of the system is given by
The similar relationship is obtained for rotating systems, where TA and ωA are the torque and
angular velocity of the input and TB and ωB 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 P(t) is the instantaneous power, measured
in watts V(t) is the potential difference 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:
where
is the resistance, measured in ohms.
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:
. The peak power is not always readily measurable,
however, and the measurement of the average power is more commonly performed by an instrument.
If one defines the energy per pulse as:
then the average power is: .
One may define the pulse length such that so that the ratios
are equal.
These ratios are called the duty cycle of the pulse train.
See also Simple machines
Mechanical advantage Motive power
Orders of magnitude Pulsed power
Intensity — in the radiative sense, power per area
Power gain — for linear, two-port networks.
References ^ Halliday and Resnick. "6.
Power".
Fundamentals of Physics.
^ Chapter 13, § 3, pp 13-2,3 The Feynman Lectures on Physics Volume I, 1963
^ Burning coal produces around 15-30 megajoules per kilogram, while detonating TNT produces
about 4.7 megajoules per kilogram.
For the coal value, see Fisher, Juliya.
"Energy Density of Coal".
The Physics Factbook.
Retrieved 30 May 2011.
For the TNT value, see the article TNT equivalent.
The coal value does not include the weight of oxygen used during combustion, while the
TNT number if TNT only.