Power | |
---|---|

Common symbols | P |

SI unit | watt (W) |

In SI base units |
kg⋅
m^{2}⋅
s^{−3} |

Derivations from other quantities | |

Dimension |

Part of a series on |

Classical mechanics |
---|

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*.^{
[1]}^{
[2]}^{
[3]} Power is a
scalar quantity.

Specifying power in particular systems may require attention 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 of the vehicle. The output power of a
motor is the product of the
torque 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 across the element.^{
[4]}^{
[5]}

Power is the rate with respect to time at which work is done; it is the time derivative of work:

where P is power, W is work, and t is time.

We will now show that the mechanical power generated by a force F on a body moving at the velocity v can be expressed as the product:

- 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
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.

As a simple example, burning one kilogram of
coal releases more energy than detonating a kilogram of
TNT,^{
[6]} but because the TNT reaction releases energy more quickly, it delivers 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. Average power is often called "power" when the context makes it clear.

Instantaneous power is the limiting value of the average power as the time interval Δ*t* approaches zero.

When power *P* is constant, the amount of work performed in time period t can be calculated as

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

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

In fluid power systems such as hydraulic actuators, power is given by

where p is
pressure in
pascals or N/m

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} 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

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.

The instantaneous electrical power *P* delivered to a component is given by

where

- is the instantaneous power, measured in watts ( joules per second),
- is the potential difference (or voltage drop) across the component, measured in volts, and
- 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
electrical resistance, measured in
ohms.

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

Power is related to intensity at a radius ; the power emitted by a source can be written as:^{[
citation needed]}

- 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

Wikimedia Commons has media related to
Power (physics).

Wikiquote has quotations related to **
Power (physics)**.

**^**Fowle, Frederick E., ed. (1921).*Smithsonian Physical Tables*(7th revised ed.). Washington, D.C.: Smithsonian Institution. OCLC 1142734534. Archived from the original on 23 April 2020.**Power or Activity**is the time rate of doing work, or if*W*represents work and*P*power,*P*=*dw*/*dt*. (p. xxviii) ... ACTIVITY. Power or rate of doing work; unit, the watt. (p. 435)**^**Heron, C. A. (1906). "Electrical Calculations for Rallway Motors".*Purdue Eng. Rev.*(2): 77–93. Archived from the original on 23 April 2020. Retrieved 23 April 2020.The activity of a motor is the work done per second, ... Where the joule is employed as the unit of work, the international unit of activity is the joule-per-second, or, as it is commonly called, the watt. (p. 78)

**^**"Societies and Academies".*Nature*.**66**(1700): 118–120. 1902. Bibcode: 1902Natur..66R.118.. doi: 10.1038/066118b0.If the watt is assumed as unit of activity...

**^**David Halliday; Robert Resnick (1974). "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 (2003). "Energy Density of Coal".*The Physics Factbook*. Retrieved 30 May 2011. For the TNT value, see the article TNT equivalent. Neither value includes the weight of oxygen from the air used during combustion.