Power | |
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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 |
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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.

- Definition
- Units
- Average power and instantaneous power
- Mechanical power
- Mechanical advantage
- Electrical power
- Peak power and duty cycle
- Radiant power
- See also
- References

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 **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* is angular frequency, 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/m^{2}, and Q is volumetric flow rate in m^{3}/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} 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 *duty cycle* of the pulse train.

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

In mechanics, **acceleration** is the rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are vector quantities. The orientation of an object's acceleration is given by the orientation of the *net* force acting on that object. The magnitude of an object's acceleration, as described by Newton's Second Law, is the combined effect of two causes:

A **centripetal force** is a force that makes a body follow a curved path. The direction of the centripetal force is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". In the theory of Newtonian mechanics, gravity provides the centripetal force causing astronomical orbits.

In physics, **potential energy** is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. The term *potential energy* was introduced by the 19th-century Scottish engineer and physicist William Rankine, although it has links to the ancient Greek philosopher Aristotle's concept of *potentiality*.

In physics and mechanics, **torque** is the rotational analogue of linear force. It is also referred to as the **moment of force**. It describes the rate of change of angular momentum that would be imparted to an isolated body.

The **Navier–Stokes equations** are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).

**Kinematics** is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.

In physics, **work** is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. A force is said to do *positive work* if when applied it has a component in the direction of the displacement of the point of application. A force does *negative work* if it has a component opposite to the direction of the displacement at the point of application of the force.

In statistical mechanics and information theory, the **Fokker–Planck equation** is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well. The Fokker-Planck equation has multiple applications in information theory, graph theory, data science, finance, economics etc.

In mathematical physics, **scalar potential**, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one position to the other. It is a scalar field in three-space: a directionless value (scalar) that depends only on its location. A familiar example is potential energy due to gravity.

In the physical science of dynamics, **rigid-body dynamics** studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are *rigid* simplifies analysis, by reducing the parameters that describe the configuration of the system to the translation and rotation of reference frames attached to each body. This excludes bodies that display fluid, highly elastic, and plastic behavior.

**Particle velocity** is the velocity of a particle in a medium as it transmits a wave. The SI unit of particle velocity is the metre per second (m/s). In many cases this is a longitudinal wave of pressure as with sound, but it can also be a transverse wave as with the vibration of a taut string.

In mechanics, **virtual work** arises in the application of the *principle of least action* to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different for different displacements. Among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action. This displacement is therefore the displacement followed by the particle according to the principle of least action.

The work of a force on a particle along a virtual displacement is known as the virtual work.

In a compressible sound transmission medium - mainly air - air particles get an accelerated motion: the **particle acceleration** or sound acceleration with the symbol a in metre/second^{2}. In acoustics or physics, **acceleration** is defined as the rate of change of velocity. It is thus a vector quantity with dimension length/time^{2}. In SI units, this is m/s^{2}.

The **Vlasov equation** is a differential equation describing time evolution of the distribution function of plasma consisting of charged particles with long-range interaction, such as the Coulomb interaction. The equation was first suggested for the description of plasma by Anatoly Vlasov in 1938 and later discussed by him in detail in a monograph.

In calculus, the **Leibniz integral rule** for differentiation under the integral sign states that for an integral of the form

In differential calculus, the **Reynolds transport theorem**, or simply the **Reynolds theorem**, named after Osborne Reynolds (1842–1912), is a three-dimensional generalization of the Leibniz integral rule. It is used to recast time derivatives of integrated quantities and is useful in formulating the basic equations of continuum mechanics.

**Linear motion**, also called **rectilinear motion**, is one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: **uniform linear motion**, with constant velocity ; and **non-uniform linear motion**, with variable velocity. The motion of a particle along a line can be described by its position , which varies with (time). An example of linear motion is an athlete running a 100-meter dash along a straight track.

In physics, **Lagrangian mechanics** is a formulation of classical mechanics founded on the stationary-action principle. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the Turin Academy of Science in 1760 culminating in his 1788 grand opus, *Mécanique analytique*.

In classical mechanics, the **central-force problem** is to determine the motion of a particle in a single central potential field. A central force is a force that points from the particle directly towards a fixed point in space, the center, and whose magnitude only depends on the distance of the object to the center. In a few important cases, the problem can be solved analytically, i.e., in terms of well-studied functions such as trigonometric functions.

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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.

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