# Power (physics)

Last updated

Power
Common symbols
P
SI unit watt (W)
In SI base units kgm 2s −3
Derivations from
other quantities
Dimension ${\mathsf {L}}^{2}{\mathsf {M}}{\mathsf {T}}^{-3}$ 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 quantity.

## Contents

Power is related to other quantities; for example, the power involved in moving a ground vehicle is the product of the 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.  

## Definition

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

$P={\frac {dW}{dt}}$ 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 $W=\mathbf {F} \cdot \mathbf {x}$ . In this case, power can be written as:

$P={\frac {dW}{dt}}={\frac {d}{dt}}\left(\mathbf {F} \cdot \mathbf {x} \right)=\mathbf {F} \cdot {\frac {d\mathbf {x} }{dt}}=\mathbf {F} \cdot \mathbf {v}$ If instead the force is variable over a three-dimensional curve C, then the work is expressed in terms of the line integral:

$W=\int _{C}\mathbf {F} \cdot d\mathbf {r} =\int _{\Delta t}\mathbf {F} \cdot {\frac {d\mathbf {r} }{dt}}\ dt=\int _{\Delta t}\mathbf {F} \cdot \mathbf {v} \,dt$ From the fundamental theorem of calculus, we know that

$P={\frac {dW}{dt}}={\frac {d}{dt}}\int _{\Delta t}\mathbf {F} \cdot \mathbf {v} \,dt=\mathbf {F} \cdot \mathbf {v} .$ 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 (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.

## Average power

As a simple example, burning one kilogram of coal releases much more energy than 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:

$P_{\mathrm {avg} }={\frac {\Delta W}{\Delta t}}$ 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.

$P=\lim _{\Delta t\to 0}P_{\mathrm {avg} }=\lim _{\Delta t\to 0}{\frac {\Delta W}{\Delta t}}={\frac {dW}{dt}}$ In the case of constant power P, the amount of work performed during a period of duration t is given by:

$W=Pt$ In the context of energy conversion, it is more customary to use the symbol E rather than W.

## Mechanical power One metric horsepower is needed to lift 75  kilograms by 1  metre in 1  second.

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:

$W_{C}=\int _{C}\mathbf {F} \cdot \mathbf {v} \,dt=\int _{C}\mathbf {F} \cdot d\mathbf {x}$ 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:

$W_{C}=U(A)-U(B)$ 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:

$P(t)={\frac {dW}{dt}}=\mathbf {F} \cdot \mathbf {v} =-{\frac {dU}{dt}}$ In one dimension, this can be simplified to:

$P(t)=F\cdot v$ In rotational systems, power is the product of the torque τ and angular velocity ω,

$P(t)={\boldsymbol {\tau }}\cdot {\boldsymbol {\omega }}$ where ω measured in radians per second. The $\cdot$ represents scalar product.

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

$P(t)=pQ$ where p is pressure in pascals, or N/m2 and Q is volumetric flow rate in m3/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 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

$P=F_{\text{B}}v_{\text{B}}=F_{\text{A}}v_{\text{A}},$ and the mechanical advantage of the system (output force per input force) is given by

$\mathrm {MA} ={\frac {F_{\text{B}}}{F_{\text{A}}}}={\frac {v_{\text{A}}}{v_{\text{B}}}}.$ 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

$P=T_{\text{A}}\omega _{\text{A}}=T_{\text{B}}\omega _{\text{B}},$ which yields the mechanical advantage

$\mathrm {MA} ={\frac {T_{\text{B}}}{T_{\text{A}}}}={\frac {\omega _{\text{A}}}{\omega _{\text{B}}}}.$ 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

$P(t)=I(t)\cdot V(t)$ where

• $P(t)$ is the instantaneous power, measured in watts (joules per second)
• $V(t)$ is the potential difference (or voltage drop) 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:

$P=I\cdot V=I^{2}\cdot R={\frac {V^{2}}{R}}$ where

$R={\frac {V}{I}}$ is the electrical resistance, measured in ohms.

## Peak power and duty cycle In a train of identical pulses, the instantaneous power is a periodic function of time. The ratio of the pulse duration to the period is equal to the ratio of the average power to the peak power. It is also called the duty cycle (see text for definitions).

In the case of a periodic signal $s(t)$ of period $T$ , like a train of identical pulses, the instantaneous power ${\textstyle p(t)=|s(t)|^{2}}$ is also a periodic function of period $T$ . The peak power is simply defined by:

$P_{0}=\max[p(t)]$ The peak power is not always readily measurable, however, and the measurement of the average power $P_{\mathrm {avg} }$ is more commonly performed by an instrument. If one defines the energy per pulse as:

$\varepsilon _{\mathrm {pulse} }=\int _{0}^{T}p(t)\,dt$ then the average power is:

$P_{\mathrm {avg} }={\frac {1}{T}}\int _{0}^{T}p(t)\,dt={\frac {\varepsilon _{\mathrm {pulse} }}{T}}$ One may define the pulse length $\tau$ such that $P_{0}\tau =\varepsilon _{\mathrm {pulse} }$ so that the ratios

${\frac {P_{\mathrm {avg} }}{P_{0}}}={\frac {\tau }{T}}$ are equal. These ratios are called the duty cycle of the pulse train.

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

$P(r)=I(4\pi r^{2})$ ## Related Research Articles In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. 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:

Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century.

A centripetal force is a force that makes a body follow a curved path. Its direction 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 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. In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment, moment of force, rotational force or turning effect, depending on the field of study. It represents the capability of a force to produce change in the rotational motion of the body. The concept originated with the studies by Archimedes of the usage of levers. Just as a linear force is a push or a pull, a torque can be thought of as a twist to an object around a specific axis. Torque is defined as the product of the magnitude of the force and the perpendicular distance of the line of action of a force from the axis of rotation. The symbol for torque is typically , the lowercase Greek letter tau. When being referred to as moment of force, it is commonly denoted by M. In physics, the Navier–Stokes equations are certain partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-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, angular velocity or rotational velocity, also known as angular frequency vector, is a pseudovector representation of how fast the angular position or orientation of an object changes with time. The magnitude of the pseudovector represents the angular speed, the rate at which the object rotates or revolves, and its direction is normal to the instantaneous plane of rotation or angular displacement. The orientation of angular velocity is conventionally specified by the right-hand rule. In physics, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, it is often represented as the product of force and displacement. A force is said to do positive work if 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. The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation. 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.

Particle displacement or displacement amplitude is a measurement of distance of the movement of a sound particle from its equilibrium position in a medium as it transmits a sound wave. The SI unit of particle displacement is the metre (m). In most cases this is a longitudinal wave of pressure, but it can also be a transverse wave, such as the vibration of a taut string. In the case of a sound wave travelling through air, the particle displacement is evident in the oscillations of air molecules with, and against, the direction in which the sound wave is travelling.

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/second2. In acoustics or physics, acceleration is defined as the rate of change of velocity. It is thus a vector quantity with dimension length/time2. In SI units, this is m/s2.

In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, 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. In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional area at a given point in space, its direction being that of the motion of the positive charges at this point. In SI base units, the electric current density is measured in amperes per square metre.

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 or zero acceleration; and non-uniform linear motion with variable velocity or non-zero acceleration. 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 100m along a straight track.

1. 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)
2. 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)
3. "Societies and Academies". Nature. 66 (1700): 118–120. 1902. Bibcode:1902Natur..66R.118.. doi:. If the watt is assumed as unit of activity...
4. Halliday and Resnick (1974). "6. Power". Fundamentals of Physics.{{cite book}}: CS1 maint: uses authors parameter (link)
5. Chapter 13, § 3, pp 13-2,3 The Feynman Lectures on Physics Volume I, 1963
6. 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.