Power (physics)

Power
Common symbols	P
SI unit	watt(W)
InSI base units	kg⋅m2⋅s−3
Derivations from other quantities	P=E/t P=F·v P=V·I P=τ·ω
Dimension	${\mathsf {M}}{\mathsf {L}}^{2}{\mathsf {T}}^{-3}$

Inphysics,poweris the amount ofenergytransferred or converted per unit time. In theInternational System of Units,the unit of power is thewatt,equal to onejouleper second. Power is ascalarquantity.

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 theaerodynamic dragplustraction forceon the wheels, and thevelocityof the vehicle. The output power of amotoris the product of thetorquethat the motor generates and theangular velocityof its output shaft. Likewise, the power dissipated in anelectrical elementof acircuitis the product of thecurrentflowing through the element and of thevoltageacross the element.

Definition

Power is theratewith respect to time at which work is done; it is the timederivativeofwork: $$P={\frac {dW}{dt}},$$ wherePis power,Wis work, andtis 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:$$P={\frac {dW}{dt}}=\mathbf {F} \cdot \mathbf {v}$$

If aconstantforceFis applied throughout adistancex,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 isvariable 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 thefundamental 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.

In older works, power is sometimes calledactivity.

Units

The dimension of power is energy divided by time. In theInternational System of Units(SI), the unit of power is thewatt(W), which is equal to onejouleper second. Other common and traditional measures arehorsepower(hp), comparing to the power of a horse; onemechanical horsepowerequals about 745.7 watts. Other units of power includeergsper second (erg/s),foot-poundsper minute,dBm,a logarithmic measure relative to a reference of 1 milliwatt,caloriesper hour,BTUper hour (BTU/h), andtons of refrigeration.

Average power and instantaneous power

As a simple example, burning one kilogram ofcoalreleases more energy than detonating a kilogram ofTNT,but because the TNT reaction releases energy more quickly, it delivers more power than the coal. IfΔWis the amount ofworkperformed during a period oftimeof durationΔt,the average powerP_avgover 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. 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 intervalΔtapproaches zero. $$P=\lim _{\Delta t\to 0}P_{\mathrm {avg} }=\lim _{\Delta t\to 0}{\frac {\Delta W}{\Delta t}}={\frac {dW}{dt}}.$$

When powerPis constant, the amount of work performed in time periodtcan be calculated as $$W=Pt.$$

In the context of energy conversion, it is more customary to use the symbolErather thanW.

Mechanical power

Onemetric horsepoweris needed to lift 75kilogramsby 1metrein 1second.

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. Inmechanics,theworkdone by a forceFon an object that travels along a curveCis given by theline integral: $$W_{C}=\int _{C}\mathbf {F} \cdot \mathbf {v} \,dt=\int _{C}\mathbf {F} \cdot d\mathbf {x},$$ wherexdefines the pathCandvis the velocity along this path.

If the forceFis derivable from a potential (conservative), then applying thegradient theorem(and remembering that force is the negative of thegradientof the potential energy) yields: $$W_{C}=U(A)-U(B),$$ whereAandBare the beginning and end of the path along which the work was done.

The power at any point along the curveCis 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 thetorqueτandangular velocityω, $$P(t)={\boldsymbol {\tau }}\cdot {\boldsymbol {\ Omega }},$$ whereωisangular frequency,measured inradians per second.The$\cdot$representsscalar product.

In fluid power systems such ashydraulicactuators, power is given by$$P(t)=pQ,$$wherepispressureinpascalsor N/m²,andQisvolumetric flow ratein 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 themechanical advantageof the system.

Let the input power to a device be a forceF_Aacting on a point that moves with velocityv_Aand the output power be a forceF_Bacts on a point that moves with velocityv_B.If there are no losses in the system, then $$P=F_{\text{B}}v_{\text{B}}=F_{\text{A}}v_{\text{A}},$$ and themechanical advantageof 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, whereT_Aandω_Aare the torque and angular velocity of the input andT_Bandω_Bare 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 themechanical 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 ofvelocity ratiosdetermined by its physical dimensions. See for examplegear ratios.

Electrical power

Main article:Electric power

Ansel Adamsphotograph of electrical wires of the Boulder Dam Power Units, 1941–1942

The instantaneous electrical powerPdelivered to a component is given by $$P(t)=I(t)\cdot V(t),$$ where

• $P(t)$is the instantaneous power, measured inwatts(joulespersecond),
• $V(t)$is thepotential difference(or voltage drop) across the component, measured involts,and
• $I(t)$is thecurrentthrough it, measured inamperes.

If the component is aresistorwith time-invariantvoltagetocurrentratio, then: $$P=I\cdot V=I^{2}\cdot R={\frac {V^{2}}{R}},$$ where $$R={\frac {V}{I}}$$ is theelectrical resistance,measured inohms.

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$T$.Thepeak poweris 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 theduty cycleof the pulse train.

Radiant power

Power is related to intensity at a radius$r$;the power emitted by a source can be written as: $$P(r)=I(4\pi r^{2}).$$

See also

• 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