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Power

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Quick
Power is the time rate at which work is done.


Details

The definition of work makes no reference to the passage of time. If a barbell weighing 350 N is lifted through a vertical distance of 0.4 m at constant velocity, 140 J of work is done, whether it took 1 second, 1 hour, or 1 year to do it. But often it is necessary to know how quickly work is done. This is described in terms of power. In ordinary conversation the word "power" may be synonymous with "energy" or "force." In the scientific field a much more precise definition is used: power is the time rate at which work is done. Like work and energy, power is a scalar quantity.

When a quantity of work ΔW is done during a time interval Δt, the average work done per unit time or average power P_avg is defined to be:

(Eq1)

Pavg  =  ΔW Δ t

The rate at which work is done might not be constant. Even when it varies, the instantaneous power P can be defined as the limit of the quotient in Eq1 as Δt approaches zero:

(Eq2)

P =   lim Δt→0 ΔW Δ t = dW dt

In mechanics the power can be expressed in terms of force and velocity. Suppose that a force

F

acts on a body while it undergoes a vector displacement Δ

s

. If F_tan is the component of

F

tangent to the path (parallel to Δ

s

), then the work done by the force is ΔW = F_tanΔs, and the average power is:

(Eq3)

Pavg  =  FtanΔs Δ t   =  Ftanvavg

Instantaneous power P is the limit of this as Δt → 0:

(Eq4)

P = Ftanv

where v is the magnitude of the instantaneous velocity. Eq4 can also be expressed in terms of the scalar product:

(Eq5)

P = F ⋅ v

which is the instantaneous rate at which force

F

does work on a particle.


Units
The SI unit of power is the watt (W). One watt equals one joule per second (1 W = 1 J/s). The kilowatt (1 kW = 10³ W) and the megawatt (1 MW = 10⁶ W) are also commonly used. In the British system, work is expressed in foot-pounds, and the unit of power is the foot-pound per second. A larger unit called horsepower (hp) is also used:

1 hp = 550 ftlb/s = 33,000 ftlb/min

That is, a 1 hp motor running at full load does 33,000 ftlb of work every minute. A useful conversion factor is:

1 hp = 746 W = 0.746 kW

or 1 horsepower equals about 3/4 of a kilowatt.

The watt is a familiar unit of electrical power; a 100 W bulb converts 100 J of electrical energy into light and heat each second. But there's nothing inherently electrical about a watt or a kilowatt. A light bulb could be rated in horsepower, and some automobile manufacturers rate their engines in kilowatts rather than horsepower.

The units of power can be used to define new units of work or energy. The kilowatt-hour (kWh) is the usual commercial unit of electrical energy. One kilowatt-hour is the total work done in 1 hour (3600 s) when the power is 1 kilowatt (10³ J/s), so:

1 kWh = (10³ J/s)(3600 s) = 3.6 × 10⁶ J = 3.6 MJ

The kilowatt-hour is a unit of work or energy, not power.

power equals energy (or work) over time which equals the quantity energy over charge times the quantity charge over time which equals voltage times current:

p =

dw dt

=

(

dw dq

)

*

(

dq dt

)

= vi

where,
p = power in watts
w = energy in joules
t = time in seconds
v = voltage in volts
i = current in amperes
q = charge in coulombs