Power (physics)

Power
Power
Common symbols	P
SI unit	watt (W)
In SI base units	kg⋅m2⋅s−3
Derivations from; other quantities	P = E/t; P = F·v; P = V·I; P = τ·ω;
Dimension

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]

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.

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 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 and instantaneous power

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

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 interval $Δ t$ approaches zero.

P=\lim _{\Delta t\to 0}P_{\mathrm {avg} }=\lim _{\Delta t\to 0}{\frac {\Delta W}{\Delta t}}={\frac {dW}{dt}}.

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

W=Pt.

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

Mechanical power

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: