PlanetPhysics/Power

Power
Power is the rate of energy transfer. Since there are several forms of energy, there are several ways of describing power. In general terms of energy, power is defined as

$$P = \frac{dE}{dt}.$$

Mechanical Power
The energy transfer in mechanical systems where work is done by an applied force

$$ P = \frac{dE}{dt} = \frac{dW}{dt}.$$

Using the relation between work and force

$$ dW = {\mathbf F} \cdot d{\mathbf r}$$

and then differentiating to get power,

$$P = \frac{dW}{dt} = {\mathbf F} \cdot \frac{d{\mathbf r}}{dt} = {\mathbf F} \cdot {\mathbf v}.$$

The corresponding form of power in rotation is

$$P = {\mathbf M} \cdot {\mathbf \omega},$$

where $${\mathbf M}$$ is the torque and $${\mathbf \omega}$$ the angular velocity vector.

Electrical Power
Since energy is transfering from a device storing electrical energy to another device in the circuit that converts to another form of energy, power is the rate of change of electrical potential energy. For a DC circuit

$$P = \frac{dU}{dt} = i V.$$

Units
The SI unit of the power is one joule per second, which is called watt :

$$\frac{\mathrm{J}}{\mathrm{s}} := \mathrm{W}.$$

The watt is equal to $$\mathrm{kg \cdot m^2/s^3}$$ in the base units.

The english units of power are

$$ 1 \left [horsepower \right] = 1 \left [hp \right] = 550 \left [\frac{ft \, lb}{s} \right]$$