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 (symbol W), equal to one joule per second (J/s). Power is a scalar quantity. 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 or, more generally, the rate of change of total mechanical energy. It is given by:
P = d E d t , {\displaystyle P={\frac {dE}{dt}},}
where P is power, E is the total mechanical energy (sum of kinetic and potential energy), and t is time. For cases where only work is considered, power is also expressed as:
P = d W d t , {\displaystyle P={\frac {dW}{dt}},}
where W is the work done on the system. However, in systems where potential energy changes without explicit work being done (e.g., changing fields or conservative forces), the total energy definition is more general. We will now show that the mechanical power generated by a force F {\textstyle \mathbf {F} } on a body moving at the velocity v {\textstyle \mathbf {v} } can be expressed as the product: P = d W d t = F ⋅ v {\displaystyle P={\frac {dW}{dt}}=\mathbf {F} \cdot \mathbf {v} }
If a constant force F {\textstyle \mathbf {F} } is applied throughout a distance x {\textstyle \mathbf {x} } , the work done is defined as W = F ⋅ x {\displaystyle W=\mathbf {F} \cdot \mathbf {x} } . In this case, power can be written as:
P = d W d t = d d t ( F ⋅ x ) = F ⋅ d x d t = F ⋅ v . {\displaystyle 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 {\textstyle C} , then the work is expressed in terms of the line integral:
W = ∫ C F ⋅ d r = ∫ Δ t F ⋅ d r d t d t = ∫ Δ t F ⋅ v d t . {\displaystyle 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 = d W d t = d d t ∫ Δ t F ⋅ v d t = F ⋅ v . {\displaystyle 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 called activity.
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.
… excerpt ends here. Continue reading the full article.
