If a constant force F is applied throughout a distance x, the work done is defined as W = F ⋅ x {\displaystyle W=\mathbf {F} \cdot \mathbf {x} } . displaystyle P=\lim _{\Delta t\to 0}P_{\mathrm {avg} }=\lim _{\Delta t\to 0}{\frac {\Delta W}{\Delta t}}={\frac {dW}{dt}}. The output power of a motor is the product of the torque that the motor generates and the angular velocity of its output shaft. 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} . displaystyle \mathrm {MA} ={\frac {T_{\text{B}}}{T_{\text{A}}}}={\frac {\omega _{\text{A}}}{\omega _{\text{B}}}}.

Other common and traditional measures are horsepower (hp), comparing to the power of a horse; one mechanical horsepower equals about 745. displaystyle \mathrm {MA} ={\frac {F_{\text{B}}}{F_{\text{A}}}}={\frac {v_{\text{A}}}{v_{\text{B}}}}. The ratio of the pulse duration to the period is equal to the ratio of the average power to the peak power. Peak power and duty cycle [ edit ] In a train of identical pulses, the instantaneous power is a periodic function of time. displaystyle P={\frac {dW}{dt}}={\frac {d}{dt}}\int _{\Delta t}\mathbf {F} \cdot \mathbf {v} \,dt=\mathbf {F} \cdot \mathbf {v} .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. 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. The similar relationship is obtained for rotating systems, where T A and ω A are the torque and angular velocity of the input and T B and ω B are the torque and angular velocity of the output. Mechanical power [ edit ] One metric horsepower is needed to lift 75 kilograms by 1 metre in 1 second.

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. 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. 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. Instantaneous power is the limiting value of the average power as the time interval Δ t approaches zero. These relations are important because they define the maximum performance of a device in terms of velocity ratios determined by its physical dimensions.

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. In the International System of Units (SI), the unit of power is the watt (W), which is equal to one joule per second. P ( t ) = p Q , {\displaystyle P(t)=pQ,} where p is pressure in pascals or N/m 2, and Q is volumetric flow rate in m 3/s in SI units. In the case of a periodic signal s ( t ) {\displaystyle s(t)} of period T {\displaystyle T} , like a train of identical pulses, the instantaneous power p ( t ) = | s ( t ) | 2 {\textstyle p(t)=|s(t)| Let the input power to a device be a force F A acting on a point that moves with velocity v A and the output power be a force F B acts on a point that moves with velocity v B.