Editing Power (physics) (section)

==Mechanical power==
[[File:Horsepower plain.svg|thumb|One ''metric horsepower'' is needed to lift 75&nbsp;[[kilogram]]s by 1&nbsp;[[metre]] in 1&nbsp;[[second]].]]
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 [[mechanical work|work]] done by a force {{math|'''F'''}} on an object that travels along a curve {{mvar|C}} is given by the [[line integral]]:
<math display="block">W_C = \int_C \mathbf{F} \cdot \mathbf{v} \, dt = \int_C \mathbf{F} \cdot d\mathbf{x},</math>
where {{math|'''x'''}} defines the path {{mvar|C}} and {{math|'''v'''}} is the velocity along this path.

If the force {{math|'''F'''}} is derivable from a potential ([[Conservative force|conservative]]), then applying the [[gradient theorem]] (and remembering that force is the negative of the [[gradient]] of the potential energy) yields:
<math display="block">W_C = U(A) - U(B),</math>
where {{mvar|A}} and {{mvar|B}} are the beginning and end of the path along which the work was done.

The power at any point along the curve {{mvar|C}} is the time derivative:
<math display="block">P(t) = \frac{dW}{dt} = \mathbf{F} \cdot \mathbf{v} = -\frac{dU}{dt}.</math>

In one dimension, this can be simplified to:
<math display="block">P(t) = F \cdot v.</math>

In rotational systems, power is the product of the [[torque]] {{math|'''τ'''}} and [[angular velocity]] {{math|'''ω'''}},
<math display="block">P(t) = \boldsymbol{\tau} \cdot \boldsymbol{\omega},</math>
where {{math|'''''ω'''''}} is [[angular frequency]], measured in [[radians per second]].  The <math> \cdot </math> represents [[scalar product]].

In fluid power systems such as [[hydraulic]] actuators, power is given by <math display="block"> P(t) = pQ,</math> where {{mvar|p}} is [[pressure]] in [[pascal (unit)|pascals]] or N/m<sup>2</sup>, and {{mvar|Q}} is [[volumetric flow rate]] in m<sup>3</sup>/s in SI units.

===Mechanical advantage===
If a mechanical system has no losses, then the input power must equal the output power. This provides a simple formula for the [[mechanical advantage]] of the system.

Let the input power to a device be a force {{math|''F''<sub>A</sub>}} acting on a point that moves with velocity {{math|''v''<sub>A</sub>}} and the output power be a force {{math|''F''<sub>B</sub>}} acts on a point that moves with velocity {{math|''v''<sub>B</sub>}}.  If there are no losses in the system, then
<math display="block">P = F_\text{B} v_\text{B} = F_\text{A} v_\text{A},</math>
and the [[mechanical advantage]] of the system (output force per input force) is given by
<math display="block"> \mathrm{MA} = \frac{F_\text{B}}{F_\text{A}} = \frac{v_\text{A}}{v_\text{B}}.</math>

The similar relationship is obtained for rotating systems, where {{math|''T''<sub>A</sub>}} and {{math|''ω''<sub>A</sub>}} are the torque and angular velocity of the input and {{math|''T''<sub>B</sub>}} and {{math|''ω''<sub>B</sub>}} are the torque and angular velocity of the output.  If there are no losses in the system, then
<math display="block">P = T_\text{A} \omega_\text{A} = T_\text{B} \omega_\text{B},</math>
which yields the [[mechanical advantage]]
<math display="block"> \mathrm{MA} = \frac{T_\text{B}}{T_\text{A}} = \frac{\omega_\text{A}}{\omega_\text{B}}.</math>

These relations are important because they define the maximum performance of a device in terms of [[velocity ratio]]s determined by its physical dimensions.  See for example [[gear ratio]]s.