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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 = ∫ C F ⋅ v d t = ∫ C F ⋅ d x, {\displaystyle 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: W C = U (A) − U (B), {\displaystyle W_{C}=U(A)-U(B),} where A and B are the beginning and end of the path along which the work was done.

The power at any point along the curve C is the time derivative: P (t) = d W d t = F ⋅ v = − d U d t. {\displaystyle P(t)={\frac {dW}{dt}}=\mathbf {F} \cdot \mathbf {v} =-{\frac {dU}{dt}}.}

In one dimension, this can be simplified to: P (t) = F ⋅ v. {\displaystyle P(t)=F\cdot v.}

In rotational systems, power is the product of the torque τ and angular velocity ω, P (t) = τ ⋅ ω, {\displaystyle P(t)={\boldsymbol {\tau }}\cdot {\boldsymbol {\omega }},} where ω is angular frequency, measured in radians per second. The ⋅ {\displaystyle \cdot } represents scalar product.

In fluid power systems such as hydraulic actuators, power is given by P (t) = p Q, {\displaystyle P(t)=pQ,} where p is pressure in pascals or N/m, and Q is volumetric flow rate in m /s in SI units.