19.1: B.1- Definitions of Work and Power

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Work is defined for translational motion as the product of a force times a distance through which the force moves. For translation in one dimension, denoted here as \(x\), the differential work of force \(f_{x}(x)\) moving through distance \(dx\) at position \(x\) is

\[d W=f_{x}(x) \times d x\label{eqn:B.1} \]

So the work by a spatially varying force moving from position \(x_1\) to position \(x_2\) is

\[W=\int_{x=x_{1}}^{x=x_{2}} d W=\int_{x=x_{1}}^{x=x_{2}} f_{x}(x) d x\label{eqn:B.2} \]

with units of lb-ft, lb-inch, or N-m \(\equiv\) J (for joule).

The velocity of one-dimensional motion is \(v_{x}=d x / d t \Rightarrow d x=v_{x} d t\). Therefore, alternative expressions for work are

\[d W=f_{x} \times v_{x} d t \Rightarrow W=\int_{t=t_{1}}^{t=t_{2}} f_{x} v_{x} d t\label{eqn:B.3} \]

This leads to the definition of power, the time rate of work:

\[P \equiv \frac{d W}{d t}=f_{x} \times v_{x}\label{eqn:B.4} \]

with units of lb-ft/s, lb-inch/s, or N-m/s = J/s \(\equiv\) W (for watt).

For rotation in one dimension, denoted here as \(\theta\), the differential work of moment \(M\) moving through angle \(d \theta\) at position \(\theta\) is \(d W=M \times d \theta\), and the differential work of moment \(M\) moving with velocity \(\dot{\theta}\) during interval \(dt\) is \(d W=M \times \dot{\theta} d t\). Therefore, the work of a possibly varying moment between states 1 and 2 can be expressed in either of the two forms:

\[W=\int_{\theta=\theta_{1}}^{\theta=\theta_{2}} M d \theta=\int_{t=t_{1}}^{t=t_{2}} M \dot{\theta} d t\label{eqn:B.5} \]

with units of lb-ft, lb-inch, or J. The associated definition of power is

\[P=M \times \dot{\theta}\label{eqn:B.6} \]

with units of lb-ft/s, hp (for horsepower, 1 hp ≡ 550 lb-ft/s), lb-inch/s, or W.

At least some of the work that is done on a real system might be stored in some form of recoverable energy, or work done might be completely dissipated and lost. For all real engineering systems, at least part of the input work that is intentionally done on a system is lost irretrievably, not used for the intended purposes.