Definition 6.6.1. Work.
The work done by a single force \(\vec{F}\) on an object moving from \(\vec{r}_i\) to \(\vec{r}_f\) is:
\begin{equation*}
W = \int_{i}^{f}\vec{F} \cdot \vec{dr}
\end{equation*}
Section 6.6 Work for Non-Constant Forces
Your previous definition for Work by a Constant Force is only valid for constant forces. If a force is changing, you can instead use an integral to find the total work.
Subsubsection Activities
Activity 6.6.1. Doing Work.
You are pulling the box shown below from \(x = 0\) to \(x = L\text{.}\) You find that as the box moves, the tension in the rope is not constant; instead, it starts at \(5T_0\text{,}\) but by the time you reach \(x = L\) it has changed to \(T_0\text{.}\) Assume the tension decreases linearly.
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Sketch a graph of the tension vs. position \(x\)
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Write an equation for the tension as a function of \(x\)
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Use the Chop-Multiply-Add strategy to find the total work the rope does on the box:Chop: Divide space into appropriate, infinitesimal pieces; label one of these pieces on your graphMultiply: Find the work done on the box during one infinitesimal piece of spaceAdd: Find the total work done on the box
Activity 6.6.2. Non-Constant Winch.
You have a winch that can be set to exert a changing force. You decide to set your winch to produce the force
1
a machine that uses a rope to pull on another object
\begin{equation*}
\vec{F}^{winch} = \frac{F_o}{L^2} (xL - x^2)\hat{x}
\end{equation*}
where \(F_o\) and \(L\) are constants and \(x\) is the position of the object being pulled. Sketch a graph of this force vs. position and use it to find the work done by the winch between \(x_i = 0\) and \(x_f = L\text{.}\)
