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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 6.6 Chop-Multiply-Add: Work for Non-Constant Forces

You have previously learned the Chop-Multiply-Add strategy for constructing physics integrals. You can use this strategy to help find the work done by non-constant forces.
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Chop: Work involves multiplying force and displacement along some path traveled by an object. You can start by chopping up the path into small steps. Each step along the path is given by an infinitesimal vector displacement \(\vec{dr}\text{.}\)
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Multiply: You can then multiply the force acting on the object by the displacement step above to get the infinitesimal amount of work done along only that step \(dW = \vec{F} \cdot \vec{dr}\text{.}\)
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Add: Last, you add up all the infinitesimal works with an integral along the entire path, resulting in the expression below!
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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*}
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Exercises Activities

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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Figure 6.6.2. A person pulling a heavy box.
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  1. Sketch a graph of the tension vs. position \(x\)
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  2. Write an equation for the tension as a function of \(x\)
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  3. Use the Chop-Multiply-Add strategy to find the total work the rope does on the box:
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    Chop: Divide space into appropriate, infinitesimal pieces; label one of these pieces on your graph
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    Multiply: Find the work done on the box during one infinitesimal piece of space
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    Add: Find the total work done on the box
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2. Non-Constant Winch.

You have a winch
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a machine that uses a rope to pull on another object
that can be set to exert a changing force. You decide to set your winch to produce the force
\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 \(x_f = L\text{.}\)
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Answer.
\begin{equation*} W_{winch} = \frac{F_o L}{6} \end{equation*}
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