Solutions - 0120-03

(1) Integral formula:

$$\begin{array}{c}W=\int dW\gg \gg {\int }_a^{b}F(y)dy\end{array}$$

Let $y=0$ at the ground and increase going up.

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(2) Compute force:

The force on the rope (at the top) when the bucket is at height $y$ is:

$$F=F_{\text{bucket}}+F_{\text{water}}+F_{\text{chain}}$$

We know $F_{\text{bucket}}=4\mathrm{lb}$.

Water is leaking at ${\displaystyle \frac{0.3}{3}}\mathrm{l}\mathrm{b}/\mathrm{f}\mathrm{t}$. Therefore:

$$F_{\text{water}}=30-\frac{0.3}{3}y$$

The weight of chain remaining is:

$$F_{\text{chain}}=0.25\cdot (50-y)$$

Put together:

$$\begin{array}{c}F=4+30-\frac{0.3}{3}y+0.25\cdot (50-y)\\ \\ \gg \gg 46.5-0.35y\end{array}$$

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(3) Integrate:

$$W={\int }_0^{50}(46.5-0.35y)dy\gg \gg 1887.5\mathrm{f}\mathrm{t}\cdot \mathrm{l}\mathrm{b}$$