Chapter 2, Problem 435RE

True or False? Justify your answer with a proof or a counterexample.

435. The amount of work to pump the water out of a half-full cylinder is half the amount of work to pump the water out of the full cylinder.

To determine

To check: whether the statement, “The amount of work to pump the water out of a half-full cylinder is half the amount of work to pump the water out of the full cylinder” is true or false.

Answer to Problem 435RE

The given statement is false.

Explanation of Solution

Given information:

The given statement is, “The amount of work to pump the water out of a half-full cylinder is half the amount of work to pump the water out of the full cylinder”.

Calculation:

Consider a cylinder of radius $r$ and height $h$ .

Let $x$ be the vertical distance below the top of the tank and the density of water is $\rho$ .

So, the work done to empty the completely filled cylinder is given by,

$\begin{array}{c}{W}_1={\displaystyle \underset{0}{\overset{h}{\int }}\rho \left(\pi r^2\right)}xdx\\ =\rho \left(\pi r^2\right){\displaystyle \underset{0}{\overset{h}{\int }}xdx}\\ =\rho \left(\pi r^2\right){\left[\frac{x^2}{2}\right]}_0^h\left[\text{Using }{\displaystyle \int x^{n}dx}=\frac{x^{n+1}}{n+1}\right]\\ =\frac{1}{2}\pi \rho r^2{h}^2\end{array}$

If the cylinder is half filled, then the work done to empty the half-filled cylinder is given by

$\begin{array}{c}{W}_2={\displaystyle \underset{\frac{h}{2}}{\overset{h}{\int }}\rho \left(\pi r^2\right)}xdx\\ =\rho \left(\pi r^2\right){\displaystyle \underset{\frac{h}{2}}{\overset{h}{\int }}xdx}\\ =\rho \left(\pi r^2\right){\left[\frac{x^2}{2}\right]}_{\frac{h}{2}}^h\left[\text{Using }{\displaystyle \int x^{n}dx}=\frac{x^{n+1}}{n+1}\right]\\ =\frac{1}{2}\pi \rho r^2\left[h^2-\frac{h^2}{4}\right]\\ =\frac{3}{8}\pi \rho r^2{h}^2\end{array}$

Thus,

$\begin{array}{l}\frac{W_1}{W_2}=\frac{\frac{1}{2}\pi \rho r^2{h}^2}{\frac{3}{8}\pi \rho r^2{h}^2}=\frac{4}{3}\\ \Rightarrow W_2=\frac{3}{4}{W}_1\end{array}$

Therefore, the amount of work to pump the water out of a half-full cylinder is three-fourth the amount of work to pump the water out of the full cylinder.

Conclusion:

Hence, the given statement is false.