Chapter 6.6, Problem 16E

To determine

To Calculate: The work done to lift the lower end of a vertical hanging rope to the level of the upper end.

Answer to Problem 16E

The work done to raise the lower end of the chain to the level of its upper end is $62.5lbft$ .

Explanation of Solution

Given Information: The weight of the chain is $25lb$ and the length of the chain is $10ft$ .

Concept Used: The work done by a force $F$ in moving a particle from $x_1$ to $x_2$ is given by

  $W={{\displaystyle \int }}_{x_1}^{x_2}F\left(x\right)dx$

Calculation:The weight per unit length of the chain is $\frac{weight}{length}=\frac{25}{10}=2.5$

An infinitesimally small element of length $dx$ having a weight $2.5 \times dx$ at a distance $x$ below the mid-point of the rope reaches a position which is at a distance $x$ above the midpoint of the rope. So,it will have risen by a distance $2x$ .

The work done to lift this infinitesimally small element of length $dx$ is

  $dW=weight\cdot distance=2.5\times dx\times 2x=5xdx$

The total work done is the integral of this infinitesimally small work from $x=0$ (at the mid-point) to $x=\frac{length}{2}=5$ (at the bottom)

So

  $W={{\displaystyle \int }}_0^{5}dw={{\displaystyle \int }}_0^{5}5xdx$

  $={5\frac{x^2}{2}]}_0^5=(5\times \frac{25}{2})-0$

  $=62.5$

Therefore, the total work done is $62.5lb\cdot ft$ .