Chapter 6, Problem 28RE

To determine

To calculate: the work done to raise the elevator from the basement to the third floor by 30ft height.

Answer to Problem 28RE

The total work required to raise the elevator form basement to the third floor is $\underset{\_}{\text{103,500}\text{ft-lb}}$.

Explanation of Solution

Given information:

The weight of elevator is 1,600 lb.

The weighs of cable is $10\text{lb/ft}$.

Length of cable is 200 ft.

The height between basement and third floor is 30 ft.

Calculation:

Calculate the work done by the elevator $w_1$ using the formula.

$w_1=Wd$ (1)

Here, W is weight of elevator and d is the distance from basement to third floor.

Substitute 1600 lb for W and 30 ft for d in Equation (1).

$\begin{array}{c}{w}_1=1600\times 30\\ =\text{48,000}\text{ft}-\text{lb}\end{array}$

Determine the work needed to raise the bottom of the cable:

$w_2=d_l\times W_c\times d$

Here, $d_l$ is length of the cable above the third floor, $W_c$ is weight of cable, and d is the distance from basement to third floor.

Substitute (200-30) ft for $d_l$, $10\text{lb/ft}$ for $W_c$, and 30 ft for d.

$\begin{array}{c}{w}_2=\left(200-30\right)\text{×}10\times 30\\ =170\times 10\times 30\\ =51,000\text{ft-lb}\end{array}$

Expression to find the work required to raise the elevator by the cable at the third floor

(0 ft to 30 ft):

$w_3={\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx}$ (2)

Here, $f\left(x\right)$ is weight of cable.

Substitute 0 for a, 30 for b, and $10x$ for $f\left(x\right)$ in Equation (2).

$\begin{array}{c}{\text{w}}_3={\displaystyle \underset{0}{\overset{30}{\int }}10xdx}\\ ={\left[\frac{10x^2}{2}\right]}_0^{30}\\ ={\left[5x^2\right]}_0^{30}\\ =\left[5{\left(30\right)}^2-0\right]\end{array}$

$=4500\text{ft-lb}$

Find the total work required as shown below:

$w_{total}=w_1+w_2+w_3$ (3)

Substitute $\text{48,000}\text{ft-lb}$ for $w_1$, $51,000\text{ft-lb}$ for $w_2$, and $4500\text{ft-lb}$ for $w_3$ in Equation (2).

$\begin{array}{c}{w}_{total}=48,000+51,000+\text{4,500}\\ =\text{103,500}\text{ft-lb}\end{array}$

Therefore, total work required to raise the elevator form basement to third floor is $\underset{\_}{\text{103,500}\text{ft-lb}}$.