Question

A 30-foot length of steel chain weighing 12 pounds per foot is hanging from the top of a building.

how many foot-pounds of work are required to pull half of the chain to the top?

A: 2080ft-lbs B: 1250ft-lbs C:4050ft-lbs D:5200ft-lbs

Answer 1

$$Solution$$

Divide the 30-ft steel chain into small sections of length $\Delta x$ , which are $x$ units below the top of the building.

The force by gravity on each section is

$$\Delta x ft*12lb/ft=12\Delta x lb$$

$W=Fd$

The work done lifting the pieces in the upper half of the steel chain is

$$\Delta W_1=12\Delta x*x$$

The work done lifting the pieces in the lower half of the steel chain is

$$\Delta W_1=12\Delta x*15ft$$

To find the total work done, we will find the sum of all such sections

$$W=W_1+W_2=lim_{n\to \infty}\sum_{i=0}^n 12x \Delta x+lim_{n\to \infty}\sum_{i=0}^n 180 \Delta x$$

$$W=\int_0^{15}12xdx+\int_{15}^{30}180dx$$

$$=[\frac{12x^2}{2}]_0^{15}+[180x]_{15}^{30}=[6(15)^2-0]+180(30-15)$$

$$\implies 1350+2700=4050ft-lbs$$

Answer is C