How much work is done lifting a 35 pound object from the ground to the top of a 30 foot building if the cable used weighs 1.5 pounds per foot?

Answer 1

Consider the problem in 2 parts.... the weight and the rope.

The work in pulling up the 35 lb weight 30 ft up is (30)(35) = 1050 ft-lbs ( this assume that the center of gravity of the weight is pulled up 30... (this may not be quite exact but it is close depending on the size & shape of the weight!)

To calculate the work pulling the rope, the weight of an increment of rope is 1.5(delta y). That incremental piece of rope is then pulled up 30-y ft if y is measured from the ground up!

So, the integral of (30-y)(1.5)(dy) from 0 to 30 should be the work pulling the rope up (assuming 30 ft of rope) and this yields

Work = 675 ft-lbs. So the total work is about 1050 + 675 = 1725 ft-lbs

Answer 2

The work can be divided into discrete pieces.... for example, to lift the 35 lb weight only up 30 feet is work equal to

W = F*D = 35x30 = 1050 ft-lbs Now think about the rope and assume that there is 30 ft exactly to be pulled up (probably a little less but we assume 30!)

Then to pull the entire rope up 1 ft requires 1ft x 1.5 lbs/ft x 30 ft of rope = 45 ft-lbs

Then for the next pull, there is 29 ft of rope so 1 ft x 1.5 lbs/ft x 29 ft of rope = 43.5

To sum up each pull of 1 ft at a time, the expression would be

1.5 x 1 x 30 + 1.5 x 1 x 29 + 1.5 x 1 x 28 + ........... + 1.5 x 1 x 1 and this factors to

(1.5 x 1)(30 + 29 + 28 + 27 + .......... + 1) = (1.5)(31 x 30)(.5) = 697.5 ft - lbs

So, the total for the rope and weight is 697.5 + 1050 = 1747.5 ft-lbs