Cable Work - Bring That Load Up

Cable Work - Bring That Load Up
TECTechnics Classroom TECTechnics Overview

A cable weighing 2 lbs/ft is used to lift a 300 lbs load from the bottom of a well 100 feet deep.
Calculate the work done to raise the load to the surface.

The string:
(a) S₇P₃A₃₁ (Force - Pull).
The math:
Cable Work Pull That Load
Pj Problem of interest is of type force. Work problems are generally of type force because force is the doer of work. In the case of white-collar work, intellectual force actuates materiality.
Consider the above diagram (fig. CW.1):
Let W be the work done to raise the load x feet.
Let k be the work done in raising the load h feet.
k = [300 + 2(100 - x_avg)]h -------(1). x_avg is the representative average of the partitions of the distance BC.
So, the rate of change of k with respect to h is:
k/h = [300 + 2(100 - x_avg)] = 500 -2x_avg
limit (k/h) as h → 0 = dW/dx (instantaneous change of W with respect to x)
So, limit (k/h) as h → 0 = 500 -2x. Since x_avg → x.
So, dW/dx = 500 -2x.
So, W = 500x - x² + C---------(2). Where C is a constant.
When x = 0, W = 0. So C = 0.
So, W = 500x - x².
So, for x = 100,
W = 500(100) - (100)(100) = 40,000 ft-lb.

Math

The point . is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.
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