a circular swimming pool has diameter of 24ft, sides are 5 ft high, and the depth of the water is 4 ft. How much work is required to pump all of the water out over the side? water weighs 62.5 lb per cubic foot
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dW = ρgAx dx
..... ρ∙g given as 62.5 lb/ft³
..... A = πr²
so we are doing Force* distance = work.
we are going to move cross sectional slabs of water with radius 12,
pi*12^2, each slab has a different height to cover. for example the slab on the bottom has to cover 4 feet and then another foot. this is the x part. and all the slabs need to cover one foot so x + 1
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the 1 foot extra on top
so 62.5 * pi *12^2 integral ( x+1) from 0 to 4 is your work
whats important is that the force is given as pounds. dont worry about the pg part. im not so sure about that
well g is gravitational constant, and weight is mass * g , and you have density in there as well. but it all works out
yes just trying to make sense of it all. im still not understanding the x+1 since the bottom has to clear 5 ft not 1 ft
well it clears 4 feet, then +1 foot for the top .
the next slab say has to clear 3 feet then +1 foot for top, so on and so forth.
the toppest slab has to clear just +1 foot.
but each slab has that extra +1 foot to clear