I have a question involving calculus and Mechanics. It's from AP french, question 10-12.(adsbygoogle = window.adsbygoogle || []).push({});

There's a pyramid with a square base of side, b, length 230m, and height, h, 150m. The density, p, of the pyramid is 2500kg/m^3. We are asked to find the total potential energy, PE of the pyramid taking the zero potential height as the base of the pyramid.

Here's what I did. I realized that the problem is to take a slice of volume, find it's mass and then multiply by gravity and the height of the slice. Then sum all the slices from height 0 to height 150.

It seems that mass is dependant on volume, which is dependant on height and base length. And potential energy is dependant on mass and height.

m = p * tripleint(dV)

= p * tripleint(dA dh)

PE = mg int(dh)

Therefore,

PE = p * tripleint(dA dh)g * int(dh)

= pg * int(tripleint(dA dh)dh)

Then I saw by inspection that tripleint(dA dh) = some function V(h), the volume of the pyramid in terms of height.

So then subbing in V(h), we find

PE = pg * int(V(h)dh)

Then I noted from a thought experiment of cutting a pyramid out of a box, that the volume of pyramid as a function of height, V(h), is,

V(h) = (1/3)(b^2)h, so I rewrote potential energy by taking out (1/3)(b^2) as constants and keeping 'h' under the integral as:

PE = (1/3) * pg * (b^2) * int(h dh), with the 'h' bounds from 0 to 150m.

After integrating I get:

(1/6) * pg * (b^2) * (h^2), with h going from 0 to 150.

Plugging all the given values for p, g, b, and h I find:

(1/6)(2500)(9.8)(230^2)(150^2 - 0^2)

= 4.860 x 10^12 Joules.

Unfortunately, this answer is off by a factor of 1/2!

The answer given in the back of the book is '2.4 x 10^12 Joules'.

(If you're not familiar with AP French, his answers are always approximations, so the other digits besides '2.4' are not known, and could be non-zero).

Why am I off by a factor of two? I'm really truly not seeing why my method is incorrect.

:(

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# Potential energy, a pyramid, and calculus

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