Potential energy, a pyramid, and calculus

  • #1
Vihsadas
3
0
I have a question involving calculus and Mechanics. It's from AP french, question 10-12.

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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  • #2
So that I can just give you a quick reply, I am not going to look for your factor of 2, but here is what I would do, very similar to what you did.

Cut the pyramid into slices of thickness dh. The volume of a slice is dV = s^2dh where s(h) is a linear function of h that is b when h = 0 and 0 when h = 150. All you have to do is integrate rho*g*h*s^2*dh from h = 0 to 150
 
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  • #3
I went to see my prof today for help, and he didn't actually go through the problem with me, but I think the gist of what he was saying is that the way that I was approaching the problem by integrating dV to find the mass, would be fine, if I realized that the height I was dealing with is no longer 150. It's instead the height to the center of mass of the pyramid.

This sounds about right? (If this were the case, then I think that the center of mass of the pyramid would be at h/sqrt(2).)
Is this right?
 
  • #4
Vihsadas said:
I went to see my prof today for help, and he didn't actually go through the problem with me, but I think the gist of what he was saying is that the way that I was approaching the problem by integrating dV to find the mass, would be fine, if I realized that the height I was dealing with is no longer 150. It's instead the height to the center of mass of the pyramid.

This sounds about right? (If this were the case, then I think that the center of mass of the pyramid would be at h/sqrt(2).)
Is this right?
The integral I described (and the one I think you were trying to do in the first place) is exactly the integral that needs to be done to find the center of mass. If you knew where the center of mass was (call the height H) then the GPE would just be mgH. Looking back at your first post, it appears to me that where you have int(dh) you should just have h, but the h belongs inside the triple integral you are doing. You should have

PE = p * tripleint(dAhdh)g
instead of
PE = p * tripleint(dA dh)g * int(dh)

The integral of dA is the s(h)^2 I was talking about

s(h) = 230 - (230/150)h

PE = pg * int(s^2hdh)

Just expand the s^2 and do the integral.
 
  • #5
You give density as
You don't have two different integrals. Imagine a thin layer at height y above the base. It has the shape of a square, of side length x, say, and so area x2. Since the sides are planes, x is a linear function of y: x= ay+ b. At the base, where y= 0, x= b= 230. At the top, where y= 150, x= a(150)+ 230, so a= -230/150= -23/15. At each height y, the square has area (230- (23/15)x)2. Taking thickness dy, the thin layer has volume (230- (23/15)x)2dy and so mass equal to 2500(230- (23/15)x)2dy kg.
Of course, at height y, the potential energy is mgy so here it is (9.81)(2500)(230- (23/15)x)2dy. That integral
[tex](9.81)(2500)\int_0^{150}(230- (23/15)x)^2dy[/tex]
gives the total potential energy.
 
  • #6
Ohhhhhh. I get it.
I see the problem now. Thank you both for helping me to actually understand this problem. I can do the problem now! :)

My troubles came from 1) not properly understanding how to set up the integral, and 2) Not understanding how to find the appropriate s(h).
 
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Related to Potential energy, a pyramid, and calculus

1. What is potential energy?

Potential energy is the energy that an object possesses due to its position or state. It is also known as stored energy because it has the potential to do work in the future.

2. How does a pyramid represent potential energy?

A pyramid can represent potential energy because the higher an object is placed on the pyramid, the more potential energy it has. This is similar to how an object placed at a higher elevation has more potential energy due to its position.

3. What is the relationship between potential energy and calculus?

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. Potential energy and calculus are related because potential energy can be mathematically calculated using the formula PE = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object.

4. Can potential energy be negative?

Yes, potential energy can be negative. This usually occurs when the object's position is below a reference point, such as the ground. In this case, the potential energy is calculated as a negative value.

5. How is potential energy related to other forms of energy?

Potential energy can be converted into other forms of energy, such as kinetic energy, thermal energy, or electrical energy. For example, when a ball is dropped from a height, its potential energy is converted into kinetic energy as it falls. It can also be converted back into potential energy, such as when a pendulum swings back and forth.

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