Center (in y direction) of Mass of 3D pyramid

[012anonymousx]

Homework Statement

A monument is made from stone blocks of density 3800kg/m^3. The monument is 15.7m high, 64.8m wide at the base, and 3.6m thick from front th back. How much work was required to build the monument? (Hint: find ycm).

Homework Equations

ycm = (1/M) * ∫ydm, M = mass total

The Attempt at a Solution

Take a cross section and get a rectangle.
The length will be: (64.8/15.7) * y
The width will be: (3.6/15.7) * y

Thus, dm = density * length * width * dy

M total = density * volume = 6958742.8

ycm = (1/M) * ∫y * density * length * width * dy (from 0 to 15.7)

Okay, so this solution is wrong. It gives the the ycm as 7.85m. But ycm is actually a third of the height (as it is for triangles).

My question is: what is fundamentally wrong with my approach?

I have the solution though. I don't need that.

 

[gneill]

Your formulas for the length and width have them both zero for y = 0. Unless the monument is supposed to be standing on its head, that could be problematical

 

[012anonymousx]

That shouldn't matter.

Take the center of mass from 7.85m from the top.

Interestingly. 7.85m is the middle of the triangle.

 

[012anonymousx]

Got it got it got it! 3.6 is constant. Brb

[EDIT] Works. You made me think of it. I was looking at integrating from the top and wrote out eqn to reverse and realized... wait a minute, I'm scaling the width but its constant...

TYVM.

 

[Nickie]

I need to know why my solution is wrong.

Your approach is incorrect because you are assuming that the cross section of the pyramid is a rectangle, when in fact it is a triangle. This means that the length and width of the cross section will change as you move up the pyramid. Additionally, your integral should be from 0 to 15.7/2 since you are only considering the top half of the pyramid.

To correctly find the ycm, you would need to integrate over the entire pyramid, taking into account the changing dimensions of the cross section. This can be done using calculus and the formula for the center of mass of a triangle.

Overall, your approach is not fundamentally wrong, but it does not accurately reflect the shape of the pyramid and therefore does not give the correct answer.