Great Pyramid of Cheops (Center of Mass Problem)

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Homework Help Overview

The discussion revolves around calculating the center of mass and the work required to lift the blocks of the Great Pyramid of Cheops, focusing on its geometric and physical properties. The problem involves concepts from calculus and physics, particularly in three-dimensional space.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to understand how to find the center of mass in three dimensions and questions whether integration is necessary. They also express uncertainty about using the center of mass height for calculating work.
  • Some participants suggest that integration is indeed required and provide insights into the relationship between height and mass elements.
  • Others question the understanding of mass distribution and the calculations involved, indicating confusion about the integration process.

Discussion Status

The discussion is ongoing, with participants providing guidance on the need for integration and clarifying the relationship between mass elements and height. There is a lack of consensus on the original poster's understanding, as they express confusion and urgency for further information.

Contextual Notes

The original poster mentions a time constraint, needing answers by the same night, which may influence the depth of the discussion and the responses provided.

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The Great Pyramid of Cheops at El Gizeh, Egypt, had a height H = 148.6 m before its topmost stone fell. Its base is a square with edge length L = 227.4 m. Its volume V is equal (L^2)H/3. Assuming that it has uniform density p(rho) = 1.8e3 kg/m^3.
(a) What is the original height of its center of mass above the base?
(b) What is the work required to lift all the blocks into place from the base level?

Kay, so I'm good with finding center of mass in one dimension, but in two and three dimensions, it is confusing the hell out of me. Am I supposed to integrate? If so, what? I just need a push in the right direction (I think).

Also, for finding the work, I'm guessing I'm supposed to use the CM height as the distance for all of the blocks traveled, but I'm not sure about that either. Any help would be appreciated!
 
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Yes, you need to integrate.

If z is the height above the base of an element of mass of thickness dz then the element of mass at height z is proportional to [itex](h-z)^2 dz[/itex]. You need to find the average of z which means summing over all elements of mass - and dividing by the total mass.
 
But wouldn't that just be dividing the total mass by the total mass? I'm really confused. I need to have the answer tonight, so if anyone can give me a bit more information, it would really help.
 
No. The integrand of the integral in the numerator will have an additional factor of z in it so the integrals are not the same.
 

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