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Center of Mass of 3D Pyramid

I need three different algebraic solutions of the center of mass of a square-based pyramid. Finding the center of mass of a square-based 3D pyramid is usually done in calculus, but our professor wants us to work it out in algebra. One hint he gave us (for one way of approaching this problem) is to split the pyramid into small blocks; this is a way of solving a calculus-problem with algebraic methods.

Solution Preview

Before the first two proofs, we do some preliminaries common for the two (and which can also be made common for the third, but not done here, to keep the 3rd close to the hint by the professor).

Preliminaries for proofs 2 and 3 (in 3D, but can be extended to any dimension)
We take the plane of the base to be the XY plane and Z the direction of the height.

Lemma 1
The center of mass (COM) of any pyramid with polygonal base is on the straight line connecting the top with the COM of the bottom.

Proof
Split pyramid into many thin slices parallel to the base. From the similarity between the slices we see that the COM of all the slices form a straight line passing through both the top and the COM of the base. As we can replace every slice by a point of the same mass in its COM, the COM of all the slices taken together must be on that line.

This lemma reduces the question of finding the COM of a pyramid to finding the height of the COM from the base.

Lemma 2
The centers of mass of two triangular base pyramids (tetrahedrons) with different triangular bases but of the same height of the top are both at the same height.

Proof
A linear transformation in the XY plane (leaving Z intact) can transform any horizontal triangle into any other triangle in the same plane. Imagining that the volume mass density of the material of which a pyramid is made remains homogeneous and changes with the linear transformation so that the mass of the pyramid or of any horizontal slice of it remains unchanged, we see that the distribution of mass on the line connecting the top with the COM of the base remains the same after a linear transformation in the XY plane. Therefore, as we can transform in this way one triangular pyramid into another, the heights of the COM of the pyramids are the same.

Lemma 3
The height of the COM of a polygonal pyramid is the same as the height of the COM of a triangular pyramid of the same top hight. ...

Solution Summary

The Center of Mass of a 3-D Pyramid is found without using calculus. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

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