To build a pyramid

This was written by me as a freshman undergrad at Case Western Reserve University in 2006 for PHYS 123, Physics I Honors. It is reprinted in full, original form, with no attempt made to make the conclusion more realistic. Some of my assumptions were silly and made the problem trivial, but these were never graded for strict accuracy. 

The Great Pyramid was the tallest structure in the world from about 2570 BC to AD 1300 (when it was surpassed by the Lincoln Cathedral in England).  Its specifications are given below:

Length of one side of base (base is square) = 230.4 m

Height (original, estimated) = 146.6 m

Number of stones = 2.4 million

Total mass = 5.9 million tonnes

Average density = 2300 kg/m³

Egyptologists, from the Ancient Greeks who subjugated the old empire of the Pharaohs, to the British archaeologists who had such an interest in the Egyptian colony, to the present day scholars, have consistently marveled at the investment of labor and planning that must have gone into producing such a marvelous creation more than four and a half millennia past. 

How much work did it REALLY take to build the pyramid?  How many workers were involved?  What was the power of the labor machine that created it?  To put things in perspective, how much would this building cost today?

for W_x is the work involved in transporting the blocks horizontally across the desert, and W_y is the work involved in getting the bricks to their locations in height on the pyramid.

where the force is in opposition to dragging the blocks from their quarry (friction) and the displacement is how far from the pyramid building site the blocks must be dragged (the stones came from various far-away source; the average distance is probably around 500 miles).  The Egyptians had no means of locomotion for these stones except ropes and muscle.  Let us say that the force of friction was approximately equal to the normal force, due to the incredibly high friction generated by rocks on sand without lubrication.  Set F equal to the force of gravity and solve for W_x.

W_x = (2.6 million tonnes)*(9.81 m/s²)*(500 miles)

W_x = 2.1E+16 J

The pyramid has an angle with the normal provided by:

Take the average height of a block (we may be getting into rough territory here) to be 2.0m.  By the total height of the pyramid, we make the deduction that the pyramid is 73 layers high, and that for every layer the angle still holds true (that is to say, the slant height of the pyramid is a straight line).  The height and area of each level depends upon which numerical level it is, so our result is going to be a sum of works required for each level; work will be the volume of the layer multiplied by the density of the pyramid times the gravitational acceleration times the height.  To simplify that expression:

where h is the height at point n, d is one side of the base of the level at point n, ρ is the density of the rock, and g is the gravitational acceleration.  When values are given appropriately:

The sum of the two energies yields:

This amount of energy is approximately equal to what is released from a 5 megaton bomb.  If you wanted to fund a labor force of this size, consider that Egyptologists project that 30,000 workers on average were needed for 20 years, provided that they worked 10 hours a day every day.  If you think you can pay average wages of 10 dollars per day without mutiny, then you too can have your own pyramid for a mere 2.2 billion dollars.