Alternative Homework Assignment: The Great Pyramid at Giza
The object of this problem is to estimate the number of people required to build the Great Pyramid at Giza in Egypt. This pyramid is nearly 150 m tall* and 230 m wide at the base. The average block of stone used in building the pyramid has dimensions .66 m x 1.00 m x 1.50 m. These blocks were quarried at a site on the Giza Plateau that shall be taken to be about 5 km away although it may have been closer. The ancient Egyptians transported these blocks from the quarry to the pyramid on wooden sleds. They lubricated the ground in front of the sled's skids to make it easier to pull the blocks. In order to pull the blocks to the required heights, the Egyptians built earthen ramps that greatly reduced the slope of the pyramid side. In this fashion, Egyptian's working for the glorification of their gods and king, built the pyramid in 20 years. They were mostly farmers and needed to tend their crops most of the year so they could only work for 3 months a year when there was nothing to be done on their farms. It is beyond the scope of this problem to calculate the number of people required to quarry the stones, make and repair the ropes, apply the limestone seal, design and build the chambers, and perform the countless other acts of art and craftsmanship that were required to make the pyramid the marvel that it is. Instead, it is limited to calculating the number of workers needed to get the stones to the site and to lift them to the required heights. It will be left as an estimate to decide how many other people were required.
a) With the dimensions of the pyramid and blocks given above, how many blocks were required to build the pyramid?
b) The blocks are made of sandstone with a density of about 2.00 g/cm3. What is the mass of the average block?
c) The Egyptian's sled system had a rather high coefficient of static friction (m= 0.7) but a much lower coefficient of kinetic friction (m= 0.3**). How much force is required to start a block moving?
d) Estimate the amount of force that one person can sustain for ten hours of work. (Think about the weight of a hiker's backpack.) Using this value, how many people does it take to pull a block at constant velocity.
e) Estimate a comfortable walking speed for an average person. Assume that the workers could only walk at half this speed while they were pulling a block, but that they could walk at this speed on the trip from the pyramid back to the quarry. At these speeds, how many blocks can one crew, of the size calculated above, move the 5 km from quarry to pyramid in one ten hour work day? Truncate this number to an integer.
f) If each worker can generate 5x their usual force for a short time, by bracing their feet against ridges or pegs that are fixed in place, and pulling with all of their might, how long does it take to accelerate a block to the walking speed? Are the forces experienced by the worker reasonable?
a) Now consider the process of lifting the blocks to the required heights. If a work crew double the size of the crew used to move the block across flat land was used to move the block up a ramp, what is the steepest angle the ramp could have? At this angle, how long would the ramp to the top of the pyramid have to be? How long would it take the crew to move the block to this height?
b) Use the results of your calculations and the information provided about the pyramid to devise a model for the time required for one crew to raise all of the blocks to the required heights. Use a simple model, but make certain that the estimate is as accurate as the estimate for the time to transport the blocks from the quarry.
c) Use your answers to 1a, 1e, and 2b to calculate a total number of worker hours required to build the pyramid.
d) Considering that the pyramid was built in 20 years, and the workers only worked on it for three months out of each year, use the previous assumption of a ten hour work day to calculate the total number of people required. For a better estimate, guess at the number of people that were needed in addition to those that have been considered in this problem.
* Most pyramid data obtained or calculated from Encyclopedia Smithsonian (www.si.edu/welcome/faq/pyramid.htm).
** Estimate found in Physics, Ohanian 1989
W. W. Norton & Company Inc.
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