2. (20 points) In the construction of pyramids and other wondrous monuments of a

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2. (20 points) In the construction of pyramids and other wondrous monuments of ancient Egypt, it has been well known by modem archeologists that the ancient Egyptians transported massive stones and statues across desert sands using sleds that large numbers of workers dragged by rope. Some of the evidence for this i presented in figure 2 which depicts a painting on the wall of the tomb of Djehuti Hotep (c. 1932-1842 B.C.) This tomb was discovered in the 1800's. It shows an enormous statue being dragged on a sled by a total of 172 men. The statue is 22.3 ft high and the statue and sled have a combined weight of 127.800 pounds. The statue and sled have a combined mass of 58.00 metrie tons (1 metric ton 1000 kg). These men are arranged in four double rows with four ropes. Because of a lack of knowledge of perspective art drawing on the part of the ancient Egyptians, the four ropes are not actually attached to a common point at the base of the sled as the figure shows, but rather are attached to the front of the sled at four equally spaced points across the width of the base with the ropes being entirely parallel to each other along their length. Assuming that each man can exert a pulling force of 50.00 lbs (222.4 N), and using the number of men pulling as depicted in the scene of figure 2, actual modern attempts at reproducing this sled dragging technique of transportation have failed, resulting in a plethora of conspiracy theories that the ancient Egyptians were really aided in construction of their great works by "UFO space aliens" and "ancient astronauts However, quite recently, in May 2014, a group of physicists lead by Professor Daniel Bonn from the University of Amsterdam discovered the secret. Notice in figure 2 the man standing on the bottom right base of the statue. This man is pouring water into the sand ahead of the sled. Since the discovery of the tomb of Djehuti Hotep in the 1800's, it had been believed by archaeologists that the pouring of the water was part of a holy religious ceremony to bless the sand upon which the great statue would pass. However, Professor Bonn, being a well trained physicist like you, was more knowledgeable than the archaeologists who, after all. just know about digging in dirt. Professor Bonn realized that the water made the sand rigid and smooth, thus reducing the coefficient of kinetic friction between the sled and sand. His team proved this by building a small scale xperimental apparatus and showed that adding the right amount of water at the right rate to the sand allowed the sled to glide across the top of the sand with a much lower coefficient of kinetic friction. Hence, dragging a large massive object on sled across desert sand by a finite number of men is now known to be feasible. No need for "space aliens a) Based on the information above, and assuming constant speed and massless ropes, calculate the coefficient of kinetic friction between the sled and wet sand that must have existed for the scene depicted in figure 2. Make a free body diagram as part of your solution. Neglect the mass of the men standing on the statue. Solve the problem algebraically first! Figure 2. Depiction of painting on the wall of the tomb of Djehuti Hotep for Problem 2a.

Explanation / Answer

a. given
F by each man = 50 lbs = 222.4 N
weight of statue = W = 127,800 lbs
number of men, n = 172 men
let coefifcneit of friciton be k
then for statue moving with constant speed
nF = k*W
hence
k = nF/W = 0.06729

b. for the case whrn spool is used
form moment balanec
Tstatue = kW
also
Tmen*R1 = Tstatue*R2
R1 = 3 m
R2 =2.5 m
hence
Tmen = kW*R2/R1
numberof men required = n
n*50 = 0.06729*127800*2.5/3
n = 143.33 men

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