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Feb 20, 2024

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The Mystery of the Great Pyramid of Giza By: Susie Mohamed (Physics 231-001 final project) Background The Great Pyramid of Giza is one of the most iconic and mysterious structures in the world. It is the oldest and largest of the three pyramids in Giza and is believed to have been built as a tomb for the Pharaoh Khufu. The pyramid was constructed over a period of more than 20 years and is made up of over 2 million stone blocks, some weighing as much as 80 tons. The Great Pyramid of Giza is a revelation in many ways. Its sheer size and complexity are a testament to the incredible skills and knowledge of the ancient Egyptians. The pyramid is also a marvel of engineering, with its precise measurements and precise alignment to the cardinal points of the compass. But the Great Pyramid of Giza is also a mystery. Despite centuries of study and exploration, we still don't know exactly how the ancient Egyptians were able to build such an enormous and complex structure with the technology available at the time. There are many theories about how the pyramid was constructed, including the use of ramps, pulleys, and levers, but none have been definitively proven. Additionally, there are many mysteries surrounding the purpose of the Great Pyramid of Giza. While it is widely believed to have been built as a tomb for the Pharaoh Khufu, there is little evidence to support this theory. Some researchers have suggested that the pyramid may have had other uses, such as an astronomical observatory or a power plant. Overall, the Great Pyramid of Giza is a revelation and a mystery that continues to fascinate and intrigue people around the world. Its incredible size and complexity, combined with the many unsolved mysteries surrounding its construction and purpose, make it a true wonder of the ancient world. https://www.brown.edu/academics/archaeology/publications/building-great-pyramid-giza- investigating-ramp-models#:~:text=Many%20scholars%20believe%20that%20the,and%20the %20spiral%20ramp%20models . Susie Mohamed Physics 231 Final Project

Physics to Analyze in this Report:  Rolling Resistance  LO 6: Modeling and solving forces on objects including spring, normal, tension, gravity , friction , and other resistive forces  LO 17: I can relate work and kinetic energy of an object  LO 15: I can use heat capacity to relate temperature and internal energy involving heat transfer between two or more objects  LO 16: I can describe the rate of heat transfer through materials Modeling and solving forces of a block in the Pyramid going up a ramp, and the work and energy of the workers to get it up the ramp: The construction of the pyramids of Giza, which were built over 4,500 years ago, is an impressive feat of engineering and human ingenuity. The largest of the pyramids, also known as the Great Pyramid of Khufu, is estimated to have been built from around 2.3 million stone blocks, some of which weigh up to 80 tons. The average weight of the blocks used is around 2 tons. Moving such enormous stones without the use of modern machinery is a remarkable achievement, and it is not entirely clear how the ancient Egyptians managed it. One theory is that they used ramps to drag the stones up to the construction site, but this would have required an enormous amount of manpower. That is what will be modeled here by solving rolling resistant (form of friction), gravity, and normal forces. This will also include use of those forces to calculate the work of a worker to move the average stone made of limestone. Knowns, unknowns, and equations for the different forces involved and Work and Energy: Knowns: Block: Mass = 2 tons Gravity= 9.8 m/s 2 Pyramid: Height = 530 m Ramp: Xo: 530 m Yo: 48.9 m Unknowns: Ɵ = ? x wagon = ? W = ? Equations: F = CrrN Tan-1 = (x/y) Frr = Frr coefficient * FG * cos Ɵ W = F * d To begin solving for the forces, we need to identify what forces are acting on the block as it is sliding up the ramp: Susie Mohamed Physics 231 Final Project FN person>block FN ground>block

FN = Normal Force FG= Force of Gravity Frr = Rolling Resistance Force Since all are the same, we can use the equation for rolling resistance to find the Force: F = CrrN F is the rolling resistance force, Crr is the coefficient of rolling friction, and N is the Normal force, which is the force perpendicular to the surface on which the wheel is rolling. N will also take into account the x & y directions of the ramp given dimensions. By using the same triangle above, the numbers 1 and 12 were given dimensions, representing the ADA act in which a wheelchair ramp has to have ratio for every inch of vertical rise, there must be 12 horizontal inches of ramp. With this, the inverse tangent equation can be used: https://handiramp.com/ada-guidelines/ada-ramp slope.htm#:~:text=The%20ADA%20requires%20that%20all,would%20need %2021%E2%80%B2%20of%20ramp . Tan Ɵ = (1/12) Rearrange to solve for the angle: Ɵ = tan -1 (1/12) Ɵ = 4.76 ⁰ This is a reasonable estimate of the angle of incline for the ramp since the blocks have such an impressive mass. The ramp could not be too steep or the blocks would not move up them as it would require too much energy. To calculate the force of the rolling resistance, gravity must be considered. The formula here describes this relationship: Frr = Frr coefficient * FG * cos Ɵ Frr = (0.05)(2000 kg)(9.8 m/s 2 )(cos 4.76 ) ⁰ Susie Mohamed Physics 231 Final Project Frr block>ramp 1 Will all be equal w/ constant velocity FG earth>block 12

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Frr = 977.6 N This calculation gives the work required in the x and y components to find the total energy needed to move a block by using the formula: W = F * d Where W is the work required, g represents gravity, Fg represents the force of gravity, and d represents the distance traveled. I used the distance of the ramp in the pyramid, 1609.4 m roughly, and used 1/3 times that to pick a point where the average block was 2 tons. This made an average distance of 530 m. Also used, is the average height of where this block would be, which was 1/3 up the pyramid, or roughly 48.9 m. https://www.brown.edu/academics/archaeology/sites/academics-archaeology/files/publication/ document/Rigby2016.pdf X direction: W rr = 977.6 N * 530 m W rr = 518128 Joules Y direction: W rr = (2000 kg) * (9.8 m/s 2 ) * (48.9 m) W rr = 958440 Joules Now that the components are solved for, we will use Pythagorean theorem to solve for the hypotenuse of the triangle, or the ramp: a 2 + b 2 = c 2 (518128) 2 + (958440) 2 = c 2 C= 1089524.602 Joules or roughly 1090 KJ This would be the theoretical work required to move an average block up a ramp in the Great Pyramid of Giza. Fun Fact: Now let’s look at the amount of breakfast one of the workers needed to consume that morning in order to have enough energy to move one block. From above, Work is equal to 1090 KJ to move a block, but there were 20 workers on average aiding in moving this block, all asserting energy onto the block to get it to move. This is represented by: 1090 KJ / 20 workers = 54.5 KJ Here is the conversion from KJ to Kcal: Susie Mohamed Physics 231 Final Project

One Kcal = 4.184 KJ 54.5 KJ / 4.184 KJ = 13.03 Kcal In other words, the standard Egyptian engineer that aided in assembling the Great Pyramid of Giza, theoretically needed a little bit more than teaspoon and a half of hummus to have enough energy to move one of the average sized blocks or about half a piece of falafel according to the nutritional facts of these two foods. Regardless of the method used to transport the blocks, the fact the ancient Egyptians were able to move and place these massive stones with such precision is a testament to their engineering skills and their ability to organize and manage large-scale projects. It is a remarkable achievement that has stood the test of time and continues to inspire awe and wonder today. Use of heat capacity of limestone to relate temperature and internal energy involving heat transfer between air and the limestone to overall describe the rate of transfer through materials: Temperature can have a significant effect on the structural integrity of the Great Pyramid of Giza. The pyramid is made up of millions of stone blocks, which expand and contract in response to changes in the temperature. The temperature outside the pyramid can vary greatly from morning to night, on average the temperature change was 23 C in the summer. ⁰ During the day, the stones on the outside of the pyramid can heat up and expand, while the stones on the inside remain relatively cool. This can create stress within the structure, which can cause cracking and other forms of damage over time. Similarly, at night, the stones on the outside can cool down and contract, while the stones on the inside remain warmer, creating further stress. Here is an example providing insight on the temperature change from the outside of the block facing the environment, which uses heat capacity to relative temperature of the limestone block, and the rate of transfer through materials (air to limestone). We can also use this to compare this to another material, which further argues how impressive the ancient Egyptians were to build the pyramid out of the material they did. Knowns: Block: Mass = 2000 kg Area = 1 m^2 Limestone: 0.86 J kg ¿ 0 C = c 1.3 J/smC⁰ Temperature: Morning: 30 0 C Night: 7 ⁰ C ∆T = 23 C ⁰ Unknowns: Φ=? Q = ? Tblock = ? Equations: kA d ∗ ∆T = Q t A = L * W Q = mc∆T Φ= kA d ∗ ∆T Susie Mohamed Physics 231 Final Project

The equation used to calculate heat flux is given by: kA d ∗ ∆T = Q t To find the area of the block, the average dimensions of a real block used were given (2.0 * 1.0m * 2.0m). A = L * W A = 1 m² λ = ½ L = 1 m Φ =( 1.3 J smC ⁰ ∗( 1 m 2 ) 1 m ) * 10⁰C * 21600s Φ =280800 W/m 2 This gives the heat flux which is the amount of thermal energy radiated over the surface of the limestone block facing the outside environment. This amount of heat gained or lost by the block can be calculated using the equation: Q = mc∆T = kA l ∗ t ∗( Tair − Tblock ) https://theengineeringmindset.com/specific-heat-capacity-of-materials/ https://www.naturalstoneinstitute.org/stoneprofessionals/technical-bulletins/rvalue/ Specific Heat Capacity of Limestone Heat conductivity of Limestone 0.86 J kg ¿ 0 C 1.3 J smC 0 Where m is the mass of the block, c is the specific heat of limestone: mc ∆T ∗ l kA ∗ t = Tair − Tblock Tair − mc ∆T ∗ l kA ∗ t = Tblock 30 ⁰ C − ( 2000 kg )( 0.86 J kg ¿ 0 C )( 23 0 C )( 1 m ) ( 1.3 J smC 0 ) ∗( 1 m 2 )( 21600 s ) = Tblock Susie Mohamed Physics 231 Final Project

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Tblock = 27.94 ⁰ C 30 C – 27.94 C = 2.06 C ⁰ ⁰ ⁰ The average block’s temperature on the outside facing the environment changes about 2 C, ⁰ during the hot summer days of Egypt to the cool summer nights. The ancient Egyptians were very clever to use limestone as the primary building material for the Great Pyramid of Giza and other monumental structures. Limestone is a sedimentary rock that is composed primarily of calcium carbonate, which is a relatively soft and easily workable material when it is first quarried. However, over time, limestone undergoes a process called diagenesis, which causes it to harden and become more durable. This process occurs when the calcium carbonate in the limestone is gradually replaced by other minerals, such as quartz or calcite, which are much harder and more resistant to weathering and erosion. This means that the limestone used in the construction of the Great Pyramid of Giza has become stronger and more durable over time, as the minerals in the stone have undergone diagenesis and bonded together to form a more solid and stable structure. In addition to its durability, limestone also has other advantageous properties that made it an ideal building material for the ancient Egyptians. It is readily available in the local area and can be quarried and shaped using relatively simple tools and techniques. It is also relatively lightweight, which made it easier to transport and handle during the construction process. Furthermore, limestone has good insulating properties, which helped to regulate the temperature inside the pyramid and protect the interior from the extreme heat of the Egyptian sun. This, combined with the natural ventilation provided by the pyramid's design, helped to keep the interior cool and dry, which was important for preserving the many treasures and artifacts that were stored inside. Overall, the use of limestone as the primary building material for the Great Pyramid of Giza was a testament to the Egyptians' ingenuity and resourcefulness. They recognized the unique properties of this natural material and were able to harness its strength and durability to create one of the most enduring and iconic structures in human history. Summary: The Great Pyramid of Giza is one of the most iconic and mysterious structures in the world. It is the oldest and largest of the three pyramids in Giza and is believed to have been built as a tomb for the Pharaoh Khufu. The pyramid was constructed over a period of more than 20 years and is made up of over 2 million stone blocks, some weighing as much as 80 tons. The Great Pyramid of Giza is a revelation in many ways. Its sheer size and complexity are a testament to the incredible skills and knowledge of the ancient Egyptians. The pyramid is also a marvel of engineering, with its precise measurements and precise alignment to the cardinal points of the compass. Susie Mohamed Physics 231 Final Project

But the Great Pyramid of Giza is also a mystery. Despite centuries of study and exploration, we still don't know exactly how the ancient Egyptians were able to build such an enormous and complex structure with the technology available at the time. There are many theories about how the pyramid was constructed, including the use of ramps, pulleys, and levers, but none have been definitively proven. Additionally, there are many mysteries surrounding the purpose of the Great Pyramid of Giza. While it is widely believed to have been built as a tomb for the Pharaoh Khufu, there is little evidence to support this theory. Some researchers have suggested that the pyramid may have had other uses, such as an astronomical observatory or a power plant. Overall, the Great Pyramid of Giza is a revelation and a mystery that continues to fascinate and intrigue people around the world. Its incredible size and complexity, combined with the many unsolved mysteries surrounding its construction and purpose, make it a true wonder of the ancient world. Susie Mohamed Physics 231 Final Project

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