Pyramids, by definitionon Merriam-Webster Dictionary, is a very large structure built especially inancient Egypt that has a square base and four triangular sides which form apoint at the top. Pyramids are also known as “mr” or “mir” by ancientEgyptians and was built to be a Royal tomb for the deceased pharaoh. There aremany developed theories by the Egyptologists about why the tombs of the earlypharaohs were built in the pyramid shape. First is because the pyramidrepresented the first land to appear at the beginning of time- a hill called’Ben-Ben’. Then, the pyramid had sloping sides so that the dead pharaoh couldsymbolically climb to the sky and live forever.
And lastly, the pyramidrepresented the rays of the sun. While Statics of RigidBodies is a course or study that deals solely with the mechanics of rigidbodies. A rigid body is a body whichdoes not deform under the influence of forces. In all real world applications, there is always deformation, however,many structures exhibit very small deformations under normal loadingconditions, and rigid body mechanics can be used with sufficient accuracy inthose cases. Also, the principles ofrigid body mechanics are some of the building blocks needed for the mechanicsof deformable bodies. The mechanics of rigid bodies is sub-divided into twoareas, statics and dynamics, with dynamics being further subdivided intokinematics and kinetics.
Statics is thestudy of bodies in equilibrium. Thismeans there are no unbalanced forces on the body, thus the body is either atrest or moving at a uniform velocity. The key to the lastingpyramids in Egypt is the stability only achieved if the force acted upon thepyramid and acted by the pyramid is in equilibrium. In equilibrium, astationary object will remain stationary because the resultant force andresultant moment is zero for all the forces applied. Meaning the Pyramids mustresist violent winds and earthquakes. Although ancient Egyptians doesn’t haveadvanced tools and mathematics to aid in the construction, they were able to dosomething incredible and impossible that some conspiracy theorist thought thatthere were aliens aiding the Egyptians in building the pyramids. Stability may beachieved but it is not enough to make it last, to make it durable and resistantto withering caused by nature. Materials that will be used to construct thepyramid will be a crucial factor to determine whether the pyramids will last.
Pyramid of Khufu was majorly made out of grey-yellow coloured nummulitelimestone and belong to Mokattam Formation. The lower layer of the Pyramid ofKhufu was made out of stones with dimensions of 1 meter by 2.5 meter and 1-1.5meters in height weighing 10 tons.
For upper layer, stones have dimensions of 1meter by 1 meter and 0.5 meter in height weighing 1.3 tons for easiertransport.
For calculation purposes, Egyptologists, who study the history,culture, literature, language, religion, architecture and art of ancient Egypt,uses 2.5 tons for average stone block. To analyse the pyramidbuilt by ancient Egyptian, we will take into our account the Pyramid of Khufu,one of the great Pyramids of Giza.
According to Wikipedia, Pyramid of Khufu isthe oldest and largest of the three pyramids in the Giza pyramid complexbordering what is now El Giza, Egypt. It is the oldest of the Seven Wonders ofthe Ancient World, and the only one to remain largely intact. It has a heightof 146.7 meters and its mass is estimated to be 5.9 million tonnes, volume ofroughly 2.5 million cubic meters sloping at approximately 52 degrees.
The forceexerted by a single hauler or Egyptian worker will be set to 117.7 Newton. To perform necessarycomputations and visualise the application of statics in the Pyramid of Khufu,we will draw a free body diagram. A free body diagram according to PhysicsClassroom, are diagrams used to show the relative magnitude and direction ofthe forces. It would be very useful to easily analyse and solve problems suchas this. So how did theEgyptians built the remarkable Pyramids? Imhotep, an architect from the AncientEgypt decided to do something which had been never attempted before: to build amonument made only of stone. He engineered a process wherein stoned weighingaround 2.
5 tons would be stacked on the top of each other, each level, a littlesmaller than the one beneath, creating a pyramid. This pyramid designed byImhotep became the first pyramid, known as Step Pyramid in Saqqara. If the stones are asheavy as 2.
5 tons, how do ancient Egyptians move and raise the stones withoutthe use of cranes or trucks? Let’s try to compute the force needed to move the2.5 ton stone. Assuming that an Egyptian worker can create 675 newton ofpushing force, A major problem facedby the builders as we can see in the computation, with friction taken in to theaccount, it is impossible for the ancient Egyptians to move the stone withoutany help. How did they move it? There are many theories made to try to explainhow they moved it from one place to another. One of the theory suggested is theEgyptians dragged the stones on to the sand which has been wet with water.According to this theory, the water reduced the friction between the stones andground by 40%. The question probablyresonates at you is how do water reduce friction? Let us know what friction is.
Friction, according to Wikipedia, is the force resisting the relative motion ofsolid surfaces, fluid layers, and material elements sliding against each other.Water under certain circumstances, may become in between the two surfaces, andact as a lubricant to reduce the amount of contact area between these twosurfaces hence- reducing the friction between them. Less friction means lesseffort to move objects from one place to another. Coefficient of friction isalso important to know. It is the ratio of friction force between two bodiesand the force sandwiching them together.
It depends on the material or surface forexample, ice or smooth surfaced object has lower coefficient of friction thanrubber, concrete or asphalt and also whether the object is stationary or not. Thecoefficient of kinetic friction is much lower than coefficient of staticfriction. In raising the blocksof stone from a lower elevation to higher requires enormous effort. To do so,ancient Egyptians used ramps. Ramps according to Wikipedia, is a flatsupporting surface tilted at an angle, with one end higher than the other, usedas an aid for raising or lowering a load.
The inclined plane is one of the sixclassical simple machines defined by Renaissance scientists. Using ramps, theeffort or force required to lift a heavy object over a greater distance isreduced drastically compared to lifting the object vertically. The possible type oframp used is the zigzag ramp wherein it was wrapped through the pyramid yetproblems is of course inevitable. Turning blocks at the corners of the zigzagwould be enormously difficult.
In addition, the weight of the ramp resting onan unfinished pyramid will put an immense strain to the not yet stable pyramid.Other type of ramp is the straight ramp, but also proved to be problematic.Straight ramps would have stretched over a mile away to the pyramid itself andwill consume more construction material than the main structure. Going back, In order tobuild a pyramid, the land had to be surveyed for a relatively flat area abovethe flood plain covering an area greater than the size of the proposed pyramid.Surveying too is an important process in building the pyramid for taking intoconsideration the enormous load to be placed on the proposed site, the bedrockhad to be surveyed for defects to ensure a stable foundation. This is a job ofa civil engineer. After a site was designated it would be prepared by levellingoff the rock foundation. Levelling is important to ensure the stability of thestructure to be built.
Building would begin on this prepared site. The Egyptians aregenerally using their body parts for measurements. The standard unit of measurewas the “cubit” which was a length from the elbow to the tip of the middlefinger. If measuring a small quantity, this kind of measurement is sufficient,but for The Great Pyramids, the “Royal Cubit” measurement was used. In “RoyalCubit” measurement, it is 52.5 cm and normally it is represented by a rod. Therod was used in measuring the base of the pyramids for it to have a perfectlysquare foundation.
The basic principles of statics were developed very early.The fundamentals of levers, inclined planes, and other principles were neededby early civilizations to construct huge structures such as the pyramids and inbuilding any structure, to have level surfaces is extremely important.Egyptians developed levels using the force of gravity called a “plumb level”which is an application of the well-known plumb line. Plumb lines wereappreciated for their ability to produced true vertical line. Let’s talk about theshape of the structure. Pyramid of course.
What is so special with its shapethat those structures lasted for centuries and resisted wrath of nature, fromsand storms, although some pyramids are not earthquake resistant due tovibrations that causes loosening of rocks which form the pyramid. To start,pyramids, relating to geometry, according to Wikipedia, is a polyhedron formedby connecting a polygonal base and a point, called the apex. Each base edge andapex form a triangle, called a lateral face. Pyramids of Egypt were so stablebecause it has a consistent low centre of gravity due to evenly distributedmass to its wide base. Centre of gravity of an object is the area in the objectwherein the mass of the object is concentrated. It is a very important conceptto consider in modern construction to ensure the stability of the building,ensuring that the structure won’t collapse once built.
It is recommendable forthe buildings to have lower centre of gravity to resist torque or rotationalforce cause by natural phenomena for example, the wind, or earthquake. Althoughcentre of gravity nor gravity has not been conceptualized until Newton,Egyptians were able to build an impressive structure. Although there are someerror in construction where in the west side is slightly longer than the eastside it would not be a big deal to the overall equilibrium of the pyramid. Before, there was alittle knowledge about this subject matter but the need for the ancientEgyptians to build tombs for their pharaoh to help them ascend in heaven andlive, pushed them to the boundaries of ancient civilization creating somethingthat would be deemed impossible at their time.
By the unmatched imagination ofancient architects, and determination of workers, they were able to designsomething that lasted for thousands of years. With the use of simple machines,such as ramps, they were able to move the heavy stones used for the pyramidthat if we looked today, will be impossible to perform without heavymachineries and earth movers such as trucks and cranes. By their wit, they madetheir job easier by making the sand rightfully wet so the stones can be pushedor pulled with a little less effort compared to pushing them through the drysand. With the knowledge acquired in the modern age, analysis of Pyramidsbecome possible and so as the creation of theories about the possible way ofconstructing and moving the materials used. Importance of the centre of gravityto the stability of the Pyramids of Egypt was also known.
What seemed to bework of an alien can now be explained with the help of statics. Althoughinformation how great pyramid architects thought of building a structure likethat is not that great or not yet known, one thing is for sure. That thepyramids of Egypt were built to last.