Pyramids, by definition
on Merriam-Webster Dictionary, is a very large structure built especially in
ancient Egypt that has a square base and four triangular sides which form a
point at the top. Pyramids are also known as “mr” or “mir” by ancient
Egyptians and was built to be a Royal tomb for the deceased pharaoh. There are
many developed theories by the Egyptologists about why the tombs of the early
pharaohs were built in the pyramid shape. First is because the pyramid
represented 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 could
symbolically climb to the sky and live forever. And lastly, the pyramid
represented the rays of the sun.
While Statics of Rigid
Bodies is a course or study that deals solely with the mechanics of rigid
bodies. A rigid body is a body which
does 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 loading
conditions, and rigid body mechanics can be used with sufficient accuracy in
those cases. Also, the principles of
rigid body mechanics are some of the building blocks needed for the mechanics
of deformable bodies. The mechanics of rigid bodies is sub-divided into two
areas, statics and dynamics, with dynamics being further subdivided into
kinematics and kinetics. Statics is the
study of bodies in equilibrium. This
means there are no unbalanced forces on the body, thus the body is either at
rest or moving at a uniform velocity.
The key to the lasting
pyramids in Egypt is the stability only achieved if the force acted upon the
pyramid and acted by the pyramid is in equilibrium. In equilibrium, a
stationary object will remain stationary because the resultant force and
resultant moment is zero for all the forces applied. Meaning the Pyramids must
resist violent winds and earthquakes. Although ancient Egyptians doesn’t have
advanced tools and mathematics to aid in the construction, they were able to do
something incredible and impossible that some conspiracy theorist thought that
there were aliens aiding the Egyptians in building the pyramids.
Stability may be
achieved but it is not enough to make it last, to make it durable and resistant
to withering caused by nature. Materials that will be used to construct the
pyramid will be a crucial factor to determine whether the pyramids will last.
Pyramid of Khufu was majorly made out of grey-yellow coloured nummulite
limestone and belong to Mokattam Formation. The lower layer of the Pyramid of
Khufu was made out of stones with dimensions of 1 meter by 2.5 meter and 1-1.5
meters in height weighing 10 tons. For upper layer, stones have dimensions of 1
meter by 1 meter and 0.5 meter in height weighing 1.3 tons for easier
transport. 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 pyramid
built 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 is
the oldest and largest of the three pyramids in the Giza pyramid complex
bordering what is now El Giza, Egypt. It is the oldest of the Seven Wonders of
the Ancient World, and the only one to remain largely intact. It has a height
of 146.7 meters and its mass is estimated to be 5.9 million tonnes, volume of
roughly 2.5 million cubic meters sloping at approximately 52 degrees. The force
exerted by a single hauler or Egyptian worker will be set to 117.7 Newton.
To perform necessary
computations and visualise the application of statics in the Pyramid of Khufu,
we will draw a free body diagram. A free body diagram according to Physics
Classroom, are diagrams used to show the relative magnitude and direction of
the forces. It would be very useful to easily analyse and solve problems such
So how did the
Egyptians built the remarkable Pyramids? Imhotep, an architect from the Ancient
Egypt decided to do something which had been never attempted before: to build a
monument made only of stone. He engineered a process wherein stoned weighing
around 2.5 tons would be stacked on the top of each other, each level, a little
smaller than the one beneath, creating a pyramid. This pyramid designed by
Imhotep became the first pyramid, known as Step Pyramid in Saqqara.
If the stones are as
heavy as 2.5 tons, how do ancient Egyptians move and raise the stones without
the use of cranes or trucks? Let’s try to compute the force needed to move the
2.5 ton stone. Assuming that an Egyptian worker can create 675 newton of
A major problem faced
by the builders as we can see in the computation, with friction taken in to the
account, it is impossible for the ancient Egyptians to move the stone without
any help. How did they move it? There are many theories made to try to explain
how they moved it from one place to another. One of the theory suggested is the
Egyptians 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 and
ground by 40%.
The question probably
resonates 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 of
solid surfaces, fluid layers, and material elements sliding against each other.
Water under certain circumstances, may become in between the two surfaces, and
act as a lubricant to reduce the amount of contact area between these two
surfaces hence- reducing the friction between them. Less friction means less
effort to move objects from one place to another. Coefficient of friction is
also important to know. It is the ratio of friction force between two bodies
and the force sandwiching them together. It depends on the material or surface for
example, ice or smooth surfaced object has lower coefficient of friction than
rubber, concrete or asphalt and also whether the object is stationary or not. The
coefficient of kinetic friction is much lower than coefficient of static
In raising the blocks
of stone from a lower elevation to higher requires enormous effort. To do so,
ancient Egyptians used ramps. Ramps according to Wikipedia, is a flat
supporting surface tilted at an angle, with one end higher than the other, used
as an aid for raising or lowering a load. The inclined plane is one of the six
classical simple machines defined by Renaissance scientists. Using ramps, the
effort or force required to lift a heavy object over a greater distance is
reduced drastically compared to lifting the object vertically.
The possible type of
ramp used is the zigzag ramp wherein it was wrapped through the pyramid yet
problems is of course inevitable. Turning blocks at the corners of the zigzag
would be enormously difficult. In addition, the weight of the ramp resting on
an 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 and
will consume more construction material than the main structure.
Going back, In order to
build a pyramid, the land had to be surveyed for a relatively flat area above
the 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 into
consideration the enormous load to be placed on the proposed site, the bedrock
had to be surveyed for defects to ensure a stable foundation. This is a job of
a civil engineer. After a site was designated it would be prepared by levelling
off the rock foundation. Levelling is important to ensure the stability of the
structure to be built. Building would begin on this prepared site.
The Egyptians are
generally using their body parts for measurements. The standard unit of measure
was the “cubit” which was a length from the elbow to the tip of the middle
finger. If measuring a small quantity, this kind of measurement is sufficient,
but for The Great Pyramids, the “Royal Cubit” measurement was used. In “Royal
Cubit” measurement, it is 52.5 cm and normally it is represented by a rod. The
rod was used in measuring the base of the pyramids for it to have a perfectly
square foundation. The basic principles of statics were developed very early.
The fundamentals of levers, inclined planes, and other principles were needed
by early civilizations to construct huge structures such as the pyramids and in
building 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 were
appreciated for their ability to produced true vertical line.
Let’s talk about the
shape of the structure. Pyramid of course. What is so special with its shape
that those structures lasted for centuries and resisted wrath of nature, from
sand storms, although some pyramids are not earthquake resistant due to
vibrations that causes loosening of rocks which form the pyramid. To start,
pyramids, relating to geometry, according to Wikipedia, is a polyhedron formed
by connecting a polygonal base and a point, called the apex. Each base edge and
apex form a triangle, called a lateral face. Pyramids of Egypt were so stable
because it has a consistent low centre of gravity due to evenly distributed
mass to its wide base. Centre of gravity of an object is the area in the object
wherein the mass of the object is concentrated. It is a very important concept
to consider in modern construction to ensure the stability of the building,
ensuring that the structure won’t collapse once built. It is recommendable for
the buildings to have lower centre of gravity to resist torque or rotational
force cause by natural phenomena for example, the wind, or earthquake. Although
centre of gravity nor gravity has not been conceptualized until Newton,
Egyptians were able to build an impressive structure. Although there are some
error in construction where in the west side is slightly longer than the east
side it would not be a big deal to the overall equilibrium of the pyramid.
Before, there was a
little knowledge about this subject matter but the need for the ancient
Egyptians to build tombs for their pharaoh to help them ascend in heaven and
live, pushed them to the boundaries of ancient civilization creating something
that would be deemed impossible at their time. By the unmatched imagination of
ancient architects, and determination of workers, they were able to design
something 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 pyramid
that if we looked today, will be impossible to perform without heavy
machineries and earth movers such as trucks and cranes. By their wit, they made
their job easier by making the sand rightfully wet so the stones can be pushed
or pulled with a little less effort compared to pushing them through the dry
sand. With the knowledge acquired in the modern age, analysis of Pyramids
become possible and so as the creation of theories about the possible way of
constructing and moving the materials used. Importance of the centre of gravity
to the stability of the Pyramids of Egypt was also known. What seemed to be
work of an alien can now be explained with the help of statics. Although
information how great pyramid architects thought of building a structure like
that is not that great or not yet known, one thing is for sure. That the
pyramids of Egypt were built to last.