Verification of Archimedes Principle.docx

VERIFICATION OF
ARCHIMEDES PRINCIPLE
Class-XI Project
NOVEMBER 26, 2022
ADITYA AGARWAL
Phoenix Greens School of Learning

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Acknowledgements
It gives us immense pleasure in expressing our deep sense of gratitude to Mr. Srinivas
(Physics teacher), and our Physics Lab Assistant. The project would not have been
completed without their able guidance.
I would also like to thank our Library in-charge where we were able to collect the
reference material with respect to the project topic.
I once again thank all our superiors, colleagues, parents, friends, and all those who
were directly or indirectly involved in the completion of the project.

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Contents
Acknowledgements .................................................................................................... 1
Archimedes ................................................................................................................ 3
Density ....................................................................................................................... 4
Buoyant Force............................................................................................................ 6
Fluid Mechanics ......................................................................................................... 8
The Project............................................................................................................... 10
Archimedes Principle................................................................................................ 10
Eureka .................................................................................................................. 10
Equipment we used for the experiment.................................................................... 12
The Experiment........................................................................................................ 12
Explanation .............................................................................................................. 13
Law of Floating......................................................................................................... 14
Application of Archimedes Principle ......................................................................... 15
Conclusion ............................................................................................................... 17
Trivia ........................................................................................................................ 18
Bibliography ............................................................................................................. 19

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Archimedes
Archimedes of Syracuse (c. 287 – c. 212 BC) was a Greek mathematician, physicist,
engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily.
Although few details of his life are known, he is regarded as one of the leading
scientists in classical antiquity. Considered to be the greatest mathematician of ancient
history, and one of the greatest of all time, Archimedes anticipated modern calculus
and analysis by applying the concept of the infinitely small and the method of
exhaustion to derive and rigorously prove a range of geometrical theorems, including:
the area of a circle; the surface area and volume of a sphere; area of an ellipse; the
area under a parabola; the volume of a segment of a paraboloid of revolution; the
volume of a segment of a hyperboloid of revolution; and the area of a spiral.
Archimedes' other mathematical achievements
include deriving an approximation of pi; defining
and investigating the spiral that now bears his
name; and devising a system using
exponentiation for expressing very large
numbers. He was also one of the first to apply
mathematics to physical phenomena, founding
hydrostatics and statics. Archimedes'
achievements in this area include a proof of the
principle of the lever, the widespread use of the
concept of center of gravity, and the enunciation
of the law of buoyancy. He is also credited with
designing innovative machines, such as his screw pump, compound pulleys, and
defensive war machines to protect his native Syracuse from invasion.
Archimedes died during the siege of Syracuse, when he was killed by a Roman soldier
despite orders that he should not be harmed. Cicero describes visiting Archimedes'
tomb, which was surmounted by a sphere and a cylinder, which Archimedes had
requested be placed on his tomb to represent his mathematical discoveries.

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Density
The density, or more precisely, the volumetric
mass density, of a substance is its mass per
unit volume. The symbol most often used for
density is ρ (the lower-case Greek letter rho),
although the Latin letter D can also be used.
Mathematically, density is defined as mass
divided by volume:
ρ = m/V
where ρ is the density, m is the mass, and V is the volume. In some cases (for instance,
in the United States oil and gas industry), density is loosely defined as its weight per
unit volume, although this is scientifically inaccurate – this quantity is more specifically
called specific weight. The density of a material varies with temperature and pressure.
This variation is typically small for solids and liquids but much greater for gases.
Increasing the pressure on an object decreases the volume of the object and thus
increases its density. Increasing the temperature of a substance (with a few
exceptions) decreases its density by increasing its volume. In most materials, heating
the bottom of a fluid results in convection of the heat from the bottom to the top, due
to the decrease in the density of the heated fluid. This causes it to rise relative to
denser unheated material.
The reciprocal of the density of a substance is occasionally called its specific volume,
a term sometimes used in thermodynamics. Density is an intensive property in that
increasing the amount of a substance does not increase its density; rather it increases
its mass.
Density is commonly expressed in units of grams per cubic centimeter. For example,
the density of water is 1 gram per cubic centimeter, and Earth’s density is 5.51 grams
per cubic centimeter. Density can also be expressed as kilograms per cubic meter (in
MKS or SI units). For example, the density of air is 1.2 kilograms per cubic meter. The
densities of common solids, liquids, and gases are listed in textbooks and handbooks.
Density offers a convenient means of obtaining the mass of a body from its volume or
vice versa; the mass is equal to the volume multiplied by the density (M = Vd), while

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the volume is equal to the mass divided by the density (V = M/d). The weight of a body,
which is usually of more practical interest than its mass, can be obtained by multiplying
the mass by the acceleration of gravity.

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Buoyant Force
In science, buoyancy or upthrust, is an upward
force exerted by a fluid that opposes the weight
of an immersed object. In a column of fluid,
pressure increases with depth as a result of the
weight of the overlying fluid. Thus, the pressure
at the bottom of a column of fluid is greater than
at the top of the column. Similarly, the pressure
at the bottom of an object submerged in a fluid
is greater than at the top of the object. This
pressure difference results in a net upwards force on the object. The magnitude of that
force exerted is proportional to that pressure difference, and (as explained by
Archimedes' principle) is equivalent to the weight of the fluid that would otherwise
occupy the volume of the object, i.e. the displaced fluid.
For this reason, an object whose density is greater than that of the fluid in which it is
submerged tends to sink. If the object is either less dense than the liquid or is shaped
appropriately (as in a boat), the force can keep the object afloat. This can occur only
in a non-inertial reference frame, which either has a gravitational field or is accelerating
due to a force other than gravity defining a "downward" direction. In a situation of fluid
statics, the net upward buoyancy force is equal to the magnitude of the weight of fluid
displaced by the body. The centre of buoyancy of an object is the centroid of the
displaced volume of fluid.
A simplified explanation for the integration of the pressure over the contact area may
be stated as follows: Consider a cube immersed in a fluid with the upper surface
horizontal. The sides are identical in area, and have the same depth distribution,
therefore they also have the same pressure distribution, and consequently the same
total force resulting from hydrostatic pressure, exerted perpendicular to the plane of
the surface of each side. There are two pairs of opposing sides, therefore the resultant
horizontal forces balance in both orthogonal directions, and the resultant force is zero.
The upward force on the cube is the pressure on the bottom surface integrated over
its area. The surface is at constant depth, so the pressure is constant. Therefore, the

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integral of the pressure over the area of the horizontal bottom surface of the cube is
the hydrostatic pressure at that depth multiplied by the area of the bottom surface.

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Fluid Mechanics
Fluid mechanics is a branch of physics
concerned with the mechanics of fluids
(liquids, gases, and plasmas) and the
forces on them. Fluid mechanics has a
wide range of applications, including
mechanical engineering, civil
engineering, chemical engineering,
biomedical engineering, geophysics,
astrophysics, and biology. Fluid
mechanics can be divided into fluid
statics, the study of fluids at rest; and
fluid dynamics, the study of the effect
of forces on fluid motion. It is a branch
of continuum mechanics, a subject
which model matter without using the information that it is made out of atoms; that is,
it models matter from a macroscopic viewpoint rather than from microscopic. Fluid
mechanics, especially fluid dynamics, is an active field of research with many
problems that are partly or wholly unsolved. Fluid mechanics can be mathematically
complex, and can best be solved by numerical methods, typically using computers. A
modern discipline, called computational fluid dynamics (CFD), is devoted to this
approach to solving fluid mechanics problems. Particle image velocimetry, an
experimental method for visualizing and analyzing fluid flow, also takes advantage of
the highly visual nature of fluid flow.
The study of fluid mechanics goes back at least to the days of ancient Greece, when
Archimedes investigated fluid statics and buoyancy and formulated his famous law
known now as the Archimedes' principle, which was published in his work On Floating
Bodies – generally considered to be the first major work on fluid mechanics. Rapid
advancement in fluid mechanics began with Leonardo da Vinci (observations and
experiments), Evangelista Torricelli (invented the barometer), Isaac Newton
(investigated viscosity) and Blaise Pascal (researched hydrostatics, formulated
Fluid Mechanics …
• Statics
• Dynamics
• Archimedes' principle
• Bernoulli's principle
• Navier–Stokes equations
• Poiseuille equation
• Pascal's law
• Viscosity (Newtonian · non-Newtonian)
• Buoyancy
• Mixing
• Pressure
• Surface tension
• Capillary action
• Atmosphere Boyle's law
• Charles's law
• Gay-Lussac's law
• Combined gas law

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Pascal's law), and was continued by Daniel Bernoulli with the introduction of
mathematical fluid dynamics in Hydrodynamica (1739).
Inviscid flow was further analyzed by various mathematicians (Leonhard Euler, Jean
le Rond d'Alembert, Joseph Louis Lagrange, Pierre-Simon Laplace, Siméon Denis
Poisson) and viscous flow was explored by a multitude of engineers including Jean
Léonard Marie Poiseuille and Gotthilf Hagen.

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The Project
This is a project to Experimentally Verify Archimedes Principle.
Archimedes Principle
Any fluid applies equal pressure in every direction. This pressure is the result of the
weight of the fluid. When an object is partially or completely submerged in a fluid, it
exerts an upward force on the object. This upward force is called the buoyant force.
Due to the buoyant force, there is an apparent decrease in the weight of the object.
The decreased weight is equal to the weight of the fluid, displaced by the object. This
relationship was invented by Archimedes. From large ships to small boats, aircraft,
submarines all of these operate according to the principle of buoyancy.
Eureka
Archimedes’s principle is also known as the physical law of buoyancy. Eureka is a
word popularized by Archimedes. He exclaimed Eureka when he realized he had
invented the method of detecting if something is made of pure or impure gold. In the
widespread tale, Archimedes didn’t use his principle he only used displaced water to
measure the volume of the crown, an alternative approach is applied with the use of
this principle - A scale has to be balanced after placing a crown on one side and pure
gold on the other, submerge the scale in water, According to Archimedes principle, if
the crown’s density differs from pure gold’s then the scale will get out of balance
underwater.
The Principle says, an object when immersed in a fluid is buoyed up by a force equal
to the weight of the fluid displaced by the object. So as to calculate the amount of
displaced water, a special equation has been derived –
If the displaced fluid has a weight, w = mg,
To find out the mass in terms of density and volume, we have m = ρV.
So finally, we have Fb = ρVg =Fg (replacing m by PV)
• One of the important inventions in this principle is that, the buoyant force is
irrespective of the size and shape of the object and is totally based on the
amount of liquid that is displaced. In case, the object is heavier than water, it

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sinks to the bottom. For example, the density of a metal object is more than
water, hence it always sinks.
• Another important feature of Archimedes principle is that it is applicable to all
the objects with different densities. If densities of both fluid and the object are
the same, then the object will remain in half immersed condition and will neither
sink nor float. And when the density of the object is much lighter then it will float

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Equipment we used for the experiment
 Spring Balance
 Water
 Box
 Rocks
 Plastic
The Experiment
Take a stone of known volume with density greater than water and hang it with a spring
weighing balance. Record the mass m (=W1) and volume V. Now lower the system of
stone and weighing balance into water slowly such that the stone is completely
immersed in water (weighing machine is not immersed). Note the new reading of the
weighing machine m2 (=W2). It is observed that m2<m suggesting the existence of an
upward force.
Calculate Upthrust = Difference in weights, ΔW = (m−m2)g
Calculate weight of water displaced, Ww = ρwVg; It is observed that ΔW = Ww

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Explanation
Partially Submerged
When an object is partially submerged (floating), It is in equilibrium. That means that
when we put the plastic container with a low mass in the liquid, it is in equilibrium; as
a result, the rock Fg is equal to the buoyant force (Fb=mg). There is no tension force
acting on the plastic container. In fact, the density of the container is less than the
density of the water that’s why it didn’t sink and floated.
Volume of displaced water < volume of plastic container
So, the final formula is:
F = ρgV
ρgVwater = ρgVrock
Totally Submerged
When the object is totally submerged the foam is suspended by a string from the force
probe into the tall graduated tank, about half-full of water. Adjust the string so that you
can lower the foam totally into the water, and the water level remains within the
measurable range. Try not to let the foam sample touch the sides of the graduated
tank.The fluid will exert a normal force on each face, and therefore only the forces on
the top and bottom faces will contribute to buoyancy.
The weight of the object in the fluid is reduced, because of the force acting on it, which
is called up thrust. In simple terms, the principle states that the buoyant force on an
object is equal to the weight of the fluid displaced by the object, or the density of the
fluid multiplied by the submerged volume times the gravitational constant, g
FB = m’g = ρfluidVdisplg
2 cases for Archimedes project
Totally Submerged Partially Submerged

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Law of Floating
Whether an immersed object will float or sink, depends on the magnitudes of the actual
weight W₁ of the object and the buoyant force W₂ exerted by the fluid.
• W₁ > W₂: The resultant force on the object is downwards, causing it to sink.
When the density of the object is greater than that of the fluid, this condition
arises.
• W₁ = W₂: When the densities of the object and the fluid are equal, the actual
weight and the buoyant force become equal. The object can float at any depth
in a fully submerged state.
• W₁ < W₂: The net force acts in the upward direction leading to a partially
submerged condition of the object. The density of the object is less than the
fluid in such cases.

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Application of Archimedes Principle
Using Archimedes law, the volume or density of any rigid body can be computed. The
proportions of the constituent metals of an alloy can be easily calculated using this
principle.
 Submarines operate using the Archimedes theory. It has
a large ballast tank, which controls the depth of the
marine. By adjusting the quantity of water in the ballast
tank, the actual weight of the submarine is varied and
thus the desired depth can be achieved.
 A ship floats on the surface of the
sea because the volume of water
displaced by the ship is enough to
have a weight equal to the weight
of the ship. A ship is constructed in
a way so that the shape is hollow,
to make the overall density of the ship lesser than the sea water. Therefore, the
buoyant force acting on the ship is large enough to support its weight. The
density of sea water varies with location. The PLIMSOLL LINE marked on the
body of the ship acts as a guideline to ensure that the ship is loaded within the
safety limit. A ship submerge lower in fresh water as fresh water density is
lesser than sea water. Ships will float higher in cold water as cold water has a
relatively higher density than warm water.
 A hydrometer is an instrument to measure the relative density of liquids.
It consists of a tube with a bulb at one end. Lead shots are placed in the
bulb to weigh it down and enable the hydrometer to float vertically in the
liquid. In a liquid of lesser density, a greater volume of liquid must be
displaced for the buoyant force to equal to the weight of the hydrometer
so it sinks lower. Hydrometer floats higher in a liquid of higher density. Density
is measured in the unit of g cm3.

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• The atmosphere is filled with
air that exerts buoyant force on
any object. A hot air balloon
rises and floats due to the
buoyant force (when the
surrounding air is greater than
its weight). It descends when
the balloon's weight is higher than the buoyant force. It becomes stationary
when the weight equals the buoyant force. The weight of the Hot-air balloon
can be controlled by varying the quantity of hot air in the balloon.
• Certain group of fishes uses Archimedes’
principles to go up and down the water. To go
up to the surface, the fishes will fill its swim
bladder (air sacs) with gases. The gases
diffuse from its own body to the bladder and
thus making its body lighter. This enables the
fishes to go up. To go down, the fishes will empty their bladder, this increases
its density and therefore the fish will sink.
• FLIP – Floating instrument platform: This is a
research ship that does research on waves in
deep water. It can turn horizontally or vertically.
When water is pumped into stern tanks, the ship
will flip vertically. The principle that is used in FLIP
is almost similar with the submarines. Both ships
pump water in or out of tank to rise or sink.

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Conclusion
As the project comes to an end, we have realized that some of our views and concepts
were wrong about Archimedes’ principle and fluid mechanics.
Archimedes’ principle is indeed a very important concept in today’s date, and it also
has a lot of scope in the upcoming future.
The test we did went smoothly and we had no problem, except for the fact that
Archimedes’ principle was quite an interesting and engaging topic for us.
An interesting future study might involve fluid mechanics to help breathing underwater
for human beings as well.
This project was very much educational and enlightening for us. We could conclude
from this project that the Archimedes’ principle has a wide range of applications and
we see it’s instances in day to day life as well.

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Trivia
 Archimedes of Syracuse introduced the theory of buoyancy in his book On
Floating Bodies (written in the Greek language) around 250 BC. This theory is
considered to be the cornerstone in the study of hydrostatics.
 It is reported that Archimedes called out “Eureka”, meaning “I have found (it)”
when he finally comprehended how to detect if a crown is made of impure gold
using the theory of buoyancy.
 A floating body does not have any apparent weight.
 Surface tension or capillarity effect is not incorporated with the Archimedes
principle.
 A large lunar impact crater is named after Archimedes.
 A portrait of Archimedes is engraved on the prestigious “Fields Medal”.

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Bibliography
Internet References
• https://www.vedantu.com/physics/archimedes-principle
• https://en.wikipedia.org/wiki/Archimedes
• https://www.britannica.com/biography/Archimedes
• https://mathshistory.st-andrews.ac.uk/Biographies/Archimedes/
• https://www.seminarsonly.com/Engineering-Projects/Physics/archimedes-
principle.php
• https://www.youtube.com/watch?v=IfldVIUX4sI
• https://byjus.com/physics/archimedes-principle/
References from textbooks
• NCERT Physics Textbook Part -II
• Foundations of Physics by H. C. Verma
Help from teachers and experiments done in physics laboratory

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