Archimedes' principle (Class XI Physics Investigatory Project)

Session Year: 2017-18
LAXMI INTERNTIONAL SCHOOL
Archimedes’ Principle
Physics Investigatory Project
Project Guide:
Mr. K.V. Shetty
Physics Professor
Submitted by:
Viraj Rajendra Sanap
Raunaq Singh Kalsi

• • •
Contents  1
Contents
Contents............................................................................................1
CERTIFICATE..................................................................................2
Acknowledgements.........................................................................3
Introduction......................................................................................4
Density ..............................................................................................5
Buoyant Force ..................................................................................7
Archimedes’ Principle.....................................................................9
Explanation.....................................................................................10
Formula...........................................................................................13
Problems .........................................................................................16
Principle of Floatation...................................................................18
Fluid Mechanics.............................................................................20
Uses of Archimedes’ Principle.....................................................22
Future Scope...................................................................................25
Conclusion......................................................................................27
Bibliography...................................................................................28

• • •
CERTIFICATE  2
Physics Teacher
(Prof. K. V. Shetty)
Principal
(Prof. Shaji Matthews)
CERTIFICATE
This is to certify that the Project Report entitled
Has been successfully completed by the following students
Mast. Viraj Rajendra Sanap
Mast. Raunaq Singh Kalsi
In partial fulfilment of physics practical examination
conducted by the Central Board of Secondary Education
(CBSE)

• • •
Acknowledgements  3
Acknowledgements
We together provided the primary incentive in completing
this project titled “Archimedes’ Principle”. Our ideas, difficulties, and
our hard work shaped this project completed.
However, several personalities have directly or indirectly
contributed to the completion of the project and also preparing the
report.
It gives us immense pleasure in expressing our deep sense of
gratitude to Mr. K.V. Shetty (Physics Professor), and our Physics Lab
Assistant. The project would not have been completed without their
able guidance.
Last but not the least we would like to thank the L.I.S. Senior Sec.
Library where we were able to collect the reference material with
respect to the project topic.
We once again thank all our superiors, colleagues, parents,
friends, and all those who were directly or indirectly involved in the
completion of the project.
Raunaq Singh Kalsi Viraj Rajendra Sanap
XI-A (Science) XI-A (Science)
Roll No. 23 Roll No. 40

• • •
Introduction  4
Introduction
rchimedes' principle states that the upward buoyant
force that is exerted on a body immersed in a fluid, whether
fully or partially submerged, is equal to the weight of the
fluid that the body displaces and acts in the upward direction at the
center of mass of the displaced fluid.
Archimedes' principle is a law of physics fundamental to fluid
mechanics.
In this project, we will be talking about:
• Density
• Buoyant Force
• Archimedes’ Principle
• Explanation
• Formula
• Problems
• Principle of Floatation
• Fluid Mechanics
• Uses of Archimedes’ Principle
A

• • •
Density  5
Density
he 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:
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
T

• • •
Density  6
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
centimetre. For example, the density of water is 1 gram per cubic
centimetre, and Earth’s density is 5.51 grams per cubic centimetre.
Density can also be expressed as kilograms per cubic metre (in MKS or
SI units). For example, the density of air is 1.2 kilograms per cubic
metre. 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 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.

• • •
Buoyant Force  7
Buoyant Force
n 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
I

• • •
Buoyant Force  8
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 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.

• • •
Archimedes’ Principle  9
Archimedes’
Principle
rchimedes’ principle, physical law of
buoyancy, discovered by the ancient
Greek mathematician and
inventor Archimedes, stating that anybody
completely or partially submerged in
a fluid (gas or liquid) at rest is acted upon by an
upward, or buoyant, force the magnitude of
which is equal to the weight of the fluid
displaced by the body. The volume of displaced
fluid is equivalent to the volume of an object
fully immersed in a fluid or to that fraction of
the volume below the surface for an object
partially submerged in a liquid. The weight of
the displaced portion of the fluid is equivalent
to the magnitude of the buoyant force. The
buoyant force on a body floating in a liquid or
gas is also equivalent in magnitude to the
weight of the floating object and is opposite in
direction; the object neither rises nor sinks.
A
The Story of
Archimedes
…
There is a famous
story that a crown
was once made for
King Hiero and he
wanted to know if
there was a way to
know if it was of
pure gold or silver
had been mixed in
it. He approached
his cousin
Archimedes who
got a brainwave
when he was in a
bath and ran on the
streets shouting
Eureka! (I’ve found
it!).

• • •
Explanation  10
Explanation
ractically, Archimedes' principle allows the buoyancy of an
object partially or fully immersed in a liquid to be calculated.
The downward force on the object is simply its weight. The
upward, or buoyant, force on the object is that stated by Archimedes'
principle, above. Thus, the net upward force on the object is the
difference between the buoyant force and its weight. If this net force is
positive, the object rises; if negative, the object sinks; and if zero, the
object is neutrally buoyant - that is, it remains in place without either
rising or sinking. In simple words, Archimedes' principle states that,
when a body is partially or completely immersed in a fluid, it
experiences an apparent loss in weight that is equal to the weight of
the fluid displaced by the immersed part of the body.
P

• • •
Explanation  11
If the weight of an object is less than that of the displaced fluid,
the object rises, as in the case of a block of wood that is released
beneath the surface of water or a helium-filled balloon that is let loose
in air. An object heavier than the amount of the fluid it displaces,
though it sinks when released, has an apparent weight loss equal to the
weight of the fluid displaced. In fact, in some accurate weighings, a
correction must be made in order to compensate for the buoyancy
effect of the surrounding air.
The buoyant force, which always opposes gravity, is nevertheless
caused by gravity. Fluid pressure increases with depth because of the
(gravitational) weight of the fluid above. This increasing pressure
applies a force on a submerged object that increases with depth. The
result is buoyancy.
Let's use a battleship
as an example. A
battleship is made of
steel. Right about now,
you may be saying, 'But
steel doesn't float!' So
how is it a battleship can
float?
Look at the image of the battleship. Now imagine drawing a line
where the water comes up on the hull of the ship. Then, fill the ship's
hull with water up to that line. How much do you think the water
would weigh? If you said 'a lot,' you're right. It would actually weigh
as much as the entire ship!

• • •
Explanation  12
The weight of the water to fill up the hull of the ship weighs the
same as the ship, so the water applies a buoyant force up on the ship
with this much force. Therefore, the ship made of steel floats!
Let's look at another example. If you put
an ice cube in a glass of water, the cube floats
because ice is less dense than water. So, the ice
underwater displaces that volume of water.
For example, a ship that is launched sinks into the ocean until the
weight of the water it displaces is just equal to its own weight. As the
ship is loaded, it sinks deeper, displacing more water, and so the
magnitude of the buoyant force continuously matches the weight of
the ship and its cargo.

• • •
Formula  13
Formula
onsider a cube immersed in a fluid, with its sides parallel to
the direction of gravity. 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 pressure difference between the
bottom and the top face is directly proportional to the height
(difference in depth). Multiplying the pressure difference by the area
of a face gives the net force on the cube – the buoyancy, or the weight
of the fluid displaced. By extending this reasoning to irregular shapes,
we can see that, whatever the shape of the submerged body, the
buoyant force is equal to the weight of the fluid displaced.
Apparent loss in weight of water = weight of object in air – weight of
object in water
The weight of the displaced fluid is directly proportional to the
volume of the displaced fluid (if the surrounding fluid is of uniform
density). The weight of the object in the fluid is reduced, because of the
force acting on it, which is called upthrust. 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,
C

• • •
Formula  14
g. Thus, among completely submerged objects with equal masses,
objects with greater volume have greater buoyancy.
Suppose a rock's weight is measured as 10 newtons when
suspended by a string in a vacuum with gravity acting on it. Suppose
that, when the rock is lowered into water, it displaces water of weight
3 newtons. The force it then exerts on the string from which it hangs
would be 10 newtons minus the 3 newtons of buoyant force: 10 − 3 = 7
newtons. Buoyancy reduces the apparent weight of objects that have
sunk completely to the sea floor. It is generally easier to lift an object
up through the water than it is to pull it out of the water.
For a fully submerged object, Archimedes' principle can be
reformulated as follows:
Apparent immersed weight = weight of object – weight of displaced fluid
then inserted into the quotient of weights, which has been
expanded by the mutual volume
Density of object / Density of fluid = Weight / Weight of displaced fluid

• • •
Formula  15
yields the formula below. The density of the immersed object relative
to the density of the fluid can easily be calculated without measuring
any volume is
Density of object / Density of fluid = Weight / Weight – Apparent
immersed weight
(This formula is used for example in describing the measuring
principle of a dasymeter and of hydrostatic weighing.)

• • •
Problems  16
Problems
Problem 1: A ball of mass 2 kg that has a diameter of 50 cm falls in the
pool. Compute its buoyant force and volume of water displaced.
Answer:
Known:
Mass of water, m = 2 kg,
Diameter of ball, d = 0.5 m r = 0.25 m
The force is given by F = mg. Hence buoyant force is
F = 2 kg × 9.8 m/s2 = 19.6 N
The Archimedes formula is given by F = ρρg Vdisp

• • •
Problems  17
Hence volume of given body=Volume of displaced liquid
Problem 2: Calculate the buoyant force acting on it, if a stone of mass
250 g is thrown in water?
Answer:
Known: m (Mass of stone) = 0.25 kg,
The buoyant force is given by
F = mg
= 0.25 ×× 9.8
= 2.45 N.
Thus, 2.45 N of upward force is being applied on the stone.

• • •
Principle of Floatation  18
Principle of Floatation
hen the buoyant force equals 1 ton, it will sink no farther.
When any boat displaces a weight of water equal to its
own weight, it floats. This is often called the "principle of
flotation". A floating object displaces a weight of fluid equal to its own
weight.
In other words, for an object floating on a liquid surface (like a
boat) or floating submerged in a fluid (like a submarine in water
or dirigible in air) the weight of the displaced liquid equals the weight
of the object. Thus, only in the special case of floating does the buoyant
force acting on an object equal the objects weight. Consider a 1-ton
block of solid iron. As iron is nearly eight times as dense as water, it
displaces only 1/8 ton of water when submerged, which is not enough
to keep it afloat. Suppose the same iron block is reshaped into a bowl.
It still weighs 1 ton, but when it is put in water, it displaces a greater
volume of water than when it was a block. The deeper the iron bowl is
immersed, the more water it displaces, and the greater the buoyant
force acting on it. When the buoyant force equals 1 ton, it will sink no
farther.
W

• • •
Principle of Floatation  19
When any boat displaces a weight of water equal to its own
weight, it floats. This is often called the "principle of flotation": A
floating object displaces a weight of fluid equal to its own weight.
Every ship, submarine, and dirigible must be designed to displace a
weight of fluid at least equal to its own weight. A 10,000-ton ship's hull
must be built wide enough, long enough and deep enough to displace
10,000 tons of water and still have some hull above the water to
prevent it from sinking. It needs extra hull to fight waves that would
otherwise fill it and, by increasing its mass, cause it to submerge. The
same is true for vessels in air: a dirigible that weighs 100 tons needs to
displace 100 tons of air. If it displaces more, it rises; if it displaces less,
it falls. If the dirigible displaces exactly its weight, it hovers at a
constant altitude.
It is important to realize that, while they are related to it, the
principle of flotation and the concept that a submerged object displaces
a volume of fluid equal to its own volume are not Archimedes'
principle. Archimedes' principle, as stated above, equates the buoyant
force to the weight of the fluid displaced.

• • •
Fluid Mechanics  20
Fluid Mechanics
luid 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 models 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.
F

• • •
Fluid Mechanics  21
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 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.
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

• • •
Uses of Archimedes’ Principle  22
Uses of Archimedes’
Principle
1. Submarine:
A submarine has a large ballast
tank, which is used to control
its position and depth from the
surface of the sea. A submarine
submerges by letting water into
the ballast tank so that its
weight becomes greater than the buoyant force. Conversely, it floats
by reducing water in the ballast tank. Thus its weight is less than the
buoyant force.
2. Hot-air balloon:
Th e 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.

• • •
3. Hydrometer:
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 cm-3.
4. Ship:
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.

• • •
5. Fishes:
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.
6. 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.

• • •
Future Scope  25
Future Scope
oday CFD simulations are becoming more and more
computationally demanding. In many areas of science and
industry there is a need to guarantee short turnaround times
and fast time-to-market. Such goals can be fulfilled only with huge
investments in hardware and software licenses.
Graphics Processing Units provide completely new possibilities
for significant cost savings because simulation time can be reduced on
hardware that is often less expensive than server-class CPUs. Almost
every PC contains a graphics card that supports either CUDA or
OpenCL.
The computations may be done on the CPUs and GPUs
concurrently. If there are multiple GPUs in the system, independent
computing tasks can be solved simultaneously. When cases are solved
on GPU the CPU resources are free and can be used for other tasks
such as pre- and post-processing. Moreover, the power efficiency per
simulation, is comparable for a dual-socket multicore CPU and a GPU.
Several ongoing projects on Navier-Stokes models and Lattice
Boltzman methods have shown very large speedups using CUDA-
enabled GPUs.
T

• • •
Future Scope  26
Just before the first computer ‘Eniac’ (1942), computational fluid
dynamics (CFD) studies were carried out by human computers.
Hardware was the hurdle that prevented both engineer and scientist
from developing numerical analysis.
Though ascent in the field of hardware is enormous, having
adequate facilities in accordance with hardware is still an issue. As
developments in hardware are getting faster, the curiosity of human
beings is getting faster two times higher. Despite hardware has come a
long way, it still is needed to dig future up. According to the survey
carried out by Sandler Research, global cloud CFD market to grow at a
CAGR of 10.73% during the period 2016-2020. Thus, cloud computing
is the candidate of the future trend of computational fluid dynamics, in
terms of hardware.
Appeared as an editorial in Applied Mechanics
Reviews, vol. 49, no. 3, pp. III–IV, 1996.
…
We consider here the future of some of the physical sciences which
are developed enough for problems to be well-posed mathematically
even though, due to their complexity, analytical solutions are not
possible. Such problems are typically approached through a
combination of physical and numerical experiments, the latter
increasing in scope and range as more computing power becomes
available.

• • •
Conclusion  27
Conclusion
s 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.
We think the tests we did went smoothly and we had no
problems, 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 its instances in day to day
life as well.
A

• • •
Bibliography  28
Bibliography
1. References from the internet
• www.britannica.com
• www.prezi.com
• www.wikipedia.com
• www.slideshare.net
• www.khanacademy.org
• www.byjus.com
• www.study.com
• www.mycbseguide.com
• www.sciencefare.com
• www.studymode.com
• www.sciencebuddies.org
• www.designmaths.weebly.com
• www.reference.com
• www.meritnation.com
2. References from textbooks
• NCERT Physics Textbook Part -II
• New Simplified Physics Vol. II by S. L. Arora
• Foundations of Physics by H. C. Verma
• Wiley's Halliday / Resnick / Walker Physics Vol 1
3. Help from teachers and experiments done in physics laboratory.

Archimedes' principle (Class XI Physics Investigatory Project)

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