UNIT 4TH
Main Topics
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Buoyancy
Density
Learning Goals
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Predict whether an object will sink or float when placed in a liquid, given densities of the object and liquid
Apply the definition of density to both liquids and solids
Relate the buoyant force on an object to the weight of liquid it displaces
Describe how the buoyant force is related to an object's relative density to the fluid
Predict the weight of a completely or partially submerged object of known mass and volume
Describe the forces that act on a completely or partially submerged object
Explain how an object that is more dense than water can be kept afloat by placing it on an object that is less
dense than water
What is a Fluid?
It is well known that matter is divided into solids and fluids. Fluids can be further divided into Liquids and Gases. It is
taught in schools, rightly so, that solids have a definite shape and a definite size, while the liquids have a definite size, but
no definite shape. They assume the shape of the container they are poured into. Gases on the other hand have neither a
shape nor a size. They can fill any container fully and assume its shape. We need a more precise definition. This comes
when we consider the response of a solid or a fluid to a shear force. A solid resists a shear force while a fluid deforms
continuously under the action of a shear force.
Archimedes' Principle
Shear: resistance of fluid being deformed by stress = viscosity
(viscosidad)
 If the weight of the water displaced is less than the weight
of the object, the object will sink
 Otherwise the object will float, with the weight of the water
displaced equal to the weight of the object.
Archimedes and the Law of the Lever
"Give me a place to stand on, and I will move the earth."
quoted by Pappus of Alexandria in Synagoge, Book VIII, c. AD 340
The 140 lb boy 2 feet from the fulcrum (center of
gravity) balances his 70 pound sister 4 feet from the
fulcrum: 2 x 140 = 4 x 70
Archimedes & his theory of displacement:
http://hilaroad.com/video/
http://www.youtube.com/watch?v=ZUYkBeAW5hc
Archimedes' Principle video: http://www.youtube.com/watch?v=eQsmq3Hu9HA
http://www.onr.navy.mil/focus/blowballast/sub/work2.htm:
Submarines: How They Work - Archimedes' Principle
Archimedes' principle is the law of buoyancy. It states that "any body partially or completely submerged in a fluid is
buoyed up by a force equal to the weight of the fluid displaced by the body." The weight of an object acts downward, and
the buoyant force provided by the displaced fluid acts upward. If these two forces are equal, the object floats. Density is
defined as weight per volume. If the density of an object exceeds the density of water, the object will sink.
http://phet.colorado.edu/en/simulation/buoyancy
Objects of equal volume experience equal buoyant forces
Equal Volumes Feel Equal Buoyant Forces
Suppose you had equal sized balls of cork, aluminum and lead, with respective specific gravities of 0.2, 2.7, and 11.3. If
the volume of each is 10 cubic centimeters then their masses are 2, 27, and 113 gm.
Each would displace 10 grams of water,
yielding apparent masses of -8 (the cork
would accelerate upward), 17 and 103
grams respectively.
The behavior of the three balls would
certainly be different upon release from
rest in the water. The cork would bob (go
up and down) up, the aluminum would
sink, and the lead would sink more rapidly.
But the buoyant force on each is the same because of identical pressure environments and equal water displacement.
The difference in behavior comes from the comparison of that buoyant force with the weight of the object.
Hmm! The crown seems lighter under water!
The buoyant force on a submerged object is equal to the weight of the liquid displaced by the object. For water, with a
density of one gram per cubic centimeter, this provides a convenient way to determine the volume of an irregularly
shaped object and then to determine its density.
Archimedes' Principle
The buoyant force on a submerged object is equal to the weight of the fluid displaced. This principle is useful for
determining the volume and therefore the density of an irregularly shaped object by measuring its mass in air and its
effective mass when submerged in water (density = 1 gram per cubic centimeter). This effective mass under water will be
its actual mass minus the mass of the fluid displaced. The difference between the real and effective mass therefore gives
the mass of water displaced and allows the calculation of the volume of the irregularly shaped object (like the king's
crown in the Archimedes story). The mass divided by the volume thus determined gives a measure of the average density
of the object. Archimedes found that the density of the king's supposedly gold crown was actually much less than the
density of gold -- implying that it was either hollow or filled with a less dense substance.
Examination of the nature of buoyancy shows that the buoyant force on a volume of water and a submerged object of the
same volume is the same. Since it exactly supports the volume of water, it follows that the buoyant force on any
submerged object is equal to the weight of the water displaced. This is the essence of Archimedes principle.
Application to determining density
Static Fluid Pressure
The pressure exerted by a static fluid depends only upon the depth of the fluid, the density of the fluid, and the
acceleration of gravity.
The pressure in a static fluid arises from the weight of the fluid and is given by the expression
Pstatic fluid = ρgh where
ρ = m/V = fluid density
g = acceleration of gravity
h = depth of fluid
The pressure from the weight of a column of liquid of area A and height h is
The most remarkable thing about this expression is what it does not include. The fluid pressure at a given depth does not
depend upon the total mass or total volume of the liquid. The above pressure expression is easy to see for the straight,
unobstructed column, but not obvious for the cases of different geometry which are shown.
Because of the ease of visualizing a column height of a known liquid, it has become common practice to state all kinds of
pressures in column height units, like mmHg or cm H 2O, etc. Pressures are often measured by manometers in terms of a
liquid column height.
Pascal's Principle
http://www.tutorvista.com
Pressure is transmitted undiminished in an enclosed static fluid.
Any externally applied pressure is transmitted to all parts of the enclosed fluid, making possible a large multiplication of
force (hydraulic press principle). The pressure at the bottom of the jug is equal to the externally applied pressure on the
top of the fluid plus the static fluid pressure from the weight of the liquid.
A multiplication of force can be achieved by the application of fluid pressure according to Pascal's principle, which for the
two pistons implies P1 = P2
This allows the lifting of a heavy load with a small force, as in an auto hydraulic lift, but of course there can be no
multiplication of work, so in an ideal case with no frictional loss: Winput = Woutput
Hydraulic Press