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Department of Physical and Mathematical Sciences
Olawumi R. Kaka
3.0
BOUYANCY, ARCHIMEDES' PRINCIPLES , SURFACE TENSION AND
HYDRODYNAMICS
3.1
BOUYANCY AND ARCHIMEDES' PRINCIPLES
When an object is submerged in a fluid, it appears to weigh less than they do when outside
the fluid. For example, a large rock at the bottom of a stream would be easily lifted compare
to lifting it from the ground. As the rock breaks through the surface of the water, it becomes
heavier. This phenomenon is as a result of upward force called the buoyant force (upthrust)
acting on the rock plus the downward gravitational force.
The buoyant force occurs because the pressure in fluid increases with depth. Thus, the
upward pressure at the bottom surface of a submerged object is greater than the downward
pressure at its top surface. Hence, a buoyant force is the upward force exerted by water on
any immersed object. To see the effect of this, consider a cylinder of height h whose top and
bottom ends have an area A and completely submerged in a fluid of density
as shown in
figure .
Therefore,
The pressure exerted by the fluid at the top surface of the cylinder is
The downward force due to this pressure on top of the cylinder is
The upward force on the bottom of the cylinder is
The net force due to the fluid pressure, which is the buoyant force,
, acts upward with a
magnitude:
1
*2
Where:
is the volume of the cylinder
Since
is the density of the fluid, then
is the weight of fluid which takes up a
volume equal to the volume of the cylinder. Thus, buoyant force on the cylinder equals the
weight of fluid displaced by the cylinder. The mode in which buoyant forces act is
summarised is known as Archimedes’s principle; which states the buoyant force on a body
immersed in a fluid is equal to the weight of the fluid displaced by the object.
3.1.1 Forces Acting on a Totally Submerged Object and a Floating Object (Partly
Submerged)
3.1.1.1
Totally Submerged Object
When an object is totally submerged in a fluid of density
, the magnitude of the upward
buoyant force is ;
Where
Vo is the volume of the object
If the object has a mass M and density
, its weight is
The net force acting on the object is
=
3.1.1.2
)
Floating Object
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Consider an object of volume Vo in a static equilibrium floating on a fluid (i.e an object that is
partially submerged). In this case, the upward buoyant force is balanced by the downward
gravitational force acting on the object. If Vf is the volume of the fluid displaced by the object
(the part of the object beneath the fluid level).
.
The buoyant force has a magnitude;
Because the weight of the object is:
Since
Therefore;
;
Example on Buoyancy:
A 15.0kg solid gold statue is being raised from a sunken ship. (i) What is the buoyant force
when the statue is immersed in a fluid (ii) the force when it is out of the water? Hence, find
the tension in the hoisting cable when the statue is (iii) at rest and completely immersed (iv)
at rest and out of water. Density of gold is
Solution
i.
The buoyant force when the statue is immersed in a fluid, we use
But
Therefore,
ii. The force when it is out of the water
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iii. The tension when the object is completely immersed
Since the statue is at rest the net external force acting on it is zero
Therefore,
iv. The tension when the statue is out of water = Statue weight
3.2
SURFACE TENSION
You would have noticed this interesting behaviour that can take place at the surfaces of
liquids. According to Archimedes' principle, a steel needle should sink in water if its density
is greater than density of water. A needle placed carefully on water, however, can be
supported by the surface tension because the liquid responds in a way similar to a stretched
membrane.
Also, water is often used for cleaning, but the surface tension makes it hard for water to
penetrate into small crevices or openings, such as are found in clothes. Soap is added to water
to reduce the surface tension, so clothes (or whatever else) get much cleaner.
The cohesive forces between liquid molecules are responsible for the phenomenon known as
surface tension. The cohesive forces between molecules down into a liquid are shared with all
neighbouring atoms but those on the surface have no neighbouring atoms above, and thus
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exhibit stronger attractive forces upon their nearest neighbours on the surface. This
enhancement of the intermolecular attractive forces at the surface is called surface tension.
3.3
HYDRODYNAMICS:
Description of the properties of a fluid as it moves at each point as a function of time is
termed hydrodynamics.
3.3.1 Characteristics of a Fluid
Considering the motion of real fluids, it is very complex and not fully understood. When fluid
is in motion, its flow can be characterized as being one of 2 main types. Steady or laminar
flow. The flow is steady if the overall flow pattern does not change with time. In this type of
flow every element passing through a given point follows the same flow line i.e. the velocity
of the fluid at any point remains constant in time. In a laminar flow, each particle of the fluid
follows a smooth path, such that the paths of different particles never cross each other. Above
a certain critical speed, fluid flow becomes turbulent; turbulent flow is irregular flow
characterized by small whirlpool-like regions.
The term viscosity is commonly used in the description of fluid flow to characterize the
degree of internal friction in the fluid. This internal friction, or viscous force, is associated
with the resistance that two adjacent layers of fluid have to moving relative to each other.
Figure 1a: Hot gases from a cigarette made visible by smoke particles. The
smoke first moves in laminar flow at the bottom and then in turbulent flow above.
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Figure 1b: Laminar flow around an automobile in a test wind tunnel.
Other characteristics involve the streamline the path taken by a fluid particle under steady
flow. The velocity of the particle is always tangent to the streamline, as shown in Figure 2. A
set of streamlines like the ones shown in Figure 2 form a flow tube. Note that fluid particles
cannot flow into or out of the sides of this tube; if they could, then the streamlines would
cross each other.
along its
Figure 2: A particle in laminar flow follows a streamline, and at each point
path the particle’s velocity is tangent to the streamline
3.3.2 Properties of an Ideal Fluid
Looking at the above, some simplifying assumptions could be said in the model of an ideal
fluid as follows:
1. The fluid is non-viscous. In a non-viscous fluid, internal friction is neglected. An
object moving through the fluid experiences no viscous force.
2. The flow is steady. In steady (laminar) flow, the velocity of the fluid at each point
remains constant.
3. The fluid is incompressible. The density of an incompressible fluid is constant.
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4. The flow is irrotational. In an irrotational flow, the fluid has no angular momentum
about any point. If a small paddle wheel placed anywhere in the fluid does not rotate
about the wheel’s center of mass, then the flow is irrotational
3.3.3 Equation of Continuity
We now know that the moving fluid doesn’t change as it flows, this leads to an important
quantitative relationship called continuity equation.
Consider an ideal fluid flowing through a pipe of non-uniform size, as illustrated in Figure 3.
The particles in the fluid move along streamlines in steady flow.
In a time t, the fluid at the bottom end of the pipe moves a distance,
.
If A1 is the cross-sectional area in this region, then the mass of fluid contained in the left
shaded region in Figure 3 is
where
is the (non-changing) density of the ideal fluid.
Figure 3: A fluid moving with steady flow through a
pipe of varying cross-sectional area
Similarly, the fluid that moves through the upper end of the pipe in the time t has a mass
However, because mass is conserved and because the flow is steady, the mass that crosses A1
in a time t must equal the mass that crosses A2 in the time t. That is,
Therefore,
And,
*3
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The product Av, which has the dimensions of volume per unit time, is called either the volume
flux or the flow rate.
Equation *3 represents equation of continuity for an incompressible fluid and it states that the
product of the area and the fluid speed at all points along the pipe is a constant
3.3.4 BERNOULLI’S EQUATION
The relationship between fluid speed, pressure and elevation was first derived in 1738 by
Swiss physicist, Daniel Bernoulli. To derive Bernoulli’s equation, consider the flow of an
ideal fluid through a non-uniform pipe in a time t. Figure 4
Figure 4: A fluid in laminar flow through a constricted pipe. The volume of the shaded section
on the left is equal to the volume of the shaded section on the right.
The force exerted by the fluid in section 1 has a magnitude:
1
The workdone by this force in a time t is
=
2
Where V is the volume of section 1
Similarly, the workdone by the fluid in section 2 is the same time t is;
=
3
The volume that passes through section 1 in a time t equal the volume that passes through
section in the same time.W2 is negative because the fluid force opposes the displacement.
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Thus, the net workdone by these forces in the time t is
4
Part of this work goes into changing the kinetic energy of the fluid and part goes into
changing the gravitational potential energy. If m is the mass that enters one and leaves the
other end in a time t, then the change in the kinetic energy of mass m is
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The change in the gravitational P.E is
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From equation 4, Total Energy,
is
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If we divide each term by V and recall that
this expression reduces to
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Rearranging,
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This is Bernoulli’s equation as applied to an ideal fluid. It is often expressed as
*4
This expression specifies that, in a laminar flow, the sum of the pressure (P),
kinetic energy per unit volume and gravitational potential energy per unit
volume has the same value at all points along a streamline.
When the fluid is at rest
equation *4 becomes
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