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Kaysons Education

Fluid

Chapter

Fluid

1

Day – 1
1
...
We drink them, breathe them, in them
...
Airplanes fly through them; ships float
in them
...
We
usually think of a gas as easily compressed and a liquid as nearly incompressible, although there
are exceptional cases
...

Like other equilibrium situations, it is based on Newton's first and third laws
...
Fluid dynamics, the study of fluids in motion, is
much more complex; indeed, it is one of the most complex branches of mechanics
...
Even so, we will barely scratch the surface of
this broad and interesting topic
...
2 Density
An important property of any material is its density, defined as its mass per unit volume
...
We use the Greek
letter (rho) for density
...
That's because the ratio of mass to volume is the same for
both objects
...
The cgs unit, the gram
per cubic centimeter (1 g/ cm3), is also widely used:
1 g/cm3 = 1000 kg/m3
The densities of several common substances at ordinary temperatures are given
...
The densest material found on earth is the metal osmium
but its density pales by comparison to the densities of exotic astronomical objects such as
white dwarf stars and neutron stars
...
0°C, 1000
kg/m3; it is a pure number without units
...
7
...


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Kaysons Education

Fluid

Densities of Some Common Substances
MATERIAL

DENSITY (kg/m3)

MATERIAL

DENSITY (kg/m3)

Air (I atm, 20C)
Ethanol
Benzene
Ice
Water
Seawater
Blood
Glycerine
Concrete
Aluminum

1
...
81 × 103
0
...
92 × 103
1
...
03 × 103
1
...
26 × 103
2 × 103
2
...
8 × 103
8
...
9 × 103
10
...
3 × 103
13
...
3 × 103
21
...

Measuring density is an important analytical technique
...

As the battery discharges, the sulfuric acid (H2S04) combines with lead in the battery plates to
form insoluble lead sulfate (PbS04), decreasing the concentration of the solution
...
30 × 103 kg/m3 for a fully charged battery to l
...

Another automotive example is permanent-type antifreeze, which is usually a solution of ethylene
glycol (ρ = 1
...
The freezing point of the solution depends on the glycol
concentration, which can be determined by measuring the specific gravity
...


1
...
This is the force that you
feel pressing on your legs when you dangle them in a swimming pool
...

If we think of an imaginary surface within the fluid, the fluid on the two sides of the
surface exerts equal and opposite forces on the surface
...
) Consider a small surface of area dA centered on a point in
the fluid; the normal force exerted by the fluid on each side is
show in figure
...
The SI unit of pressure is the pascal,
I pascal = 1 Pa = N/m2

Two related units, used principally in meteorology, are the bar, equal to 105 Pa, and the millibar,
equal to 100 Pa
...
This pressure varies with weather changes
and with elevation
...
To four significant figures,
(Pa)av = 1 atm = 1
...
013 bar = 1013 millibar = 14
...
2
Note:- Don't confuse pressure and force In everyday language the words "pressure" and "force"
mean pretty much the same thing
...
Fluid pressure acts perpendicular to any surface in the
fluid, no matter how that surface is oriented show in figure
...
By contrast, force is a
vector with a definite direction
...
As shows in
figure, a surface with twice the area has twice as much force exerted on it by the fluid, so the
pressure is the same
...
4 Pressure, Depth, and Pascal’s Law
If the weight of the fluid can be neglected, the pressure in a fluid is the same throughout its
volume
...
But often the
fluid's weight is not negligible
...
When you dive into deep water,
your ears tell you that the pressure increases rapidly with increasing depth below the surface
...
We'll assume that the density has the same value throughout the
fluid (that is, the density is uniform), as does the acceleration due to gravity g
...
Consider a thin element of fluid with
thickness dy
...
The volume of the fluid element is dV = A dy, its mass is
dm = dV = A dy, and its weight is dw = dmg = gA dy
...
Call the pressure at the bottom sUlface p;
the total y-component of upward force on this surface is pA
...
The fluid
element is in equilibrium, so the total y-component of force, including the weight and the forces at
the bottom and top surfaces, must be zero:

When we divide out the area A and rearrange, we get

This equation shows that when y increases, p decreases; that is, as we move upward in the fluid,
pressure decreases, as we expect
...
Take point 1
at any level in the fluid and let P represent the pressure at this point
...
The depth of point 1 below
the surface is h = y2 – yl, is becomes

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Fluid

The pressure p at a depth h is greater than the pressure p0 at the surface by an amount pgh
...
The shape of the
container does not matter show in figure
...
This fact was recognized
in
1653 by the French scientist Blaise Pascal (1623-1662) and is called Pascal's law
...

The hydraulic lift shown schematically Pascal's law
...
The applied pressure p = FJAI
is
transmitted through the connecting pipe to a larger piston of area A2
...
Dentist's chairs, car lifts and jacks many elevators, and hydraulic brakes
all use this principle
...
In a room with a ceiling height of 3
...
2 kg/m3,the difference in pressure between floor and ceiling, given is

or about 0
...
But between sea level and the summit of Mount
Everest (8882 m) the density of air changes by nearly a factor of three , and in this case we cannot
use
...
A pressure of several hundred
atmospheres will cause only a few percent increase in the density of most liquids
...
Introduction
Ripples on a pond, musical sounds, seismic tremors triggered by an earthquake- all these are wave
phenomena
...
As a wave propagates,
it carries energy
...

This chapter and the next are about mechanical waves-waves that travel within some material
called medium
...
To help us understand waves in general, we'll look at the simple case of
waves that travel on a stretched string or rope
...
When a musician strums a guitar or
bows a violin, she makes waves that travel in opposite directions along the instrument's strings
...
We'll
discover that sinusoidal waves can occur on a guitar or violin string only for certain special
frequencies, called normal-mode frequencies, determined by the properties of the string
...
(In the next chapter we'll find that interference also helps explain the
pitches of wind instruments such as flutes and pipe organs
...
Electromagnetic waves-including light, radio
waves, infrared and ultraviolet radiation, and x rays-can propagate even in empty space, where
there is no medium
...


1
...
As the wave travels through the medium, the particles that make up the
medium undergo displacements of various kinds, depending on the nature of the wave
...
The medium is a string or rope under tension
...

Successive sections of string go through the same motion that we gave to the end, but at
successively later times
...


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Wave

The medium is a liquid or gas in a tube with a rigid wall at the right end and a movable piston a
the left end
...
This time the motions of the particles of the
medium are back and forth along the same direction that the wave travels
...

The medium is a liquid in a channel, such as water in an irrigation ditch or canal
...
In this case the displacements of the water have both longitudinal and transverse
components
...
For the stretched string it is the state in which the
system is at rest, stretched out along a straight line
...
And for the liquid in a trough it is a smooth, level water
surface
...
And in each case there are forces that tend to restore the
system to its equilibrium position when it is displaced, just as the force of gravity tends to pull a
pendulum toward its straight-down equilibrium position when it is displaced
...
First, in each case the disturbance travels
or propagates with a definite speed through the medium
...
Its value is determined in each case by the mechanical
properties of the medium
...
(The wave speed is not the
same as the speed with which particles move when they are disturbed by the wave
...
The overall pattern of the wave disturbance
is what travels
...
The wave motion transports this energy from one region of the
medium to another
...
3 Periodic Waves
The transverse wave on a stretched string in is an example of a wave pulse
...
The result is a single
"wiggle," or pulse, that travels along the length of the string
...

A more interesting situation develops when we give the free end of the string a repetitive,
or periodic, motion
...

In particular, suppose we move the string up and down with simple harmonic motion (SHM) with
amplitude A, frequency 1, angular frequency
and period
Figure
shows one way to do this
...

As we will see, periodic waves with simple harmonic motion are particularly easy to analyze;
we call them sinusoidal waves
...
So this particular kind of wave motion is worth special attention
...
Figure shows the shape of a part of the string near the left end at time intervals of
of a period, for a total time of one period
...
As the wave moves, any point on the string (any of the red
dots, for example) oscillates up and down about its equilibrium position with simple harmonic
motion
...

Wave motion vs
...
The wave
moves with constant speed v along the length of the string,
while the motion of the particle is simple harmonic and
transverse (perpendicular) to the length of the string
...
The length of one complete

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Kaysons Education

Wave

Wave pattern is the distance from one crest to the next, or
From one trough to the next, or from any point to the
Corresponding point on the next repetition of the
wave shape
...

The wave pattern travels with constant speed u and
advances a distance of one wavelength in a time
interval of one period T
...
The frequency is a property
of the entire periodic wave because all points on the
string oscillate with the same frequency of waves on a
String propagate in just one dimension
...

Shows a wave propagating in two dimensions on the
surface of a tank of water
...
In many important situations including waves on a string, wave speed v is
determined entirely by the mechanical properties of the medium
...
In this chapter we will consider only waves of this kind
...
)
To understand the mechanics of a periodic longitudinal wave, we consider a long tube
filled with a fluid, with a piston at the left end as in Fig
...
This region then pushes against the
neighboring region of fluid, and so on, and a wave pulse moves along the tube
...
This motion forms regions in the fluid where the pressure and density are greater or less
than the equilibrium values
...
Shows compressions as darkly shaded areas and rarefactions as
lightly shaded areas
...


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Kaysons Education

Wave

The wave propagating in the fluid-filled tube at time
intervals of 1/8 of a period, for a total time of one period
...
Each particle in the fluid
oscillates in SHM parallel to the direction of wave
propagation (that is, left and right) with the same amplitude
A and period T as the piston
...

Just like the sinusoidal transverse wave, in one period T the
longitudinal wave travels one wavelength to the right
...
Just as for transverse waves, in this chapter
and the next we will consider only situations in which the
speed of longitudinal waves does not depend on the
frequency
...
4 Wave Function for a sinusoidal wave
Let's see how to determine the form of the wave function for a sinusoidal wave
...
Every
particle of the string oscillates with simple harmonic motion with the same amplitude and
frequency
...
The particle at point B is at its maximum positive value of y at t = 0 and returns to y =
0 at t = 2/8 T; these same events occur for a particle at point A or point C at t = 4/8 T and t = 6/8T,
exactly one half-period later
...

Hence the cyclic motions of various points on the string are out of step with each other by
various fractions of a cycle
...
For example, if one point has its maximum positive
displacement at the same time that another has its maximum negative displacement, the two are a
half cycle out of phase
...
)
Suppose that the displacement of a particle at the left end of the string (x = 0), where the
wave originates, is given by –

That is, the particle oscillates in simple harmonic motion with amplitude A, frequency f, and
angular frequency
The notation y (x = 0, t) reminds us that the motion of this particle is
a special case of the wave function y (x, t) that describes the entire wave
...
Introduction
Many kinds of motion repeat themselves over and over: the vibration of a quartz crystal in a
watch, the swinging pendulum of a grandfather clock, the sound vibrations produced by a clarinet
or an organ pipe, and the back-and-forth motion of the pistons in a car engine
...
Understanding
periodic motion will be essential for our later study of waves, sound, alternating electric currents,
and light
...
When it
is moved away from this position and released, a force or torque comes into play to pull it back
toward equilibrium
...

Picture a ball rolling back and forth in a round bowl or a pendulum that swings back and forth past
its straight-down position
...
We will also study why oscillations often tend to
die out with time and why some oscillations can build up to greater and greater displacements
from equilibrium when periodically varying forces act
...
Simple Harmonic Motion
The simplest kind of oscillation occurs when the restoring force F, is directly proportional to the
displacement from equilibrium x
...
The constant of proportionality between F and x is the force constant k
...
On either side of the
equilibrium position, F, and x always have opposite signs
...
3 we represented the force
acting on a stretched ideal spring as F, = kx
...
The force constant k is always positive and has units of N/m (a useful alternative set of units
is kg/s2)
...
When the
restoring force is directly proportional to the displacement from equilibrium, as given by the

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Kaysons Education

Simple Harmonic Motion

oscillation is called simple harmonic motion, abbreviated SHM
...
This
`acceleration is not constant, so don't even think of using the constant- acceleration equations
...
A
body that undergoes simple harmonic motion is called a harmonic oscillator
...
But in many systems the restoring force is approximately proportional to
displacement if the displacement is sufficiently small
...
Thus we can use SHM as an approximate model for many different periodic
motions, such as the vibration of the quartz crystal in a watch, the motion of a tuning fork, the
electric current in an alternating-current circuit, and the oscillations of atoms in molecules and
solids
...
1 Equations of simple harmonic motion
To explore the properties of simple harmonic motion, we must express the displacement x of the
oscillating body as a function of time, x(t)
...
As we mentioned, the formulas for constant
acceleration because the acceleration changes constantly as the displacement x changes
...

Figure shows a top view of a horizontal disk of radius A with a ball attached to its rim at point Q
...
A horizontal light beam shines on the rotating disk and casts a shadow of the ball
on a screen
...
We then
arrange a body attached to an ideal spring, like the combination shown in figures
...
We will prove that the motion of the body and the motion of the
ball's shadow are identical if the amplitude of the body's oscillation is equal to the disk radius A,

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Kaysons Education

Simple Harmonic Motion

and if the angular frequency 27πf of the oscillating body is equal to the angular speed w of the
rotating disk
...


We can verify this remarkable statement by finding the acceleration of the shadow at P and
comparing it to the acceleration of a body undergoing SHM
...
The reference point
...
At time t the vector OQ from the origin to the reference
point Q makes an angle the positive x-axis
...
Such a rotating
vector is called a phasor
...
The phasor method for analyzing oscillations is useful in many areas of
physics
...
Hence
the x-velocity of the shadow P along the x-axis is equal to the x-component of the velocity vector
of the reference point Q, and the x-acceleration of P is equal to the x-component of the
acceleration vector of Q
...
Furthermore, the magnitude of
speed squared times the radius of the circle
...
These are precisely the hallmarks of simple harmonic motion
...
The reason is that these quantities are equal
...
During time T the point Q
moves through
radians, so its angular speed is
But this is just for the angular
frequency of the point P, which verifies our statement about the two interpretations of w
...
So we reinterpret an expression for the angular frequency of simple harmonic
motion for a body of mass m, acted on by a restoring force with force constant k:

When you start a body oscillating in SHM, the value of w is not yours to choose; it is
predetermined by the values of k and m
...
When we take the square root we get S-1, or more properly rad/s because this is an angular
frequency (recall that a radian is not a true unit)
...
that a larger mass m, with its greater inertia, will have less acceleration, move
more slowly, and take a longer time for a complete cycle
...

Equations show that the period and frequency of simple harmonic motion are completely
determined by the mass m
...
In simple harmonic motion
...
For given values of m and k, the time of one
complete oscillation is the same whether the amplitude is large or small
...
Larger A means that the body reaches larger values of |x| and is subjected to
larger restoring forces
...
The
oscillations of a tuning fork are essentially simple harmonic motion, which means that it always
vibrates with the same frequency, independent of amplitude
...
If it were not for this characteristic of simple harmonic motion, it
would be impossible to make familiar types of mechanical and electronic clocks run accurately or
to play most musical instruments in tune
...


3
...
When the point mass is pulled to one side of its straight-down equilibrium
position and released, it oscillates about the equilibrium position
...
The
path of the point mass (sometimes called a pendulum bob) is not a straight line but the arc of a
circle with radius L equal to the length of the string
...
If the motion is simple harmonic, the restoring force must be directly
proportional to x or (because
) to Is it?
We represent the forces on the mass in terms of tangential and radial components
...
The restoring force is proportional not to but to sine, so the motion is not simple
harmonic
...
For example,
when = 0
...
0998, a difference of only 0
...
With this approximation,

The restoring force is then proportional to the coordinate for small displacements, and the force
constant is k = mg/L
...
This is because the restoring
force, a component of the particle's weight, is proportional to m
...
(This is the same physics that explains why bodies of
different masses fall with the same acceleration in a vacuum
...
Newton's Law of Gravitation
The example of gravitational attraction that's probably most familiar to you is your weight, the
force that attracts you toward the earth
...
Along with his three laws of motion, Newton published the law of gravitation in
1687
...

Translating this into an equation, we have
Where Fg is the magnitude of the gravitational force on either particle, ml and m2 are their masses,
r is the distance between them, and G is a fundamental physical constant called the gravitational
constant
...
That the gravitational
force between two particles decreases with increasing distance r: If the distance is doubled, the
force is only one-fourth as great, and so on
...

Don't confuse g and G because the symbols g and G are so similar, it's common to confuse the two
very different gravitational quantities that these symbols represent
...
g
...
By
contrast, capital G relates the gravitational force between any two bodies to their masses and the
distance between them
...
In the next section we'll see how the values of g
and G are related
...
Even when the masses of the particles are different, the two interaction forces have
equal magnitude Figure
...
When you fall from a diving board into a
swimming pool, the entire earth rises up to meet you! (You don't notice this because the earth's
mass is greater than yours by a factor of about 10 23
...
)

We have stated the law of gravitation in terms of the interaction between two particles
...
Thus, if we model the earth as a spherically symmetric
body with mass mE the force it exerts on a particle or a spherically symmetric body with mass m,
at a distance r between centers, is

provided that the body lies outside the earth
...
At points inside the earth the situation is different
...
As the body
enters the interior of the earth (or other spherical body), some of the earth's mass is on the side of
the body opposite from the center and pulls in the opposite direction
...

Spherically symmetric bodies are an important case because moons, planets, and stars all tend to
be spherical
...
As a result, the body naturally tends to assume a
spherical shape, just as a lump of clay forms into a sphere if you squeeze it with equal forces on all
sides
...


1
...
The force is extremely
small for bodies that are small enough to be brought into the laboratory, but it can be measured
with an instrument called a torsion balance, which Sir Henry Cavendish used in 1798 to
deterrn
...
A modern version of the Cavendish torsion balance is shown in figure
...
Two small spheres,
each of mass ml, are mounted at the ends of the horizontal arms of the T
...
To measure this angle, we shine a beam of light on a mirror fastened to the
T
...


After calibrating the Cavendish balance, we can measure gravitational forces and thus determine
G
...
Gravitational forces combine
vectorially
...


2
...
This led to the expression U
= mgy
...
For problems in which r changes enough that the
gravitational force can't be considered constant, we need a more general expression for
gravitational potential energy
...
We consider a body of mass
m outside the earth, and first compute the work Wgrav done by the gravitational force when the
body moves directly away from or toward the center of the earth from r = r l to r = r2, as in figure
...
Because points directly inward toward the center
of the earth, Fr is negative
...
By an
argument similar to that work depends only on the initial and final values of r, not on the path
taken
...
We now define the
corresponding potential energy U so that Wgrav = U1 – U2
...
But in
fact you've seen negative values of V before
...
We have chosen U to be zero when the body of mass m is infinitely far from the earth (r
= )
...

If we wanted, we could make U = 0 at the surface of the earth, where r > RE, by simply adding the
quantity GmEm/RE
...
We won't do this for two reasons:
One, it would make the expression for U more complicated; and two, the added term would not
affect the difference in potential energy between any two points, which is the only physically
significant quantity
...
If the gravitational force on the body is the only force that
does work, the total mechanical energy of the system is constant, or conserved
...


3
...
We can
now broaden our definition: The weight of a body is the total gravitational force exerted on the
body by all other bodies in the universe
...
At the surface of the moon we consider a body's weight to be the gravitational attraction
of the moon, and so on
...
m, we find

The acceleration due to gravity g is independent of the mass m of the body because m doesn't
appear in this equation
...
We can measure all the quantities for mE, so this relationship allows us to compute the
mass of the earth
...
At a point above the earth's
surface a distance r from the center of the earth (a distance r – RE above the surface), the weight of
a body is given with RE replaced by r:
The weight of a body decreases inversely with the square of its distance from the earth's center
...
The apparent weight of a body on earth differs slightly from the earth's
gravitational force because the earth rotates and is therefore not precisely an inertial frame of
reference
...
We will return to the effect of the earth's rotation
...
But this does not mean that the earth is uniform

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