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Fluids at Rest
Density and Pressure - Pascal's Law
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Volume of solid, liquid or gas depends on the stress or pressure on it.
Solids and liquids have lower compressibility compared to gases.
Pressure
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Average pressure (Pav) is normal force (F) acting per unit area (A).
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Pav =
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For very small area:
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Force exerted by a liquid on the submerged object is normal at all points on the
surface.
Pressure is a scalar quantity.
Its SI unit is Nm−2 (pascal).
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F
(Scalar quantity)
A
Thrust and Pressure
Thrust is a total force in a particular direction. The unit of thrust, therefore is the same as
that of force : Newtons (N). Pressure is the force or thrust applied per unit area.
Force (or thrust)
Pressure (P) = 
Area
A unit of pressure is Newtons/m2 or N/m2. The unit is also called a Pascal (Pa). There are
other units in use also but these are the units in the M.K.S. system.
To appreciate the difference between thrust and pressure, do the following experiment.
Take two trays filled with sand and one heavy rectangular block that measures 1 kg. Keep
the first block on its side in one tray and the second block on its base in the second tray.
You will notice that the first block has sunk deeper in the sand. Although the weights of the
two blocks is the same then why does this happen? Since the weights are same, the thrust
or the force applied is the same in both the cases. But the area on which this force is acting
is different in the two cases. In the first case, the area is smaller hence the pressure is
more. In the second case the area on which the force is acting is larger, hence the pressure
is less. Thus, we can explain why the first block sinks into the sand more than the second
block; in the first case the pressure applied is more.
We can see several application of pressure in
our everyday lives : cutting tool edges are
sharper than their heads, a pressure applied
on the head is magnified at the edge. The
same is true with pin heads or nails, they can
be pushed with minimum pressure on the
heads.
Units of pressure
As discussed above, height of a Hg column is a standard way of measuring pressure. At sea
level, and at standard gravitational force, the air pressure is defined as one atmosphere.
1 atmosphere = 760 mm of Hg.
There are other units of measuring pressure also. They are Newtons/m2, Pascal, bars and
torr, pounds per square inch (psi), etc.
1 atm = 14.7 psi
1atm = 101,325 N/m2
1atm = 1.01325 bar
1 torr = 1 mm of Hg
Derivation of expression for a pressure in a liquid
We have seen in the experiments above that pressure is the
same at the same depth in the liquid. At a point B in the liquid,
the pressure will depend on the weight of the liquid column
above it.
Thus
Total thrust or force at level B = weight of the liquid column above B
= Mass of the liquid column (m) x g
(g : gravitational acceleration 9.8 m/s2)
= volume x density ( density = volume/mass)
= area x h x ρ x g
force (or thrust)
Pressure P =  = h x ρ x g
area
Thus pressure at a point inside the liquid is given by its depth or height, density and g, the
gravitational acceleration. The liquid pressure on the
surface is zero as h = 0. But the liquid surface is also
exposed to atmospheric pressure, pressure of the
surrounding air on the surface.
Total pressure at any point in a liquid = atmospheric pressure + h x ρ x g
When dams are built to stop and store water, it becomes necessary to make the base of
the dam broader than the top. The base has to sustain greater pressure from the pressure
of the water column as compared to the pressure exerted by the water on the surface.
Density (ρ)
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Mass (m) per unit volume (v)
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(scalar quantity)
Its SI unit is kg m−3.
A liquid is largely incompressible and therefore, its density is nearly constant at all
pressures.
The density of water at 4°C is 1 × 103 kg m−3.
The relative density of a substance is the ratio of its density to the density of water at
4°C.
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Why liquid seeks it own level
If you fill a container with liquid, you will notice that the liquid will form a uniform level.
The same does not happen with solid particles; solid level can remain uneven. It is
interesting to observe how liquid level is adjusted when you pour coloured water in one of
the vessel arrangements shown below.
Irrespective of the shapes of the container, the
water will stand at the same level in all the vessels.
Thus liquid seeks its own level. The reason behind
this is the pressure exerted by the liquid inside
itself. If the height of the liquid level is low in one
vessel, the pressure experienced by its column is
atmospheric pressure + height of the column. In another vessel, where the height of the
liquid is larger, the pressure experienced by the water column will be more, as the height is
more. To equalize the pressure, liquid will flow from the high-pressure region to the lower
pressure region thereby making the heights of liquid in each vessel the same.
Pressure in liquids
Liquids exert pressure due to distribution of their own weight. To see how liquids exert
pressure, try the following experiment. Take three tins of different sizes or diameters. On
tin number I, make three holes at the same height. On tin numbers II and III, make three
holes at different heights. Place three long tapes to close the three holes on each on the
tins. Now fill the tin with water. Remove the tapes quickly and observe the streams coming
out of each of the holes.
You will observe the following :
Water stream will start
pouring out through the holes.
This means that water is
exerting
pressure
in
all
direction.
• In tin I, the water stream
comes out evenly irrespective
of the direction of the hole. This
means that the pressure is
•
•
equal at the same height or depth.
In Tin II and III, the water stream coming out of the lowest hole reaches the farthest.
This shows that the pressure exerted by liquid increases with depth. Also the
pressure is acting perpendicular to the liquid surface.
•
Since there is no difference between the streams coming out of tins II and III, the
pressure exerted by liquid is independent of the size of the container, but depends
only on the height or the depth of the liquid. (This is markedly different from what
happens when a solid is applying pressure or weight, as seen in the earlier section).
Pascal’s Law
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Pressure inside a fluid at rest is same at all points if they are at the same height.
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Pressure exerted in all directions in a fluid at rest is same.
Variation of Pressure with Depth and Hydraulic Machine
Fluid under Gravity
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Derivation:
Consider,
Pressure at point 1 = P1
Pressure at point 2 = P2
Mass of fluid inside cylinder = m
Area of the base of cylinder = A
Height of the cylinder = h
Density of fluid =ρ
(Volume of cylinder = V)
m = ρhA
∴P2 − P1 = ρgh
When point 1 is open to atmosphere,
P1 = Atmospheric pressure (Pa)
P2 = P (absolute pressure)
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∴ P = Pa + ρgh
Gauge pressure = P − Pa = ρgh
Pressure is same at all points at the same depth.
The liquid pressure at a point is independent of the quantity of liquid, but depends
upon the depth of point below the liquid surface. This is known as hydrostatic
paradox.
The atmospheric pressure at any point is equal to the weight of a column of air of
unit cross-sectional area, extending from that point to the top of the earth’s
atmosphere.
Atmospheric pressure at sea level is 1.013 × 105 Pa (1 atm).
Two pressure measuring devices are mercury barometer and open tube manometer.
Hydraulic Machines
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These are based on Pascal’s law for transmission of fluid pressure, which states that
external pressure applied on any part of a fluid contained in a vessel is transmitted
undiminished and equally in all directions.
Hydraulic lift, hydraulic brakes, hydraulic press are some examples of hydraulic machines
Hydraulic Lift
•
Consider,
Area of small cross-section = A1
Force exerted on it = F1
Pressure transmitted,
Area of larger piston = A2
Upward force on the piston = P × A2
Force supported by the large piston, F2 = PA2
Since A2 is greater than A1, force F2 on the larger piston will also be much larger than the
force F1 applied on the smaller piston.
Mechanical advantage of the device is
.
HYDRAULIC BRAKES
Two very useful devices
based on Pascal's law
are
hydraulic brakes and
hydraulic lift shown
overleaf. The pressure
applied by the foot on
the
break pedal is applied to the brake fluid contained in the master cylinder. This
pressure is transmitted undiminished in all directions and acts through the brake
pads on the wheel reducing the rotatory motion to a halt. Sliding friction between
the tyres and the road surface opposes the tendency of forward motion reducing the
linear momentum to zero.
Atmospheric pressure and its measurement
The air or the atmosphere around the earth extends upwards up to about 300 km from the
sea level. The density of air is maximum at the sea level and becomes rarer as we go higher
up (due to gravitational pull of the earth, air is more dense close to the surface of the
earth). The air exerts pressure on surfaces that it is in contact with. At sea level, the air
pressure is 101325 N/m2. Atmospheric pressure decreases as we go to higher and higher
altitudes, since the density of air also decreases. Atmospheric pressure at a given point is
same in all directions. This is similar to the case of liquids. The entire weather system of the
earth depends on the movement of air. Air moves from high-pressure region to low
pressure region, in order to equalize pressure.We can demonstrate two simple
experiments in order to understand the presence of atmospheric pressure.
Experiment 1 : Take a tin can and heat it with its cork in the open position. The heat will
make the air inside hot. The hot air will leave the container. Now quickly close the
container. You will notice that the tin
can
collapses. Why does this happen? The
air inside
the tin container is expelled partially on
heating.
When you close the container, there is
no way
for the air outside to enter the tin. The
air
pressure inside is less than the air
pressure
outside the tin. The air outside the tin
presses
against its surface. This crushes the tin.
The
surface area of the tin is reduced, till
such time
that the pressure inside becomes same
as the
pressure outside.The experiment shows
that air
exerts pressure.
Experiment 2 : Take a glass and fill it with water up to the
brim. No air gap should remain between the water level
and the rim of the glass. Cover the glass with a postcard or
cardboard piece. Invert the glass quickly. You will notice
that the cardboard and water are held in place. This
implies that some force is acting on the cardboard from
below, which is able to hold the weight of water on it. The
force is the atmospheric pressure acting in the upward
direction.The experiment also shows that air exerts
pressure.
Archimedes' Principle
a
Have you ever had a swim in a pool? Don’t you feel lighter? Have you ever drawn water
from a well and felt that the bucket of water feels heavier when it is out of the water? Have
you ever wondered why a ship made of iron and steel does not sink, but if the same
amount of iron and steel in the form of a sheet would sink? Well, all these phenomena
occur in all fluids including water and are due to exerted pressure.
WE WILL STUDY: 1. Archimedes Principle
1. Theoretical proof of Archimedes' Principle
2. Application of Archimedes' Principle to determine densities of liquids
1. Archimedes Principle
Take a spring balance, a piece of stone, a measuring cylinder and water. Measure the
weight of stone in air by tying the string around in a loop, and hanging it from the spring
balance. Take water in a measuring cylinder and note its volume level. Then dip the stone
in the water while it is still hanging from the spring balance. You will see that the stone is
weighing less!! If you see the water level now, you will see it has risen. Now from the
volume of the water displaced, calculate the weight of water from the following equation
for density :
Density of water
= Mass of water (in gm)/Volume of water (in cubic cm)
Density of water is 1 gm/cm3. You will see that the mass of water displaced is exactly
equal to the reduction in weight of the stone in water.
Archimedes was the first person to understand this phenomenon more than about 2,200
years ago and hence the phenomenon is named after him. Click here for an interesting
anecdote on Archimedes.
Archimedes’ Principle states that a body immersed in a liquid, wholly or partly, loses its
weight. The loss of weight is equal to the weight of the liquid displaced by the body.
2. Theoretical proof of Archimedes’ Principle
Consider the figure alongside, here a square piece of iron is immersed in liquid. The piece
of iron is experiencing forces from all sides and they are:
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The down ward force due to its weight = W
Downward force acting on the upper surface of the iron piece, due to water pressing
on it = F1
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Upward force due to the tension of the string = T
Upward force acting on the lower surface of the iron piece
due to water pressing on it = F2
Horizontal forces acting on the other surfaces due to water
pressure = H
Since the piece of iron is stationary and is not moving either up
or down or side ways, we can safely say that H=0 and
Total upward force = Total Downward force
T+ F2 = W + F1
Pressure is defined as force per unit area.
F1 = P1 (on the upper surface of the iron piece) x area
and F2 = P2 (on the lower surface of the iron piece ) x area.
Pressure at a point inside a liquid is proportional to the height at which the point is from
the surface, multiplied by the density of the liquid (ρ) and the gravitational force. In the
above figure the pressure at the top surface of the iron piece is h1ρ g and at the bottom
surface is h2ρ g.
Therefore F1 = (h1ρ g) x area and F2 = (h2ρ g) x area
W - T = (ρ g ) x volume of the iron piece
W - T = loss of the weight of the iron piece when immersed in liquid.
(ρ g ) x volume of the iron piece = (ρ g) x volume of the liquid displaced by the
iron piece
= ρ g x V = (mass of liquid displaced) x g
= weight of liquid displaced by the body
Hence we can conclude that the loss of weight of a body in a liquid is equal to the weight of
the liquid displace by the body.
The Archimedes principle holds good for irregular as well as regular bodies and any liquids.
The upward force experienced by the immersed body is also known as upthrust or
buoyancy.
3. Application of Archimedes’ Principle to determine densities
of liquids
Density of a substance is given as the mass per unit volume.
Quite often, it is easier to quote the relative density of the
substance with respect to the density of water. Hence the
relative density (R.D.) of a substance is defined as the ratio of
the density of the substance with respect to that of water.
Density of substance
R.D = -------------------------------Density of water
Density of water is 1 gm/cm3. (Density changes with temperature; density of water is 1
gm/cm3 at 4oC. It is taken as the same at all temperatures unless the temperatures are
close to 0oC or 100oC , where water changes to ice or steam respectively)
To determine the density of an unknown liquid by Archimedes’ method, please do the
following :
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Weigh a given object in air = W1
Weigh the same object in water = W2
Weigh the same object in the unknown liquid = W3
R.D
Weight
of the displaced liquid Volume of water displaced
×
Volume of the displaced liquid Weight of the water displaced
Since the volume displaced by the object in both liquid and water is same, they get
cancelled out from the above equation.
(W1 - W3 )
R.D. = ----------------(W1 - W2 )
IMPORTANT TERMS RELATED TO THE STUDY OF SURFACE TENSION OF LIQUID
(i) Force of adhesion or adhesive force. It is force of attraction acting between the
molecules of different substances. For example, water wets the surface of a glass container.
This is due to force of adhesion between water and glass molecules. Similarly, while
writing, graphite from lead pencil sticks to the paper on account of adhesive forces. Fevicol,
cement etc are useful in glueing two surfaces together again on account of adhesive forces.
Force of adhesion is different for different substances e.g. gum has a greater adhesive
force for a solid surface than water.
(ii) Force of cohesion or cohesive force. It is the force of attraction amongst the
molecules of the same substance. For example, definite shape and size of solids is due to
strong forces of cohesion amongst their molecules. Liquids have a definite volume, but no
definite shape. Therefore, cohesive forces in case of liquids must be weaker than the
cohesive forces in case of solids. Gases, on the contrary, have neither fixed volume, nor
fixed shape. Therefore, cohesive forces amongst the molecule of a gas are minimum.
Mercury does not wet the surface of a glass container because the force of cohesion
amongst molecules of mercury is stronger than the force of adhesion between molecules of
mercury and glass.
The cohesive and adhesive forces are VanderWaal forces. These forces are different
from ordinary gravitational forces and do not obey inverse square law. The cohesive or
adhesive force varies inversely as the seventh power of distance between the molecules i.e.
the cohesive or adhesive force increases rapidly with decrease in distance between the
molecules.
Molecular range. It is the maximum distance upto which a molecule can exert some
measureable attraction on other molecules. It is different for different substances. The
order of molecular range is 10-9 m in solids and liquids.
Sphere of influence. It is on imaginary sphere drawn with a molecule as centre and
molecular range as radius. All the molecules in this sphere attract the molecule at the
centre and vice versa.
Surface film. It is the top most layer of liquid at rest with thickness equal to the molecular
range.
SURFACE TENSION
It has been found that a liquid in small quantity (i.e. a liquid drop) at rest, free from
external forces like gravity, always tends to have a spherical shape. Since for a given
volume, a sphere has the least surface area, hence it shows that the free surface of every
liquid at rest has a tendency to have a least surface area. While doing so, the free surface
behaves as if covered by a stretched membrane, having tension in all directions parallel to
the surface. This tension in the free surface of liquid at rest is called the surface tension.
Thus surface tension is the property of the liquid by virtue of which the free surface of
liquid at rest tends to have minimum area and as such it behaves as if covered with a
stretched membrane.
of
by definition, is
S = F/l
Let F be the total force
acting on an imaginary line
length I, drawn tangentially
on the liquid surface at rest,
the force of surface tension S,
Units of surface tension are dyne/cm in CGS system and Nm- in S.I. The dimensional
formula of surface tension is [ML0T-2]. Surface tension is a scalar quantity because it has no
specific direction for a given liquid.
ILLUSTRATIONS OF SURFACE TENSION
(1) Rain drops are spherical in shape because each drop tends to acquire minimum surface
area due to surface tension, and for a given volume, the surface area of sphere is minimum.
(2) When mercury is split on a clean glass plate, it forms globules. Tiny globules are
spherical on account of surface tension because force of gravity is negligible. The bigger
globules get flattened from the middle but have round shape near the edges.
The globule takes spherical shape due to surface tension but in case of big mercury
globule, the force of gravity is large. The centre of gravity of the big globule is lowered due
to its heavy weight. As a result, the big globule gets flattened from the middle.
(3) When a greased iron needle is placed gently on the surface of water at rest, so that it
does not prick the water surface, the needle floats on the surface of water despite its being
heavier. A slight depression, on the surface of water is observed just below the needle,
indicating that the water surface behaves like a stretched membrane. The forces of surface
tension (S, S) act in the inclined manner. The weight of the needle is balanced by the
vertical components of the forces of surface tension. If the water surface is pricked by one
end of the needle, the needle sinks down.
(4) Let a circular metallic ring provided with handle. Form a soap film on the ring by dipping
it in soap solution. Place a cotton thread loop gently on the soap film. We note that the
thread loop takes an irregular shape. Prick the soap film with a sharp edge of the needle
from the middle portion of the loop. The thread loop now takes a circular shape, as if it
were equally pulled from all sides.
It is so because, the remaining soap film (outside the thread loop) tries to have
minimum surface area due to surface tension and for a given perimeter, the area of the
circle is maximum.
(5) When a shaving brush is dipped in water, its hair spread out. On taking out the brush
from water, the water film formed between the hairs while tending to make its surface
area minimum due to surface tension will bring the hairs closer to each other. That is why,
the hairs of shaving brush when taken out of water are pressed together.
(6) Take a wire tray of narrow mesh. Let its wire be coated with wax. Pour some water
gently into the tray. Water does not flow down. This is because in every narrow hole of the
mesh an elastic water membrane is formed. .
(7) The bits of camphor are seen dancing on the surface of water. Infact these bits floating
on clean surface, reduce the surface tension of water where they are dipping. Since the
shape of bits of camphor is irregular, unequal forces of surface tension act on them. That is
why they move helter skelter or erratically on the surface of water.
(8) Oil drop spreads on cold water. It may remain as a drop on hot water. This is due to the
fact that the surface tension of oil is less than that of cold water and is more than that of
hot water.
SURFACE
BODIES:
TENSION OF SOME
SURFACE ENERGY
We know that free surface of liquid at rest tends to contract in order to have minimum
surface area due to surface tension. Thus, if the area of free surface of the liquid has to be
increased, some work will have to be done against the force of surface tension. This work
done appears in the liquid surface film as its potential energy, which is called surface
energy.
Hence surface energy is defined as the amount of work done against the force of
surface tension, in forming the liquid surface of a given area at a constant temperature.
To obtain an expression for surface energy, take a rectangular metallic frame having a
wire AB which can slide along the sides. Dip the frame in soap solution and form a soap film
ABCD on the rectangular frame. There will be two free
surfaces of the film where air and soap are in contact. The forces of surface tension act
tangentially inwards and perpendicular to the sides on both free surfaces of the mm as. All
other sides being fixed, we can increase the surface area of soap film by displacing the
sliding wire AB outwards.
Let S = surface tension of the soap solution,
l = length of the wire PO.
Since there are two free surfaces of the film and
tension acts on both of them, hence total inward force
AB is
surface
on the wire
F = s x 2l
To increase the area of the soap film, we have to pull the sliding wire AB outwards with
a force F. Let the mm be stretched by displacing the wire AB through a small distance x to
the position A’B’.
The increase in area of the film, ABB’A’.
= a = 2 (I x x)
:. Work done in stretching the film is
E = force applied x distance moved
= (S x 2l) x x = S X (2l x)
=SXa
where 2l x = a = increase in area of the film on both sides
(I x x = increase in area of film on one side)
If temperature of the film remains constant in this process, this work done is stored in
the film as its surface energy.
S = E / a. Hence swface tension of a liquid is numerically equal to the surface energy per
unit area.
If increase in area is unity i.e. a = 1 then
S = E.
Therefore, surface tension may be defined a.f the amount of work done in increasing
the area of the liquid surface by unity against the force of surface tension, at a constant
temperature.
Angle of contact
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Angle between tangent to the liquid surface at the point of contact and solid surface
inside the liquid is called angle of contact.
Case I
(Water and oily surface interface)
Where,
Sla − Interfacial tension of liquid-air interface
Ssa − Interfacial tension of solid-air interface
Sls − Interfacial tension of liquid-solid interface
Sla cos θ + Ssl = Ssa
If Ssl > Sla, then angle of contact is an obtuse angle and the molecules of liquid are
attracted strongly to themselves and weakly to those of solids.
•
A lot of energy is used to create the liquid-solid interface.
Case II
•
If Ssl la, then the angle of contact is acute and the molecules of the liquid are strongly
attracted to those of the solid.
Drops and Bubbles
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Liquid drops are spherical as they have least area and will have minimum energy.
Spherical drop
Where,
r − Radius of drop
P0 − Pressure outside the bubble
Pi − Pressure inside the bubble
S − Surface tension of the bubble
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•
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Surface energy = 4πr2S
Let radius increase by Δ r.
Then, extra surface energy
= [4 π (r + Δr)2 − 4πr2] S
= 8πrΔrS (1)
At equilibrium, the energy used is balanced by the energy gained.
Energy gain is the pressure difference = Pi − P0
∴ Work done, W = (Pi − P0) 4πr2Δr (2)
From equations (1) and (2),
Liquid−gas interface
o Convex side has higher pressure than concave side.
For Bubble − Having two interfaces
Capillary Rise
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Angle of contact between water and glass is acute.
Surface of water in the capillary is concave.
Pressure difference between two sides of top surface, Pa − P0 = 2S/r
•
Considering points A and B, they must be at same pressure i.e.,
P0 + hρg = Pa
•
Therefore, capillary rise is due to surface tension.
Height of water rise,
EXCESS OF PRESSURE ON CURVED SURFACE OF LIQUID (PRESSURE DIFFERENCE ACROSS A
SURFACE FILM)
If the free surface of a liquid is plane, then the surface
tension acts horizontally. So, it has no component normal to the
horizontal surface. As a result, no extra pressure is communicated
to the inside or outside. Thus, the pressure Pl on the liquid side is
equal to the pressure Pv on the vapour side.
We know that a liquid molecule on the concave surface. In order
that the curved surface will be equally attracted on all the sides,
and the forces (ρ,ρ) of surface tension will be tangentially to the liquid surface in opposite
diresction. Thus the pressure on the liquid side is equal to the pressure on the vapour side
in case of plane surface.
For the curved surface of liquid in
equilibrium the pressure on concave side
of liquid will be greater than pressure on
its convex side.
EXCESS PRESSURE INSIDE A LIQUID DROP:
Initial surface area = 4πR2
Final surface area = 4π (R+ dR)2
Increase in surface area ‘dA’ = 4π (R2+ dR2 +2RdR) - 4πR2 =
8πRdR
(Ignore dR2 which is negligibly small)
Increase in surface energy of drop ‘dE’ = increase in surface area * Surface tension = dA * S
= 8πRdR * S
This increase in surface energy is due to work done dW.
dW = dE
p(4πR2)dR = 8πRdR*S
p = 2S / R
EXCESS PRESSURE INSIDE AN AIR BUBBLE IN LIQUID:
It is case similar to that of liquid drop in air. This is because it has only
one free surface. So excess pressure inside an air drop is = 2S /R
EXCESS PRESSURE INSIDE A SOAP BUBBLE:
Lets consider a soap bubble of radius ‘R’. Let S be the surface
tension of liquid. Let dW be the work done by the excess
pressue in increasing the radius of the bubble by an infinitesimally small amount ‘dR’.
Then dW = p(4πR2)dR
Soap bubble has two free surfaces.
The increase in surface area dA = 2 (8πR {dR}) = (16πR {dR})
This increase in surface energy is due to work done dW.
dW = dE
p(4πR2)dR = 16πRdR*S
p = 4S / R