Pressure Variation with depth - Mechanical Properties of Fluids Video Lecture - Class 11

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watched all the visuals from part one to
four before going ahead with part 5 now
as I told you in this slide we will
study how does pressure vary with depth
I told you in the previous slide right
that when we talk of height there is a
variation of pressure with height so
here we will study how does texture vary
with difference in height let us
consider just to understand the basic
fact it is considered or they are
assuming here that we have some
cylindrical object inside a cubed so for
this particular object let us say that
we will consider two different position
one position would be this position one
and the other one is this that this
position let us try to see how this
pressure varies with depth in this case
as we already as I already told you that
here will be leaving the studying about
fluids at rest
since this fluid is not moving so there
is no horizontal motion that is along
the horizontal direction the net force
is equal to zero
so along the horizontal direction the
net force is equal to zero now what
about the vertical direction let us look
at the vertical direction in this
vertical direction suppose we define two
positions one is position one and the
other one is position 2 so what do we
see here in the along the vertical
direction the force at position 1 will
be equal to pressure at position 1
ending area this area is the cross
sectional area now what will be the
direction of this force this force will
be 5
radicular to the cross-sectional area
that means this force is perpendicular
in cross-sectional area a segments that
is the cross-sectional area so let us
suppose this force F mine is somewhat in
this direction right so this is
perpendicular to the cross-sectional
area
similarly at position two we can say
that F 2 is nothing but V 2 into area so
here again we say that let us suppose
this is f 2 so what do we say that what
is the total force that is F let F net
would be nothing but minus F 1 plus F 2
why so because F 1 is acting along the
negative Y axis so we have taken it as
if- whereas F 2 is acting along the
positive Y axis so we have taken it as
positive so this we can write it as P 2
minus P 1 into area so this is the net
force
now this net force will be balanced by
the weight of this cylinder now we have
considered this cylinder it is suppose
this cylinder has some mass say M say M
is the mass of this cylinder
therefore this net force because let we
are considering that if this is in
equilibrium under equilibrium condition
if this net force will be equal to mass
into acceleration that is the weight of
the cylinder
now this weight of the cylinder will be
equal to nothing but the weight of the
fluid displaced so the weight of the
cylinder this is basically the weight of
cylinder
now this weight of the cylinder will be
equal to the weight of fluid displaced
so what is the weight of the fluid
displaced so in that case we can write
down this mass is nothing but density
into volume because we know that density
is equal to mass into mass by volume so
we can say intensity F is Rho into
volume of the water displaced into G now
what will be this volume this volume
will be nothing but height that is
height of the cylinder into the area
that is the cross sectional area of the
cylinder into G because volume is
nothing but what is volume volume is
nothing but height into area so we can
say P 2 minus P 1 into a is equal to Rho
into H into into G so this a and a will
get cancelled so we get P 2 minus P 1 is
equal to Rho H G that means the
difference in pressure that is the
pressure at level 1 and the pressure at
level 2 are different and they this
difference is dependent on the height of
the cylinder that means with height the
value of pressure changes now if you
take if you consider the top of the
fluid top of the cylinder rather if you
consider the top of the cylinder and if
it is exposed to air then the value of P
1 would be nothing but the atmospheric
pressure that is P a so in that case we
can say that P 2 is equal to P a plus
Rho G H that means this P 2 represents
pressure at any point at a depth H so
pressure at any point is nothing but the
atmospheric pressure plus h ro G that
means the pressure at any point at a
depth of at any depth from the sea
is greater than the atmospheric pressure
by the quantity Rho G H so here we
understood that how the value of
pressure depends on how the value of
pressure changes with depth so what is
the conclusion it sees that the pressure
P add a depth below the surface of a
liquid open to the atmosphere is greater
than atmospheric pressure by an amount
goal G H right so now let us go ahead
and look at on sets which are related to
this the next conclusion is pressure is
independent of the cross-sectional or
base area or the shape of the one
container you can see here the pressure
is dependent only on this point typical
0hg h by rho is the density G is the
acceleration due to gravity which are
both constants and H is the height or
the depth at which we are measuring
pressure so we can say that pressure is
not dependent on cross sectional area or
base area or the shape of the container
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