Transcript Lecture-21-11

Chapter 15
Fluids
Pressure
Pressure is force
per unit area
Pressure is not the same as force!
The same force applied over a
smaller area results in greater
pressure – think of poking a
balloon with your finger and
then with a needle.
Pressure is a useful concept for discussing fluids,
because fluids distribute their force over an area
Pressure and Depth
Pressure increases with depth in a fluid due to the
increasing mass of the fluid above it.
Pressure and depth
Pressure in a fluid includes pressure on the fluid
surface (usually atmospheric pressure)
Pressure depends only on depth and external pressure
(and not on shape of fluid column)
Equilibrium only when pressure is the same
Unequal pressure will cause liquid flow:
must have same
pressure at A and B
Oil is less dense, so a taller column of oil is
needed to counter a shorter column of water
Pascal’s principle
An external pressure applied to an enclosed fluid is
transmitted to every point within the fluid.
Hydraulic lift
F1 / A1 = P = F2 / A2
Assume fluid is “incompressible”
Pascal’s principle
Hydraulic lift
F1 / A1 = P = F2 / A2
Are we getting “something for nothing”?
Assume fluid is “incompressible”
so Work in = Work out!
Buoyancy
A fluid exerts a net upward force on any object it
surrounds, called the buoyant force.
This force is due to the
increased pressure at the
bottom of the object
compared to the top.
Consider a cube
with sides = L
Archimedes’ Principle
Archimedes’ Principle: An object completely immersed
in a fluid experiences an upward buoyant force equal in
magnitude to the weight of fluid displaced by the object.
Buoyant Force When a Volume V is
Submerged in a Fluid of Density ρfluid
Fb = ρfluid gV
Q: Does buoyant force
depend on depth?
a) yes
b) no
Measuring the Density
The King must know: is his crown true gold?
Get the mass from
W = T1 = mg
Get the volume from
( T1 - T2 ) = V(ρwater g)
The crown-maker makes a crown for the king. Archimedes weighs the crown and
determines that its weight in air is 5.54 N and that its weight in water is 5.05 N.
Should the crown-maker maker be paid or ???
The crown-maker makes a crown for the king. Archimedes weighs the crown and
determines that its weight in air is 5.54 N and that its weight in water is 5.05 N.
Should the crown-maker maker be paid or ???
Off with his head!!
Applications of Archimedes’ Principle
An object floats when it displaces an
amount of fluid equal to its weight.
wood
block
brass
block
equivalent
mass of water
equivalent
mass of water
Can Brass Float?
An object made of material that is denser than
water can float only if it has indentations or
pockets of air that make its average density less
than that of water.
brass
block
equivalent
mass of water
An object floats when it displaces an
amount of fluid equal to its weight.
Applications of Archimedes’ Principle
The fraction of an object that is submerged when it
is floating depends on the densities of the object
and of the fluid.
Cartesian Diver
Think of a weighted balloon submerged in water
How will the balloon change when pressure
goes up?
Did its weight change when pressure went up?
So when pressure goes up:
- will it float higher?
- or will it sink?
Wood in Water
Two beakers are filled to the brim with water. A wooden
block is placed in the beaker 2 so it floats. (Some of the
water will overflow the beaker and run off). Both beakers are
then weighed. Which scale reads a larger weight?
b
a
c
same for both
Wood in Water
Two beakers are filled to the brim with water. A wooden
block is placed in the beaker 2 so it floats. (Some of the
water will overflow the beaker and run off). Both beakers are
then weighed. Which scale reads a larger weight?
The block in 2 displaces an amount of
b
a
water equal to its weight, because it is
floating. That means that the weight
of the overflowed water is equal to the
weight of the block, and so the beaker
in 2 has the same weight as that in 1.
c
same for both
Wood in Water II
Earth
A block of wood floats in a container of
water as shown on the right. On the
Moon, how would the same block of wood
float in the container of water?
Moon
a
b
c
Wood in Water II
Earth
A block of wood floats in a container of
water as shown on the right. On the
Moon, how would the same block of wood
float in the container of water?
Moon
A floating object displaces a
weight of water equal to the
object’s weight. On the Moon,
the wooden block has less
weight, but the water itself
also has less weight.
a
b
c
A wooden block is held at the bottom of a bucket filled with water.
The system is then dropped into free fall, at the same time the force
pushing the block down is also removed. What will happen to the
block?
a) the block will float to the surface.
b) the block will stay where it is.
c) the block will oscillate between the
surface and the bottom of the bucket
A wooden block is held at the bottom of a bucket filled with water.
The system is then dropped into free fall, at the same time the force
pushing the block down is also removed. What will happen to the
block?
a) the block will float to the surface.
b) the block will stay where it is.
c) the block will oscillate between the
surface and the bottom of the bucket
Bouyant force is created by a change of pressure with depth.
Pressure is created by the weight of water being held up.
In free-fall, nothing is being held up! No apparent weight!
A wooden block of cross-sectional area A, height H, and
density ρ1 floats in a fluid of density ρf .
If the block is displaced downward and then released, it
will oscillate with simple harmonic motion. Find the
period of its motion.
h
A wooden block of cross-sectional area A, height H, and
density ρ1 floats in a fluid of density ρf .
If the block is displaced downward and then released, it
will oscillate with simple harmonic motion. Find the
period of its motion.
Vertical force: Fy = (hA)g ρf - (HA)g ρ1
at equilibrium: h0 = Hρ1/ρf
h = h0 - y
Total restoring force: Fy = -(Agρf)y
Analogous to mass on a spring, with κ = Agρf
h
Fluid Flow and Continuity
Continuity tells us that whatever the mass of fluid in a
pipe passing a particular point per second, the same mass
must pass every other point in a second. The fluid is not
accumulating or vanishing along the way.
Volume per
unit time
This means that where the pipe is
narrower, the fluid is flowing faster
Continuity and Compressibility
Most gases are easily compressible; most liquids
are not. Therefore, the density of a liquid may be
treated as constant (not true for a gas).
mass flow is
conserved
volume flow is
conserved
Bernoulli’s Equation
When a fluid moves from a wider area of a pipe to a narrower
one, its speed increases; therefore, work has been done on it.
The kinetic energy of a fluid element is:
Equating the work done to the increase in
kinetic energy gives:
Bernoulli’s Equation
Where fluid moves faster, pressure is lower
Bernoulli’s Equation
If a fluid flows in a pipe of constant diameter, but
changes its height, there is also work done on it
against the force of gravity.
Equating the work
done with the change
in potential energy
gives:
Bernoulli’s Equation
The general case, where both height and speed
may change, is described by Bernoulli’s equation:
This equation is essentially a statement of
conservation of energy in a fluid.
Dynamic lift
v high
P low
v low
P high
Aircraft wing
Applications of Bernoulli’s Equation
If a hole is punched in the side of an open
container, the outside of the hole and the top of
the fluid are both at atmospheric pressure.
Since the fluid inside
the container at the
level of the hole is at
higher pressure, the
fluid has a horizontal
velocity as it exits.