Today’s Lecture
Archimedes’ Principle
Continuity – Conservation of Mass
Bernoulli’s Principle
Static Fluid Pressure
Static fluid pressure does not depend on the
shape, total mass or surface area of the liquid.
volume = V = hA
weight = mg
The pressure on the sides of the containers results in a
force that has a vertical component. It is this vertical
component plus the force from the pressure at the bottom
that supports the total weight of the liquid!
Integrated Vertical Pressure
Force for Fluid in a Cone
For a cone of height h
and maximum radius R,
The pressure at a height z is
The area that is normal to
the gravitational force is
Integrated Vertical Pressure Force
for Fluid in a Cone
So let’s do the integral!
As expected the total vertical force is equal
to the weight of the water in the cone
Buoyant Force
In equilibrium the upward pressure
force, Fp, is equal to the weight of
the fluid, Fg. If the fluid is replaced
by a solid object, the pressure
force remains the same. In general
the gravitational force changes.
The net force on the solid object
depends on whether it is more or
less dense than the fluid.
Archimede’s principle: The buoyant
force on an object is equal to the weight
of the fluid displaced by the object.
Fp = mw g = ρ wVg
The buoyant force is applied to the center of gravity of the
displaced fluid (center of the submerged volume of the body).
Equal Volumes Feel Equal Buoyant Forces
Suppose you had equal sized balls of cork, aluminum and lead,
with respective densities of 0.2, 2.7, and 11.3 times the density of
water. If the volume of each is 10 cubic centimeters then their
masses are 2, 27, and 113 grams.
What are their apparent weights and masses? (Apparent
mass can be defined as apparent weight divided by g).
Each would displace 10 grams of water, yielding
apparent masses of -8 (the cork would accelerate
upward), 17 and 103 grams respectively.
Archimedes Principle:
The buoyant force on an object is equal to the
weight of the fluid displaced by the object.
Ship empty.
Ship loaded with
50 ton of iron.
Ship loaded with 50
ton of Styrofoam.
Volume of the submerged part of the ship (or any other floating
object) is equal to mass of the ship divided by the density of water:
Demo of Cartesian diver.
Archimedes Principle:
The buoyant force on an object is equal to the
weight of the fluid displaced by the object.
The density of ice is ρice=.917gm/cm3 .
The density of sea water is ρsea=1.027gm/cm3.
What fraction of iceberg is submerged?
So almost 90% of the iceberg is submerged,
not so good for those on the Titanic!
Archimedes Principle: Is the Crown Made of Gold?
Consider weighing a completely submerged object of mass m
and you want to determine its density.
The crown weighs 25N in air and 23.3N in water.
Hence ρ = (25/1.7)gm/cm3 = 14.7gm/cm3.
The density of Au is ρ = 19.3gm/cm3 and Ag is ρ = 10.5gm/cm3.
Crown is approximately 48%Au and 52%Ag.
Another Example of Archimedes Principle
An aerogel made from 1.00g of SiO2 weighs 7.35mN in air.
Assuming that the density of air is 1.2kg/m3, find density of
the aerogel.
We can use the expression that we derived last slide by replacing
ρwat with ρair (the aerogel is completely submerged ☺)
It is interesting to note that this aerogel is only 4 times as
dense as air!
Center of Gravity vs. Center of Buoyancy
stability of a boat against tipping over.
Gravitational force is
applied at the center of
gravity.
Buoyancy force is
applied to the center of
buoyancy, which is
located at the center of
the displaced water.
Center of gravity should
be below center of
buoyancy for stable
equilibrium.
There is something wrong with the picture on the right… What?
What About a Raft?
Is its center of gravity situated below the center of buoyancy?
How come, those people are so careless and are not afraid to turn over?
The center of buoyancy, or center of displaced water, changes as the raft tips.
Steady Flow in a River.
Velocity in each point is shown
by a vector with the length
proportional to the velocity.
Velocity gets higher, where the
river gets narrower.
Flow represented by streamlines,
that are everywhere tangent to flow
direction. Higher density of the
streamlines corresponds to higher
flow velocity.
In a steady flow there are no variations in velocity and pattern of flow
in time. Nevertheless, the actual fluid elements flowing past any
particular point in are always different. The fluid elements also get
accelerated and decelerated as they move along the streamlines.
Fluid Motion Obeys the Standard
Laws of Mechanics
Newton’s second law:
Newton’s law can often be a complicated differential equation,
particularly for fluids, it becomes the
Navier-Stokes equation:
Any way to make our life easier?!
Let’s try to use the laws of conservation!!
Motion of Fluids Obeys the Standard
Conservation Laws
Conservation of mass:
m = const
Conservation of momentum:
Conservation of energy:
2
mv
KE + PE =
+ mgh = const
2
Using the laws of conservation means doing
appropriate bookkeeping and doing algebra
instead of solving differential equations!
Flow Tube
A small tubelike region
bounded on its sides
by a continuous set of
streamlines
and on its ends by
small areas at right
angles to the
streamlines.
Cross-section areas
on the left and right
ends are:
A1 and A2.
Densities and
velocities are:
ρ1, ρ2 and v1, v2
Mass of fluid, Δm1, entering the tube
from the left over the time interval Δt
By mass conservation, over the time
interval Δt, the same mass, Δm2, is
exiting the tube from the right
Therefore ρ1A1v1 = ρ2A2v2 or ρAv = const.
everywhere along a flow tube.
This is the “Continuity Equation”.
If the fluid is incompressible and its
density, ρ, is constant we have
vA = const
Bernoulli’s Equation
How does the total energy of a small fluid element change, as it
moves inside the flow tube from cross-section 1 to cross-section 2?
Kinetic energy:
Potential energy:
1
2
2
ΔKE = m(v2 − v1 )
2
How does this change in the
total energy become possible?
There are external forces
originating from pressure of
the liquid outside the tube,
which do work on the fluid
element!
Positive work as it enters from the left
W1 = F1Δx1 = P1 A1Δx1
Negative work as it exits from the right
W2 = F2 Δx2 = − P2 A2 Δx2
The total energy balance
ΔKE + ΔPE = W1 + W2
Total Energy Balance
1
2
2
m(v2 − v1 ) + mg ( h2 − h1 ) = P1 A1Δx1 − P2 A2 Δx2
2
A1Δx1 = A2 Δx2
Incompressible fluids – constant density and volume
1 2
1 2
P1 + ρv1 + ρgh1 = P2 + ρv2 + ρgh2
2
2
1 2
P + ρv + ρgh = const
2
Bernoulli’s equation
Example: Bernoulli’s Equation
Flow velocity at bottom of tank
Supposed you have a sealed tank with a
pressure twice atmospheric pressure, 2Pa.
What is the outflow speed at the bottom
of the tank?
From Bernoulli’s equation
Example: Bernoulli’s Equation
Draining a tank
z
A tank of height h and cross sectional
area A is initially full of water. If a
small hole of area a<<A is cut in the
bottom, how long does it take, T, to
drain the tank?
Conservation of mass/volume
Bernoulli’s equation Conservation of energy
With the
result:
Simply A/a times the free
fall time.
Example of Bernoulli’s Equation
Venturi Flowmeter
Consider a Venturi flowmeter with
cross-sectional areas of A1 and A2
Solve for v1 in terms of the change in
pressure.
From Bernoulli’s equation and the Continuity equation,
Demo of Venturi tube.
Aerodynamic Lift
Does the lift come from Bernoulli effects?
or
Does the lift come from Reaction effects?
Actually both, it depends on the design of the wing!
What must happen is that there is momentum transfer
to the air stream DOWN which produces via a
reaction force an upward LIFT on the wing.
Aerodynamic Lift – Curve Balls
Does the lift come from Bernoulli effects?
Basically yes, but again it is the momentum
transfer that is important
In figure (a) the airflow is symmetric => no lift.
In figure (b) the rotation of the ball induces a momentum
transfer to the airflow and there is lift due to the reaction forces
It appears that backspin induces lift??
In baseball does backspin induce a rising effect??
Demo with styrofoam balls.
Wind Energy – Windmills
What about extracting
energy from the wind?
This is the power extracted if the efficiency were 100%.
The wind cannot come to a stop behind the windmill as flow would stop
Theoretically the best one can do is P = (2/3)3ρ v3A.
The best design is now 80% of this. For v=10m/s, P/A=285W/m2.
Fluids Summary
Pressure – Scalar quantity
- What is the origin of vector forces?
Pascal’s law – A change in pressure is experienced throughout
the fluid. Applied in hydraulic systems.
Uniform gravitational field => A pressure gradient of
The vertical component of the force normal to a surface means
Weight of the fluid
This leads to Archimedes's principle – for a submerged object
Fluids Summary
Conservation Laws: Continuity equation
Conservation of mass
For an incompressible fluid, ρ = const.
Conservation of mass/volume
Conservation Laws: Bernoulli equation
Conservation of energy