Transcript [] Chapter 14b

Chapter 14: Fluid mechanics
Reading assignment:
Chapter 15.1-15.3
Homework : (due Friday, Nov 4, 2005):
Problems:
Q5, Q10, 1, 3, 7, 12, 20,
• Fluids flow.
• Fluids are a collection of randomly arranged molecules held
together by by weak cohesive forces. (Unlike crystals (solids) which
arrange orderly on a lattice)
• Pressure, Pascal’s law
• Buoyant forces and Archimedes Principle
• Continuity equation
• Bernoulli’s equation
Buoyant forces and
Archimedes's Principle
Archimedes’s principle:
The magnitude of the buoyant
force equals the weight of the
fluid displaced by the object.
FB  m f  g   f  V f  g
Imagine holding two bricks under water.
Brick A is just beneath the surface of the
water, while brick B is at a greater depth.
The force needed to hold brick B in place is
1. larger
2. the same as
3. Smaller
than the force required to hold brick A in
place.
A 200-ton ship enters the lock of a canal.
The fit between the sides of the lock and the
ship is tight so that the weight of the water
left in the lock after it closes is much less
than 200 tons. Can the ship still float if the
quantity of water left in the lock is much less
than the ship’s weight?
1. Yes, as long as the water gets up to the
ship’s waterline.
2. No, the ship touches bottom because it
weighs more than the water in the lock.
Black board example 15.5
Archimedes’s principle
An iron cube weighs 9.80 N in air.
How much does it weigh in water.
The density of iron is 7.86·103
kg/m3. The density of water is
0.998·103 kg/m3
Buoyant forces and
Archimedes's Principle
Case 1:
Totally submerged objects.
Ftotal  B  Fg  (  f   o )Vo  g
If density of object is less than density of fluid: Object rises (accelerates up)
If density of object is greater than density of fluid: Object sinks. (accelerates down).
Archimedes’ principle can also be applied to balloons floating in
air (air can be considered a liquid)
Buoyant forces and
Archimedes's Principle
Case 2:
Floating objects.
Buoyant force of displaced liquid is balanced by
gravitational force.
Black board example 15.6
Homework 15.35
Archimedes’s principle
An ice slab has a thickness of 30.0 cm and a density of 917
kg/m3. When a 1100 kg car is resting on it the slab floats in
water ( = 998 kg/m3) with its top at the same level as the
water’s surface.
Find the area of the slab.
Two cups are filled to the same level with
water. One of the two cups has ice cubes
floating in it. Which weighs more?
1. The cup without ice cubes.
2. The glass with ice cubes.
3. The two weigh the same.
Two cups are filled to the same level with
water. One of the two cups has ice cubes
floating in it. When the ice cubes melt, in
which cup is the level of the water higher?
1. The cup without ice cubes.
2. The glass with ice cubes.
3. It is the same in both.
Two cups are filled to the same level with
water. One of the two cups has plastic balls
floating in it. If the density of the plastic balls
is less than that of ice, which of the two cups
weighs more?
1. The cup without plastic balls.
2. The cup with plastic balls.
3. The two weigh the same.
A boat carrying a large boulder is floating
on a lake.The boulder is thrown overboard
and sinks. The water level in the lake (with
respect to the shore)
1. rises.
2. drops.
3. remains the same.
A lead weight is fastened on top of a large
solid piece of Styrofoam that floats in a container
of water. Because of the weight of the
lead, the water line is flush with the top surface
of the Styrofoam. If the piece of Styrofoam
is turned upside down so that the
weight is now suspended underneath it,
1. the arrangement sinks.
2. the water line is below the top surface of
the Styrofoam.
3. the water line is still flush with the top
surface of the Styrofoam.
A lead weight is fastened to a large solid
piece of Styrofoam that floats in a container
of water. Because of the weight of the
lead, the water line is flush with the top surface
of the Styrofoam. If the piece of Styrofoam
is turned upside down, so that the
weight is now suspended underneath it, the
water level in the container
1. rises.
2. drops.
3. remains the same.
Consider an object floating in a container of
water. If the container is placed in an elevator
that accelerates upward,
1. more of the object is below water.
2. less of the object is below water.
3. there is no difference.
Consider an object that floats in water but
sinks in oil.When the object floats in water,
half of it is submerged. If we slowly pour oil
on top of the water so it completely covers
the object, the object
1. moves up.
2. stays in the same place.
3. moves down.
In the following section we assume:
- the flow of fluids is laminar (not turbulent)
 There are now vortices, eddies, turbulences. Water layers flow smoothly
over each other.
- the fluid has no viscosity (no friction).
 (Honey has high viscosity, water has low viscosity)
Equation of continuity
A1v1  A2v2  constant
For fluids flowing in a “pipe”, the product of area and
velocity is constant (big area  small velocity).
Why does the water emerging
from a faucet “neck down” as it
falls?
A circular hoop sits in a stream of water, oriented
perpendicular to the current. If the
area of the hoop is doubled, the flux (volume
of water per unit time) through it
1. decreases by a factor of 4.
2. decreases by a factor of 2.
3. remains the same.
4. increases by a factor of 2.
5. increases by a factor of 4.
Bernoulli’s
equation
Conservation of energy
1 2
P  v  gy  constant
2
1
1
2
2
P1  v1  gy1  P2  v2  gy2
2
2
Black board example 15.7
Homework 15.51
Bernoulli’s law
Water flows through a horizontal pipe, and then out into the
atmosphere at a speed of 15 m/s. The diameters of the left
and right sections of the pipe are 5.0 cm and 3.0 cm,
respectively.
(a) What volume of water flows into the atmosphere during a 10
min period?
(b) What is the flow speed of the water in the left section of the
pipe?
(c) What is the gauge pressure in the left section of the pipe?
Two hoses, one of 20-mm diameter, the other
of 15-mm diameter are connected one behind
the other to a faucet. At the open end
of the hose, the flow of water measures 10
liters per minute. Through which pipe does
the water flow faster?
1. the 20-mm hose
2. the 15-mm hose
3. The flow rate is the same in both cases.
4. The answer depends on which of the two
hoses comes first in the flow.