Physics 123 Unit #1 Review
I. Definitions & Facts
Density
Pressure
Barometer
Turbulence
Specific gravity (= material /
Atmosphere, bar, Pascal
Streamline, laminar flow
Gauge pressure
water
)
II. Mathematics & Tools
Fluids obey the equations of motion.
III. Basic Concepts
An ideal fluid: incompressible, laminar, irrotational, no viscosity.
Pascal’s principle: A pressure applied at any point in the fluid at rest is spread throughout the entire fluid.
Archimedes’ principle: The buoyant force equals the weight of the displaced liquid.
Fluid “in the air” is always at atmospheric pressure.
Bernoulli’s equation is a result of conservation of energy.
IV. Equations to Memorize
Continuity equation:
Bernoulli’s equation:
Av
P
flow rate,
gh ½ v 2
A!v1
A2v2
constant
Physics 123
Unit 1 Reading and Homework Assignments
15.1 Understand density and pressure. The device in Figure 15.2 only
works when fluid is not flowing. To measure pressure in a flow, we
have to be careful to separate the effects of the stream’s momentum
from the pressure. A better device would have a cap on it with little
holes to allow the fluid to enter.
The differential form Eq. 15.2 is not important to use, but you should
understand why it’s mentioned. Many units of pressure are in use.
Note that 1 atmosphere = 1.013 × 105 Pa, 1 bar = 1.00 × 105 Pa. In
vacuum work, we use 1 torr = 1 mm Hg = 1/760 atm.
15.2 Especially important are Eq. 15.4, Pascal’s Principle, and the
idea of hydraulics. Eq. 15.4 only works for static fluids that can’t be
compressed, so it does not work for the air.
Work Example 15.2 (without looking at the solution). If you like a challenge, try 15.4.
15.3 Know gauge pressure. We’ll see more of the barometer later.
What is the pressure at the surface level of the Hg in the air to the side of the barometer tube?
In the air just under the barometer tube?
15.4 Know Archimedes’ principle well (top of p. 466). Understand why it is true.
Do Quick Quizzes 15.6, 15.7 (same), and 15.8 (falls). Understand Examples 15.5, and 15.6.
15.5 Know what an “ideal fluid” is. Memorize the four items.
15.6 Note that what we say hereafter only applies to ideal fluids and “tubes of flow.” (15.7) is
very important. Recognize that Av is the volume flow per unit time.
15.7 Bernoulli’s equation is a consequence of energy conservation. Understand its derivation and
know it (15.9) well! It is important to remember that P + ½ v2 + gy is NOT the total pressure
(P is that). It is related to total energy/work per unit volume and has the name “total head” in
engineering.
Understand all three Examples. Try doing the “trick” in 15.9.
15.8 These are a few of many applications. You should understand all of these.
Homework 1-1 15.1-15.4
Questions:
Questions 15.5 (think of no water under the fish), 15.6, 15.7, 15.16
Essential Problems:
Problems 15.2, 15.6, 15.8, 15.21
Application Problems:
Problems 15.19, 15.25
Computational Problems:
A. The area to the right of the line x=a in the circle to the
right to the right is
A
1
R2
2
R 2 sin
1
a
R
a R2 a2
A cylindrical log of radius 10 cm is placed in water. If the density
of the wood is 0.5 g/ cm3, how much of the log is above the
surface of the water? Make a graph of this height as a function of
the density of the wood.
Contextual Problem:
B. A drink cup is placed on the seat of a car. As the car
undergoes a uniform acceleration, a, in the forward direction, what angle does the surface of the
liquid make to the horizontal? Find an expression for the pressure as a function of the distance, d,
measured perpendicularly from the surface of the liquid. (Remember that no experiment can
differentiate between a stationary elevator in a gravitational field and an elevator accelerating
upward with acceleration g. This is called the Principle of Equivalence and is the basis of
Einstein’s General Theory of Relativity.)
Homework 1-2 15.5-15.8
Questions:
Question 15.18, 15.25
A. Why does the water level in a toilet go up and down on a windy day? (A genuine
question from one of my neighbors...)
Essential Problems:
Problems 15.37
B. A tank 6 m high is drained by a hose extending horizontally from the bottom of the
tank. What is the velocity of the water coming out the hose? What is the pressure at the bottom of
the tank? What is the pressure of the water in the hose?
Application Problems:
Problems 15.35, 15.38, 15.43 (the vapor pressure is nearly zero)
Computational Problems:
C. A pipe runs horizontally along the ground. The pipe gradually tapers from a radius of 5
cm to a radius of 2 cm. Water comes out of the pipe at a speed of 1.25 m/s. Plot the speed of the
water within the nozzle as a function of the nozzle radius. Plot the pressure of the water also as a
function of the radius of the pipe.
Contextual Problems:
D. How long does it take to drain a half-gallon milk carton (filled with water) if we punch
a hole in the bottom with a pencil? You have to model the flow with a differential equation. To
do this, you will need to determine the flow rate when the water is at an arbitrary level, y, in the
container.
E. A hose comes horizontally outward from the bottom of a tank of water. When real
water flows through the hose, energy is lost in friction between the water and the hose. Assume
that the energy lost by a small volume of water, V, is proportional to the surface area of the
water in contact with the hose and is also proportional to the length of the hose.
(a) Find the surface area, A, of the water in terms of V.
(b) By appropriately modifying Bernoulli’s Equation, show that the velocity of the water coming
out the hose is:
v
2gh
r
where h is the height of the water, is a constant, is the length of the hose, and r is the radius of
the hose.
Physics 123 – Section 2
Sample Exam #1
Assume all fluids are ideal.
Possibly useful information:
The density of water is 1000 kg / m3.
1 bar = 1.00 × 105 Pa.
Use g = 9.80 m / s2.
Terminology
1. (5 points) Density
2. (5 points) Barometer
3. (5 points) What is the SI unit of pressure?
4. (5 points) What is streamline flow?
Conceptual Applications
5. (5 points) Bernoulli’s Equation, as we derived it, holds only for an ideal fluid. An ideal fluid must have each of
the following characteristics except:
A. it must have streamline flow.
B. it must be incompressible.
C. it must have no viscosity.
D. it must not have rotational motion.
E. All of the above are characteristics of an ideal fluid.
6. (5 points) An open-topped tank has water flowing out through
a hole in the bottom, as illustrated. The height of the water level
above the ground is h + H = 2.00 m. In order to increase the flow
rate through the hole, you could:
A. Increase H, leaving H + h constant.
B. Increase H, leaving h constant.
C. Increase the size of hole to allow more water to flow
through the hole.
D. Decrease the size of the hole in order to increase the
velocity of the water out of the hole.
E. None of the above.
7. (5 points) A hydraulic lift is best understood by applying
A. Pascal’s Principle.
B. Euler’s Angles.
C. Bernoulli’s Equation.
D. The continuity equation.
E. None of the above.
Equations and Tools
For each of the following equations:
A) Tell what each symbol means
B) Write a short problem which can be solved with this equation
The problem must be a word problem, not just “a=5, b=4, what is c?”
8. (5 points)
gh ½ v 2
P
9. (5 points)
P1
10. (5 points)
Av
gy1
P2
P0
gy2
Problems
We can usually evaluate how well you understand the physics of a problem by the way you attempt to solve the problem. Occasionally, however, you
may be unable to work a problem or a part of a problem even if you do understand the physics. If you cannot work a problem, please do the following
so that you may receive partial credit:
1. Discuss the basic physics concepts involved. Include enough detail that the grader will know if you understand the physics.
2. Write down the equation(s) you need to solve the problem. Explain the meaning of each symbol in the equations.
3. Explain as much of the problem as you understand.
4. Explain why you feel you have difficulties with the problem.
11. (25 points) A block of iron of dimensions 50.0 cm × 50.0 cm × 10.0 cm is placed in a large tank of mercury.
The specific gravity of iron is 7.86 and the specific gravity of mercury is 13.52.
(a) How much of the iron will float above the mercury?
(b) What is the maximum buoyant force this block would experience in mercury?
12. (25 points) A tank 1.50 m in height
is filled with water to a level of 1.32 m.
At the bottom of the tank a hose is
attached. The hose goes up at an angle of
55.0 with respect to the ground. The end
of the hose at a height of 58.0 cm above
the bottom of the tank. Atmospheric
pressure is 1.00 bar.
1.32 m
(a) Find the velocity of the water coming
out of the hose.
(b) Find the velocity of the water inside the hose at the bottom end of the hose (next to the tank).
(c) Find the pressure of the water inside the hose at the bottom end of the hose (next to the tank).
(d) Is the pressure you found in part (c) the same as the pressure at the bottom of the tank (not inside the hose)?
Briefly explain.
58.0 cm
13. (25 points) A person is drinking water from a
cylindrical tumbler by using a straw. The
dimensions of the system are shown in the diagram
to the right. The person maintains a gauge pressure
of –3000 Pa in her mouth. The diameter of the
straw is 5.00 mm.
(a) What is the flow rate of the water into the
person’s mouth? (Assume the straw is narrow
enough that you may ignore the velocity of the
water in the tumbler.)
(18 points)
(b) If the person keeps the pressure in her mouth constant while drinking, what will happen to the flow rate as the
level of water in the tumbler drops? (Give a descriptive answer, you do not need to calculate the flow rate as a
function of water level.) (7 points)
14. (25 points) A horizontal pipe consists of three sections as
shown in the diagram to the right. The right section, C, is open
to the air where water flows out at a speed of 12.00 m/s. (The
speed, of course, is the same just inside and just outside the
pipe.) Section A is not open to the air. The diameter of sections
A and C is 2.00 cm. The diameter of section B is 4.00 cm.
(a) What is the velocity of the water in section B? (8 points)
(b) What is the pressure of the water in section B? (9 points)
(c) What is the pressure of the water in section A? (8 points)
Answers.
Terminology
1. Mass per unit volume of a material.
2. A device used to measure air pressure.
3. Pascal ( or newton / m2. )
4. Flow that is not turbulent (or chaotic). Flow where fluid at a given point will always follow the same path.
Conceptual Applications
5. E.
6. C. 7.A.
Equations and Tools
8.
P
g
h
v
P
pressure
density of the fluid
gravitational acceleration
height
velocity of the flow
atmospheric pressure
If the level of water in a closed tank is 2.00 m above the ground and water leaving a hose at the ground level exits
with a speed of 4.0 m/s, what is the air pressure in the top of the tank?
9. (5 points)
P
g
y
pressure
density of the fluid
gravitational acceleration
height, vertical position
Water flows through a hose down a slope. The bottom end of the hose is open to air. If the top end of the hose is
1.24 m above the bottom end, what is the pressure at the top of the hose?
10. (5 points)
A
v
flow rate
cross sectional area of a pipe
velocity of the flow
Water flows from a 6" pipe at 4.0 m/s. How many liters of water flow through the tank in one minute?
Problems
11. (a) We know that if the iron floats, the buoyant force must be equal in magnitude to the weight of the iron.
Thus:
FB W
Hg Vdisplaced g
Fe VFe g
Hg A h
Fe A H
where A is the cross sectional area
h is the height of the block below the surface
H is the total height of the block
h
Fe
h
5.81 cm
Hg
Thus the amount out of the
water is 4.19 cm.
(b) The maximum buoyant force occurs when the block is totally submerged.
FB,max
Hg VFe g
13520 kg/ m 3 × 0.500 m× 0.500 m× 0.100 m× 9.80 m/ s 2
3.31× 103 N
12. (25 points)
(a) We apply Bernoulli’s Equation to the water surface in the tank and the end of the hose:
Total head at the surface: P0
2
Total head at the hose opening: P0
vopen
g htop
g hopen ½ vopen
2 g ( htop hopen )
2 ×9.80 ×0.740 m/ s
3.81 m/ s
(b) By the continuity
equation, the velocity of the water in the hose must be the same as in part (a), 3.81 m/s.
( c)
2
Total head at the bottom of the hose: Pbot ½ vopen
2
Pbot ½ vopen
Pbot
2
P0 ½ vopen
P0 g hopen
g hopen
1.00× 105 Pa 1000 × 9.80 × 0.580 Pa
106 kPa
(d) No, the pressure of the
hose is less than the pressure in the bottom of the tank. Since the height of the water in the bottom of the tank is
the same as in the bottom of the hose, and the velocity of the water increases as it enters the hose, the pressure
must drop to keep the total head constant.
13. (a)
First we find the velocity of the water by using Bernoulli’s Equation:
Total head at the surface: P0
g htop
g hs ½ v 2
Total head at the straw opening: P0 3000 Pa
½ v2
3000×
vopen
3000 Pa
g( hs htop )
2
2 ×9.80 ×( 0.27 0.18) m/ s
1000
2.06 m/ s
Letting A denote the cross
sectional area of he straw, the flow rate is:
A × 2.06 m/ s
0.0000196 m 2 ×2.06 m/ s
4.04 ×10
5
m3/s
(b) The difference in height between the mouth and the surface gets larger as so the flow rate will decrease. (It is
harder to pull the water up a larger distance.)
14. (a) We apply the continuity equation:
A C vc
vB
AC
AB
vC
A B vB
1
v
4 C
3.00 m/ s
(b) Note that the velocity of the water and the height of the
water are the same at C and just after the water leaves the hose. Therefore, the pressure at point A must be P0. By
Bernoulli’s Equation:
2
PB ½ vB P0 ½ vC
2
2
PB P0 ½ ( vC vB )
1.00 × 105 Pa ½ × 1000 × ( 144 9.0 ) Pa
168 kPa
velocity of the water in A must be the same as in C. Hence:
2
Total head at A: PA ½ vC
2
Total head at C: P0 ½ vC
PA P0
( c) By the continuity equation, the
Physics 123 Unit Summary
Unit #1
ID Number: ____________
Unit Score: (sum all boxes below):
1. Homework problems completed on time ________ × 10 = _________
Homework problems completed late
________ × 8 = _________
Sum of previous lines ________ ÷ 18 problems = __________, average score per problem
Homework score: Average score per problem × 2.5 =
( Maximum = 25 )
2. Work
Item
Hours
Item
Hours
__________________________
________
__________________________
________
__________________________
________
__________________________
________
__________________________
________
__________________________
________
__________________________
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__________________________
________
Total Hours (expected = 7, maximum = 12): ________
Work Score: Total Hours × 5.83 =
( Maximum = 70 )
3. Reading Checks
Reading score: 10 ×
# correct
=
3
( Maximum = 10 )
4. Quizzes
Quiz score: 10 ×
# taken
=
3
( Maximum = 10 )
5. Walk-in Labs
Lab score: = 15 × #completed =
( Maximum = 15 )