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# Introduction to Fluid Mechanics 8th (physics, engineering)

## Educators

### Problem 1

A number of common substances are
Tar$\quad$ Sand
"silly Putty"$\quad$ Jello
Modeling clay $\quad$ Toothpaste
Wax $\quad$ Shaving cream
Some of these materials exhibit characteristics of both solid and fluid behavior under different conditions. Explain and give examples.

Khoobchandra A.

### Problem 2

Give a word statement of each of the five basic conservation laws stated in Section $1.4,$ as they apply to a system.

Khoobchandra A.

### Problem 3

The barrel of a bicycle tire pump becomes quite warm during use. Explain the mechanisms responsible for the temperature increase.

Khoobchandra A.

### Problem 4

Discuss the physics of skipping a stone across the water surface of a lake. Compare these mechanisms with a stone as it bounces after being thrown along a roadway.

Khoobchandra A.

### Problem 5

Make a guess at the order of magnitude of the mass (e.g., $0.01,0.1,1.0,10,100,$ or 1000 Ibm or $\mathrm{kg}$ ) of standard air that is in a room 10 ft by 10 ft by 8 ft, and then compute this mass in Ibm and kg to see how close your estimate was.

Khoobchandra A.

### Problem 6

A spherical tank of inside diameter 16 ft contains compressed oxygen at 1000 psia and $77^{\circ} \mathrm{F}$. What is the mass of the oxygen?

Khoobchandra A.

### Problem 7

Very small particles moving in fluids are known to experience a drag force proportional to speed. Consider a particle of net weight $W$ dropped in a fluid. The particle experiences a drag force, $F_{D}=k V,$ where $V$ is the particle speed. Determine the time required for the particle to accelerate from rest to 95 percent of its terminal speed, $V_{t},$ in terms of $k$ $W,$ and $g.$

Khoobchandra A.

### Problem 8

Consider again the small particle of Problem 1.7. Express the distance required to reach 95 percent of its terminal speed in percent terms of $g, k,$ and $W.$

Khoobchandra A.

### Problem 9

A cylindrical tank must be designed to contain 5 kg of compressed nitrogen at a pressure of 200 atm (gage) and $20^{\circ} \mathrm{C} .$ The design constraints are that the length must be twice the diameter and the wall thickness must be $0.5 \mathrm{cm}$ What are the external dimensions?

Khoobchandra A.

### Problem 10

In a combustion process, gasoline particles are to be dropped in air at $200^{\circ} \mathrm{F}$. The particles must drop at least 10 in. in 1 s. Find the diameter $d$ of droplets required for this. (The drag on these particles is given by $F_{D}=\pi \mu V d,$ where $V$ is the particle speed and $\mu$ is the air viscosity. To solve this problem, use Excel's Goal Seek.)

Khoobchandra A.

### Problem 11

For a small particle of styrofoam (1 $\mathrm{lbm} / \mathrm{ft}^{3}$ ) (spherical, with diameter $d=0.3 \mathrm{mm}$ ) falling in standard air at speed $V$ the drag is given by $F_{D}=3 \pi \mu V d,$ where $\mu$ is the air viscosity. Find the maximum speed starting from rest, and the time it takes to reach 95 percent of this speed. Plot the speed as a function of time.

Khoobchandra A.

### Problem 12

In a pollution control experiment, minute solid particles (typical mass $1 \times 10^{-13}$ slug ) are dropped in air. The terminal speed of the particles is measured to be $0.2 \mathrm{ft} / \mathrm{s}$. The drag of these particles is given by $F_{D}=k V,$ where $V$ is the instantaneous particle speed. Find the value of the constant $k$. Find the time required to reach 99 percent of terminal speed.

Khoobchandra A.

### Problem 13

For Problem $1.12,$ find the distance the particles travel before reaching 99 percent of terminal speed. Plot the distance traveled as a function of time.

Khoobchandra A.

### Problem 14

A sky diver with a mass of $70 \mathrm{kg}$ jumps from an aircraft. The aerodynamic drag force acting on the sky diver is known to be $F_{D}=k V^{2},$ where $k=0.25 \mathrm{N} \cdot \mathrm{s}^{2} / \mathrm{m}^{2} .$ Determine the maximum speed of free fall for the sky diver and the speed reached after $100 \mathrm{m}$ of fall. Plot the speed of the sky diver as a function of time and as a function of distance fallen.

Khoobchandra A.

### Problem 15

For Problem 1.14, the initial horizontal speed of the sky diver is $70 \mathrm{m} / \mathrm{s}$. As she falls, the $k$ value for the vertical drag remains as before, but the value for horizontal motion is $k=0.05 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2} .$ Compute and plot the $2 \mathrm{D}$ trajectory of the sky diver.

Khoobchandra A.

### Problem 16

The English perfected the longbow as a weapon after the Medieval period. In the hands of a skilled archer, the longbow was reputed to be accurate at ranges to $100 \mathrm{m}$ or more. If the maximum altitude of an arrow is less than $h=10 \mathrm{m}$ while traveling to a target $100 \mathrm{m}$ away from the archer, and neglecting air resistance, estimate the speed and angle at which the arrow must leave the bow. Plot the required release speed and angle as a function of height $h.$

Khoobchandra A.

### Problem 17

For each quantity listed, indicate dimensions using mass as a primary dimension, and give typical SI and English units:
(a) Power
(b) Pressure
(c) Modulus of elasticity
(d) Angular velocity
(e) Energy
(f) Moment of a force
(g) Momentum
(h) Shear stress
(i) Strain
(j) Angular momentum

Khoobchandra A.

### Problem 18

For each quantity listed, indicate dimensions using force as a primary dimension, and give typical SI and English units:
(a) Power
(b) Pressure
(c) Modulus of elasticity
(d) Angular velocity
(e) Energy
(f) Momentum
(g) Shear stress
(h) Specific heat
(i) Thermal expansion coefficient
(j) Angular momentum

Khoobchandra A.

### Problem 19

Derive the following conversion factors:
(a) Convert a viscosity of $1 \mathrm{m}^{2} / \mathrm{s}$ to $\mathrm{ft}^{2} / \mathrm{s}$
(b) Convert a power of $100 \mathrm{W}$ to horsepower.
(c) Convert a specific energy of $1 \mathrm{kJ} / \mathrm{kg}$ to $\mathrm{Btu} / \mathrm{lbm}$.

Khoobchandra A.

### Problem 20

Derive the following conversion factors:
(a) Convert a pressure of 1 psi to kPa.
(b) Convert a volume of 1 liter to gallons.
(c) Convert a viscosity of 1 lbf $\cdot \mathrm{s} / \mathrm{ft}^{2}$ to $\mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}$

Khoobchandra A.

### Problem 21

Derive the following conversion factors:
(a) Convert a specific heat of $4.18 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}$ to $\mathrm{Btu} / \mathrm{lbm} \cdot^{\circ} \mathrm{R}$
(b) Convert a speed of $30 \mathrm{m} / \mathrm{s}$ to mph.
(c) Convert a volume of $5.0 \mathrm{L}$ to in $^{3}$.

Khoobchandra A.

### Problem 22

Express the following in SI units:
(a) 5 acre $\cdot f t$
(b) 150 in $^{3} / \mathrm{s}$
(c) 3 gpm
(d) $3 \mathrm{mph} / \mathrm{s}$

Khoobchandra A.

### Problem 23

Express the following in SI units:
(a) $100 \mathrm{cfm}\left(\mathrm{ft}^{3} / \mathrm{min}\right)$
(b) 5 gal
(c) $65 \mathrm{mph}$
(d) 5.4 acres

Khoobchandra A.

### Problem 24

Express the following in BG units:
(a) $50 \mathrm{m}^{2}$
(b) $250 \mathrm{cc}$
(c) $100 \mathrm{kW}$
(d) $5 \mathrm{kg} / \mathrm{m}^{2}$

Khoobchandra A.

### Problem 25

Express the following in $\mathrm{BG}$ units:
(a) $180 \mathrm{cc} / \mathrm{min}$
(b) $300 \mathrm{kW} \cdot \mathrm{hr}$
(c) $50 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}$
(d) $40 \mathrm{m}^{2} \cdot \mathrm{hr}$

Khoobchandra A.

### Problem 26

While you're waiting for the ribs to cook, you muse about the propane tank of your barbecue. You're curious about the volume of propane versus the actual tank size. Find the liquid propane volume when full (the weight of the propane is specified on the tank). Compare this to the tank volume (take some measurements, and approximate the tank shape as a cylinder with a hemisphere on each end). Explain the discrepancy.

Khoobchandra A.

### Problem 27

A farmer needs $4 \mathrm{cm}$ of rain per week on his farm, with 10 hectares of crops. If there is a drought, how much water (L/min) will have to be supplied to maintain his crops?

Khoobchandra A.

### Problem 28

Derive the following conversion factors:
(a) Convert a volume flow rate in cubic inches per minute to cubic millimeters per minute.
(b) Convert a volume flow rate in cubic meters per second to gallons per minute (gpm).
(c) Convert a volume flow rate in liters per minute to gpm.
(d) Convert a volume flow rate of air in standard cubic feet per minute (SCFM) to cubic meters per hour. A standard cubic foot of gas occupies one cubic foot at standard temperature and pressure $\left(T=15^{\circ} \mathrm{C}$ and \right. $p=101.3 \mathrm{kPa}$ absolute

Khoobchandra A.

### Problem 29

The density of mercury is given as 26.3 slug/ft $^{3}$. Calculate the specific gravity and the specific volume in $\mathrm{m}^{3} / \mathrm{kg}$ of the mercury. Calculate the specific weight in $1 \mathrm{bf} / \mathrm{ft}^{3}$ on Earth and on the moon. Acceleration of gravity on the moon is $5.47 \mathrm{ft} / \mathrm{s}^{2}.$

Khoobchandra A.

### Problem 30

The kilogram force is commonly used in Europe as a unit of force. (As in the U.S. customary system, where 1 lbf is the force exerted by a mass of 1 lbm in standard gravity, 1 kgf is the force exerted by a mass of $1 \mathrm{kg}$ in standard gravity. Moderate pressures, such as those for auto or truck tires, are conveniently expressed in units of $\mathrm{kgf} / \mathrm{cm}^{2}$. Convert 32 psig to these units.

Khoobchandra A.

### Problem 31

In Section 1.6 we learned that the Manning equation computes the flow speed $V(\mathrm{m} / \mathrm{s})$ in a canal made from unfinished concrete, given the hydraulic radius $R_{h}(\mathrm{m}),$ the channel slope $S_{0}$ and a Manning resistance coefficient constant value $n \approx 0.014$ For a canal with $R_{h}=7.5 \mathrm{m}$ and a slope of $1 / 10,$ find the flow speed. Compare this result with that obtained using the same $n$ value, but with $R_{h}$ first converted to $\mathrm{ft}$, with the answer assumed to be in $\mathrm{ft} / \mathrm{s}$. Finally, find the value of $n$ if we wish to correctly use the equation for $B G$ units (and compute $V$ to check!).

Khoobchandra A.

### Problem 32

From thermodynamics, we know that the coefficient of performance of an ideal air conditioner $\left(C O P_{\text {ideal }}\right)$ is given by
$C O P_{\text {ideal }}=\frac{T_{L}}{T_{H}-T_{L}}$
where $T_{L}$ and $T_{H}$ are the room and outside temperatures (absolute). If an $\mathrm{AC}$ is to keep a room at $20^{\circ} \mathrm{C}$ when it is $40^{\circ} \mathrm{C}$ outside, find the $C O P_{\text {ideal }}$. Convert to an $E E R$ value, and compare this to a typical Energy Star-compliant $E E R$ value.

Khoobchandra A.

### Problem 33

The maximum theoretical flow rate (slug/s) through a supersonic nozzle is
$\dot{m}_{\max }=2.38 \frac{A_{t} p_{0}}{\sqrt{T_{0}}}$
where $A_{t}\left(\mathrm{ft}^{2}\right)$ is the nozzle throat area, $p_{0}(\mathrm{psi})$ is the tank pressure, and $T_{0}\left(^{\circ} \mathrm{R}\right)$ is the tank temperature. Is this equation dimensionally correct? If not, find the units of the 2.38 term. Write the equivalent equation in SI units.

Khoobchandra A.

### Problem 34

The mean free path $\lambda$ of a molecule of gas is the average distance it travels before collision with another molecule. It is given by
$\lambda=C \frac{m}{\rho d^{2}}$
where $m$ and $d$ are the molecule's mass and diameter, respectively, and $\rho$ is the gas density. What are the dimensions of constant $C$ for a dimensionally consistent equation?

Khoobchandra A.

### Problem 35

In Chapter 9 we will study aerodynamics and learn that the drag force $F_{D}$ on a body is given by
$F_{D}=\frac{1}{2} \rho V^{2} A C_{D}$
Hence the drag depends on speed $V$, fluid density $\rho$, and body size (indicated by frontal area $A$ ) and shape (indicated by drag coefficient $C_{D}$ ). What are the dimensions of $C_{D} ?$

Khoobchandra A.

### Problem 36

A container weighs 3.5 lbf when empty. When filled with water at $90^{\circ} \mathrm{F}$, the mass of the container and its contents is 2.5 slug. Find the weight of water in the container, and its volume in cubic feet, using data from Appendix A.

Khoobchandra A.

### Problem 37

An important equation in the theory of vibrations is
$m \frac{d^{2} x}{d t^{2}}+c \frac{d x}{d t}+k x=f(t)$
where $m(\mathrm{kg})$ is the mass and $x(\mathrm{m})$ is the position at time $t(\mathrm{s})$ For a dimensionally consistent equation, what are the dimensions of $c, k,$ and $f ?$ What would be suitable units for $c$ $k,$ and $f$ in the SI and $\mathrm{BG}$ systems?

Khoobchandra A.

### Problem 38

A parameter that is often used in describing pump performance is the specific speed, $N_{S_{c a}},$ given by
$N_{s_{a_{1}}}=\frac{N(\mathrm{rpm})[Q(\mathrm{gpm})]^{1 / 2}}{[H(\mathrm{ft})]^{3 / 4}}$
What are the units of specific speed? A particular pump has a specific speed of $2000 .$ What will be the specific speed in $\mathrm{SI}$ units (angular velocity in rad/s)?

Khoobchandra A.

### Problem 39

A particular pump has an "engineering" equation form of the performance characteristic equation given by $H(\mathrm{ft})=$ $1.5-4.5 \times 10^{-5}[Q(\mathrm{gpm})]^{2},$ relating the head $H$ and flow rate $Q .$ What are the units of the coefficients 1.5 and $4.5 \times 10^{-5} ?$ Derive an SI version of this equation.

Khoobchandra A.

### Problem 40

Calculate the density of standard air in a laboratory from the ideal gas equation of state. Estimate the experimental uncertainty in the air density calculated for standard conditions $\left(29.9 \text { in. of mercury and } 59^{\circ} \mathrm{F}\right)$ if the uncertainty in measuring the barometer height is ±0.1 in. of mercury and the uncertainty in measuring temperature is $\pm 0.5^{\circ} \mathrm{F}$. (Note that 29.9 in. of mercury corresponds to 14.7 psia.)

Khoobchandra A.

### Problem 41

Repeat the calculation of uncertainty described in Problem 1.40 for air in a hot air balloon. Assume the measured barometer height is $759 \mathrm{mm}$ of mercury with an uncertainty of $\pm 1 \mathrm{mm}$ of mercury and the temperature is $60^{\circ} \mathrm{C}$ with an uncertainty of $\pm 1^{\circ} \mathrm{C}$. [Note that $759 \mathrm{mm}$ of mercury corresponds to $101 \mathrm{kPa}(\mathrm{abs}) .]$

Khoobchandra A.

### Problem 42

The mass of the standard American golf ball is $1.62 \pm$ $0.01 \mathrm{oz}$ and its mean diameter is $1.68 \pm 0.01$ in. Determine the density and specific gravity of the American golf ball. Estimate the uncertainties in the calculated values.

Khoobchandra A.

### Problem 43

A can of pet food has the following internal dimensions:
$102 \mathrm{mm}$ height and $73 \mathrm{mm}$ diameter (each $\pm 1 \mathrm{mm}$ at odds of 20 to 1 ). The label lists the mass of the contents as 397 g. Evaluate the magnitude and estimated uncertainty of the density of the pet food if the mass value is accurate to ±1 g at the same odds.

Khoobchandra A.

### Problem 44

The mass flow rate in a water flow system determined by collecting the discharge over a timed interval is $0.2 \mathrm{kg} / \mathrm{s}$. The scales used can be read to the nearest $0.05 \mathrm{kg}$ and the stopwatch is accurate to 0.2 s. Estimate the precision with which the flow rate can be calculated for time intervals of (a) $10 \mathrm{s}$ and (b) $1 \mathrm{min}$

Khoobchandra A.

### Problem 45

$1 \mathrm{kg},$ and that the timer has a least count of $0.1 \mathrm{s}$. Estimate the time intervals and uncertainties in measured mass flow rate that would result from using $100,500,$ and 1000 mL beakers. Would there be any advantage in using the largest beaker? Assume the tare mass of the empty 1000 mL beaker is 500 g.

Khoobchandra A.

### Problem 46

The mass of the standard British golf ball is $45.9 \pm 0.3 \mathrm{g}$ and its mean diameter is $41.1 \pm 0.3 \mathrm{mm}$. Determine the density and specific gravity of the British golf ball. Estimate the uncertainties in the calculated values.

Khoobchandra A.

### Problem 47

The estimated dimensions of a soda can are $D=66.0 \pm$ $0.5 \mathrm{mm}$ and $H=110 \pm 0.5 \mathrm{mm} .$ Measure the mass of a full can and an empty can using a kitchen scale or postal scale. Estimate the volume of soda contained in the can. From your measurements estimate the depth to which the can is filled and the uncertainty in the estimate. Assume the value of $\mathrm{SG}=1.055,$ as supplied by the bottler.

Khoobchandra A.

### Problem 48

From Appendix A, the viscosity $\mu\left(\mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\right)$ of water at temperature $T(\mathrm{K})$ can be computed from $\mu=A 10^{B /(T-C)}$ where $A=2.414 \times 10^{-5} \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}, B=247.8 \mathrm{K},$ and $C=140 \mathrm{K}$
Determine the viscosity of water at $30^{\circ} \mathrm{C},$ and estimate its uncertainty if the uncertainty in temperature measurement is $\pm 0.5^{\circ} \mathrm{C}$

Khoobchandra A.

### Problem 49

Using the nominal dimensions of the soda can given in Problem $1.47,$ determine the precision with which the diameter and height must be measured to estimate the volume of the can within an uncertainty of ±0.5 percent.

Khoobchandra A.

### Problem 50

An enthusiast magazine publishes data from its road tests on the lateral acceleration capability of cars. The measurements are made using a 150 -ft-diameter skid pad. Assume the vehicle path deviates from the circle by $\pm 2 \mathrm{ft}$ and that the vehicle speed is read from a fifth-wheel speed-measuring system to ±0.5 mph. Estimate the experimental uncertainty in a reported lateral acceleration of $0.7 \mathrm{g}$. How would you improve the experimental procedure to reduce the uncertainty?

Khoobchandra A.

### Problem 51

The height of a building may be estimated by measuring the horizontal distance to a point on the ground and the angle from this point to the top of the building. Assuming these measurements are $L=100 \pm 0.5 \mathrm{ft}$ and $\theta=30 \pm 0.2^{\circ},$ estimate the height $H$ of the building and the uncertainty in the estimate. For the same building height and measurement uncertainties, use Excel's Solver to determine the angle (and the corresponding distance from the building at which measurements should be made to minimize the uncertainty in estimated height. Evaluate and plot the optimum measurement angle as a function of building height for $50 \leq H \leq 1000 \mathrm{ft}$

Khoobchandra A.

### Problem 52

An American golf ball is described in Problem 1.42 Assuming the measured mass and its uncertainty as given, determine the precision to which the diameter of the ball must be measured so the density of the ball may be estimated within an uncertainty of ±1 percent.

Khoobchandra A.
A syringe pump is to dispense liquid at a flow rate of $100 \mathrm{mL} / \mathrm{min} .$ The design for the piston drive is such that the uncertainty of the piston speed is 0.001 in./min, and the cylinder bore diameter has a maximum uncertainty of 0.0005 in. Plot the uncertainty in the flow rate as a function of cylinder bore. Find the combination of piston speed and bore that minimizes the uncertainty in the flow rate.