Introduction to Fluid Mechanics

Philip J. Pritchard, Robert W. Fox

Chapter 3

Fluid Statics - all with Video Answers

Educators

+ 10 more educators

Chapter Questions

Problem 1

Compressed nitrogen $(140 \mathrm{lbm})$ is stored in a spherical tank of diameter $D=2.5$ ft at a temperature of $77^{\circ} \mathrm{F}$. What is the pressure inside the tank? If the maximum allowable stress in the tank is $30 \mathrm{ksi}$, find the minimum theoretical wall thickness of the tank.

Vipender Yadav

Vipender Yadav

Numerade Educator

Problem 2

Because the pressure falls, water boils at a lower temperature with increasing altitude. Consequently, cake mixes and boiled eggs, among other foods, must be cooked different lengths of time. Determine the boiling temperature of water at 1000 and $2000 \mathrm{m}$ elevation on a standard day, and compare with the sea-level value.

Vipender Yadav

Numerade Educator

Problem 3

Ear "popping" is an unpleasant phenomenon sometimes experienced when a change in pressure occurs, for example in a fast-moving elevator or in an airplane. If you are in a two-seater airplane at $3000 \mathrm{m}$ and a descent of $100 \mathrm{m}$ causes your ears to "pop," what is the pressure change that your ears "pop" at, in millimeters of mercury? If the airplane now rises to $8000 \mathrm{m}$ and again begins descending, how far will the airplane descend before your ears "pop" again? Assume a U.S. Standard Atmosphere.

Vipender Yadav

Numerade Educator

Problem 4

When you are on a mountain face and boil water, you notice that the water temperature is $195^{\circ} \mathrm{F}$. What is your approximate altitude? The next day, you are at a location where it boils at $185^{\circ} \mathrm{F}$. How high did you climb between the two days? Assume a U.S. Standard Atmosphere.

Vipender Yadav

Numerade Educator

Problem 5

A 125 -mL cube of solid oak is held submerged by a tether as shown. Calculate the actual force of the water on the bottom surface of the cube and the tension in the tether.

Anand Jangid

Numerade Educator

Problem 6

The tube shown is filled with mercury at $20^{\circ} \mathrm{C}$. Calculate the force applied to the piston.

Supratim Pal

Numerade Educator

Problem 7

The following pressure and temperature measurements were taken by a meteorological balloon rising through the lower atmosphere:
$$\begin{array}{llllllllllll}
p(p \sin ) & 14.71 & 14.62 & 14.53 & 14.45 & 14.36 & 14.27 & 14.18 & 14.1 & 14.01 & 13.92 & 13.84 \\
T\left(^{\circ} \mathrm{F}\right) & 53.6 & 52 & 50.9 & 50.4 & 50.2 & 50 & 50.5 & 51.4 & 52.9 & 54 & 53.8
\end{array}$$
The initial values (top of table) correspond to ground level. Using the ideal gas law $(p=\rho R T \text { with } R=53.3 \mathrm{ft} \cdot 1 \mathrm{bf} /$ Ibm $\left.\cdot^{\circ} \mathrm{R}\right),$ compute and plot the variation of air density (in $\left.\mathrm{Ibm} / \mathrm{ft}^{3}\right)$ with height.

Vipender Yadav

Numerade Educator

Problem 8

A hollow metal cube with sides $100 \mathrm{mm}$ floats at the interface between a layer of water and a layer of SAE $10 \mathrm{W}$ oil such that $10 \%$ of the cube is exposed to the oil. What is the pressure difference between the upper and lower horizontal surfaces? What is the average density of the cube?

James Kiss

Numerade Educator

Problem 9

Your pressure gage indicates that the pressure in your cold tires is $0.25 \mathrm{MPa}$ (gage) on a mountain at an elevation of $3500 \mathrm{m}$ What is the absolute pressure? After you drive down to sea level, your tires have warmed to $25^{\circ} \mathrm{C}$. What pressure does your gage now indicate? Assume a U.S. Standard Atmosphere.

Vipender Yadav

Numerade Educator

Problem 10

An air bubble, 0.3 in. in diameter, is released from the regulator of a scuba diver swimming $100 \mathrm{ft}$ below the sea surface. (The water temperature is $86^{\circ} \mathrm{F}$.) Estimate the diameter of the bubble just before it reaches the water surface.

Vipender Yadav

Numerade Educator

Problem 11

A cube with 6 in. sides is suspended in a fluid by a wire. The top of the cube is horizontal and 8 in. below the free surface. If the cube has a mass of 2 slugs and the tension in the wire is $T=50.7$ Ibf, compute the fluid specific gravity, and from this determine the fluid. What are the gage pressures on the upper and lower surfaces?

Chai Santi

Numerade Educator

Problem 12

Assuming the bulk modulus is constant for seawater, derive an expression for the density variation with depth, $h$ below the surface. Show that the result may be written
\[
\rho \approx \rho_{0}+b h
\]
where $\rho_{0}$ is the density at the surface. Evaluate the constant
$b .$ Then, using the approximation, obtain an equation for the variation of pressure with depth below the surface. Determine the depth in feet at which the error in pressure predicted by the approximate solution is 0.01 percent.

Chai Santi

Numerade Educator

Problem 13

Oceanographic research vessels have descended to $6.5 \mathrm{mi}$ below sea level. At these extreme depths, the compressibility of seawater can be significant. One may model the behavior of seawater by assuming that its bulk modulus remains constant. Using this assumption, evaluate the deviations in density and pressure compared with values computed using the incompressible assumption at a depth, $h$ of $6.5 \mathrm{mi}$ in seawater. Express your answers as a percentage. Plot the results over the range $0 \leq h \leq 7$ mi.

Chai Santi

Numerade Educator

Problem 14

An inverted cylindrical container is lowered slowly beneath the surface of a pool of water. Air trapped in the container is compressed isothermally as the hydrostatic pressure increases. Develop an expression for the water height, $y,$ inside the container in terms of the container height, $H,$ and depth of submersion, $h .$ Plot $y / H$ versus $h / H$.

Chai Santi

Numerade Educator

Problem 15

You close the top of your straw with your thumb and lift the straw out of your glass containing Coke. Holding it vertically, the total length of the straw is $45 \mathrm{cm},$ but the Coke held in the straw is in the bottom $15 \mathrm{cm} .$ What is the pressure in the straw just below your thumb? Ignore any surface tension effects.

Vipender Yadav

Numerade Educator

Problem 16

A water tank filled with water to a depth of $16 \mathrm{ft}$ has in inspection cover $(1 \text { in. } \times 1 \text { in. })$ at its base, held in place by a plastic bracket. The bracket can hold a load of 9 lbf. Is the bracket strong enough? If it is, what would the water depth have to be to cause the bracket to break?

Vipender Yadav

Numerade Educator

Problem 17

A container with two circular vertical tubes of diameters $d_{1}=39.5 \mathrm{mm}$ and $d_{2}=12.7 \mathrm{mm}$ is partially filled with mercury. The equilibrium level of the liquid is shown in the left diagram. A cylindrical object made from solid brass is placed in the larger tube so that it floats, as shown in the right diagram. The object is $D=37.5 \mathrm{mm}$ in diameter and $H=76.2$ $\mathrm{mm}$ high. Calculate the pressure at the lower surface needed to float the object. Determine the new equilibrium level, $h,$ of the mercury with the brass cylinder in place.

Penny Riley

Numerade Educator

Problem 18

A partitioned tank as shown contains water and mercury. What is the gage pressure in the air trapped in the left chamber? What pressure would the air on the left need to be pumped to in order to bring the water and mercury free surfaces level?

Vipender Yadav

Numerade Educator

Problem 19

In the tank of Problem $3.18,$ if the opening to atmosphere on the right chamber is first sealed, what pressure would the air on the left now need to be pumped to in order to bring the water and mercury free surfaces level? (Assume the air trapped in the right chamber behaves isothermally.)

Keshav Singh

Numerade Educator

Problem 20

Consider the two-fluid manometer shown. Calculate the applied pressure difference.

Vipender Yadav

Numerade Educator

Problem 21

A manometer is formed from glass tubing with uniform inside diameter, $D=6.35 \mathrm{mm},$ as shown. The U-tube is partially filled with water. Then $\psi=3.25 \mathrm{cm}^{3}$ of Meriam red oil is added to the left side. Calculate the equilibrium height, $H,$ when both legs of the U-tube are open to the atmosphere.

Narayan Hari

Numerade Educator

Problem 22

The manometer shown contains water and kerosene. With both tubes open to the atmosphere, the free-surface elevations differ by $H_{0}=20.0 \mathrm{mm} .$ Determine the elevation difference when a pressure of $98.0 \mathrm{Pa}$ (gage) is applied to the right tube.

Chai Santi

Numerade Educator

Problem 23

The manometer shown contains two liquids. Liquid $A$ has $\mathrm{SG}=0.88$ and liquid $B$ has $\mathrm{SG}=2.95 .$ Calculate the deflection, $h,$ when the applied pressure difference is $p_{1}-p_{2}=18 \mathrm{lbf} / \mathrm{ft}^{2}$.

Vipender Yadav

Numerade Educator

Problem 24

Determine the gage pressure in kPa at point $a$, if liquid $A$ has $\mathrm{SG}=1.20$ and liquid $B$ has $\mathrm{SG}=0.75$. The liquid surrounding point $a$ is water, and the tank on the left is open to the atmosphere.

Vipender Yadav

Numerade Educator

Problem 25

An engineering research company is evaluating using a sophisticated $\$ 80,000$ laser system between two large water storage tanks. You suggest that the job can be done with a $\$ 200$ manometer arrangement. Oil less dense than water can be used to give a significant amplification of meniscus movement; a small difference in level between the tanks will cause a much larger deflection in the oil levels in the manometer. If you set up a rig using Meriam red oil as the manometer fluid, determine the amplification factor that will be seen in the rig.

Narayan Hari

Numerade Educator

Problem 26

Water flows downward along a pipe that is inclined at $30^{\circ}$ below the horizontal, as shown. Pressure difference $p_{A}-p_{B}$ is due partly to gravity and partly to friction. Derive an algebraic expression for the pressure difference. Evaluate the pressure difference if $L=5 \mathrm{ft}$ and $h=6$ in.

Narayan Hari

Numerade Educator

Problem 27

Consider a tank containing mercury, water, benzene, and air as shown. Find the air pressure (gage). If an opening is made in the top of the tank, find the equilibrium level of the mercury in the manometer.

Keshav Singh

Numerade Educator

Problem 28

A reservoir manometer has vertical tubes of diameter $D=18 \mathrm{mm}$ and $d=6 \mathrm{mm} .$ The manometer liquid is Meriam red oil. Develop an algebraic expression for liquid deflection $L$ in the small tube when gage pressure $\Delta p$ is applied to the reservoir. Evaluate the liquid deflection when the applied pressure is equivalent to $25 \mathrm{mm}$ of water (gage).

Narayan Hari

Numerade Educator

Problem 29

A rectangular tank, open to the atmosphere, is filled with water to a depth of $2.5 \mathrm{m}$ as shown. A U-tube manometer is connected to the tank at a location $0.7 \mathrm{m}$ above the tank bottom. If the zero level of the Meriam blue manometer fluid is $0.2 \mathrm{m}$ below the connection, determine the deflection $l$ after the manometer is connected and all air has been removed from the connecting leg.

Vipender Yadav

Numerade Educator

Problem 30

A reservoir manometer is calibrated for use with a liquid of specific gravity 0.827 . The reservoir diameter is $5 / 8$ in. and the (vertical) tube diameter is $3 / 16$ in. Calculate the required distance between marks on the vertical scale for 1 in. of water pressure difference.

Keshav Singh

Numerade Educator

Problem 31

The manometer fluid of Problem 3.29 is replaced with mercury (same zero level). The tank is sealed and the air pressure is increased to a gage pressure of 0.5 atm. Determine the deflection $l$.

Keshav Singh

Numerade Educator

Problem 32

The inclined-tube manometer shown has $D=96 \mathrm{mm}$ and $d=8 \mathrm{mm}$. Determine the angle, $\theta$, required to provide a 5: 1 increase in liquid deflection, $L,$ compared with the total deflection in a regular U-tube manometer. Evaluate the sensitivity of this inclined-tube manometer.

Keshav Singh

Numerade Educator

Problem 33

The inclined-tube manometer shown has $D=76 \mathrm{mm}$ and $d=8 \mathrm{mm},$ and is filled with Meriam red oil. Compute the angle, $\theta,$ that will give a 15 -cm oil deflection along the inclined tube for an applied pressure of $25 \mathrm{mm}$ of water (gage). Determine the sensitivity of this manometer.

Chai Santi

Numerade Educator

Problem 34

A barometer accidentally contains 6.5 inches of water on top of the mercury column (so there is also water vapor instead of a vacuum at the top of the barometer). On a day when the temperature is $70^{\circ} \mathrm{F}$, the mercury column height is 28.35 inches (corrected for thermal expansion). Determine the barometric pressure in psia. If the ambient temperature increased to $85^{\circ} \mathrm{F}$ and the barometric pressure did not change, would the mercury column be longer, be shorter, or remain the same length? Justify your answer.

Keshav Singh

Numerade Educator

Problem 35

A student wishes to design a manometer with better sensitivity than a water-filled U-tube of constant diameter. The student's concept involves using tubes with different diameters and two liquids, as shown. Evaluate the deflection $h$ of this manometer, if the applied pressure difference is $\Delta p=250 \mathrm{N} / \mathrm{m}^{2}$.

Chai Santi

Numerade Educator

Problem 36

A water column stands $50 \mathrm{mm}$ high in a 2.5 -mm diameter glass tube. What would be the column height if the surface tension were zero? What would be the column height in a 1.0 -mm diameter tube?

Keshav Singh

Numerade Educator

Problem 37

If the tank of Problem 3.29 is sealed tightly and water drains slowly from the bottom of the tank, determine the deflection, $l$, after the system has attained equilibrium.

Chai Santi

Numerade Educator

Problem 38

Consider a small-diameter open-ended tube inserted at the interface between two immiscible fluids of different densities. Derive an expression for the height difference $\Delta h$ between the interface level inside and outside the tube in terms of tube diameter $D,$ the two fluid densities $\rho_{1}$ and $\rho_{2},$ and the surface tension $\sigma$ and angle $\theta$ for the two fluids' interface. If the two fluids are water and mercury, find the height difference if the tube diameter is 40 mils $(1 \mathrm{mil}=0.001 \text { in. })$.

Keshav Singh

Numerade Educator

Problem 39

You have a manometer consisting of a tube that is 0.5 in. inner diameter (ID). On one side, the manometer leg contains mercury, 0.6 in. $^{3}$ of an oil $(\mathrm{SG}=1.4),$ and 0.2 in. $^{3}$ of air as a bubble in the oil. The other leg contains only mercury. Both legs are open to the atmosphere and are in a static condition. An accident occurs in which 0.2 in. $^{3}$ of the oil and the air bubble are removed from one leg. How much do the mercury height levels change?

Nicholas Mogoi

Numerade Educator

Problem 40

Compare the height due to capillary action of water exposed to air in a circular tube of diameter $D=0.5 \mathrm{mm}$ and between two infinite vertical parallel plates of gap
\[
a=0.5 \mathrm{mm}
\].

Keshav Singh

Numerade Educator

Problem 41

Two vertical glass plates 12 in. $\times 12$ in. are placed in an open tank containing water. At one end the gap between the plates is 0.004 in., and at the other it is 0.080 in. Plot the curve of water height between the plates from one end of the pair to the other.

Chai Santi

Numerade Educator

Problem 42

Based on the atmospheric temperature data of the U.S. Standard Atmosphere of Fig. $3.3,$ compute and plot the pressure variation with altitude, and compare with the pressure data of Table A.3.

Chai Santi

Numerade Educator

Problem 43

On a certain calm day, a mild inversion causes the atmospheric temperature to remain constant at $30^{\circ} \mathrm{C}$ between sea level and 5000 -m altitude. Under these conditions, (a) calculate the elevation change for which a 3 percent reduction in air pressure occurs, (b) determine the change of elevation necessary to effect a 5 percent reduction in density, and (c) plot $p_{2} / p_{1}$ and $\rho_{2} / \rho_{1}$ as a function of $\Delta z$.

Keshav Singh

Numerade Educator

Problem 44

At ground level in Denver, Colorado, the atmospheric pressure and temperature are $83.2 \mathrm{kPa}$ and $25^{\circ} \mathrm{C}$. Calculate the pressure on Pike's Peak at an elevation of $2690 \mathrm{m}$ above the city assuming (a) an incompressible and (b) an adiabatic atmosphere. Plot the ratio of pressure to ground level pressure in Denver as a function of elevation for both cases.

Chai Santi

Numerade Educator

Problem 45

The Martian atmosphere behaves as an ideal gas with mean molecular mass of 32.0 and constant temperature of $200 \mathrm{K} .$ The atmospheric density at the planet surface is $\rho=0.015 \mathrm{kg} / \mathrm{m}^{3}$ and Martian gravity is $3.92 \mathrm{m} / \mathrm{s}^{2}$. Calculate the density of the Martian atmosphere at height $z=20 \mathrm{km}$ above the surface. Plot the ratio of density to surface density as a function of elevation. Compare with that for data on the Earth's atmosphere.

Keshav Singh

Numerade Educator

Problem 46

A door $1 \mathrm{m}$ wide and $1.5 \mathrm{m}$ high is located in a plane vertical wall of a water tank. The door is hinged along its upper edge, which is $1 \mathrm{m}$ below the water surface. Atmospheric pressure acts on the outer surface of the door and at the water surface.
(a) Determine the magnitude and line of action of the total resultant force from all fluids acting on the door.
(b) If the water surface gage pressure is raised to 0.3 atm, what is the resultant force and where is its line of action?
(c) Plot the ratios $F / F_{9}$ and $y^{\prime} / y_{c}$ for different values of the surface pressure ratio $p_{s} / p_{\text {atm }}$. $\left(F_{0}\right.$ is the resultant force when $\left.p_{s}=p_{\text {atm }}\right)$.

Chai Santi

Numerade Educator

Problem 47

A door $1 \mathrm{m}$ wide and $1.5 \mathrm{m}$ high is located in a plane vertical wall of a water tank. The door is hinged along its upper edge, which is $1 \mathrm{m}$ below the water surface. Atmospheric pressure acts on the outer surface of the door. (a) If the pressure at the water surface is atmospheric, what force must be applied at the lower edge of the door in order to keep the door from opening? (b) If the water surface gage pressure is raised to 0.5 atm, what force must be applied at the lower edge of the door to keep the door from opening? (c) Find the ratio $F / F_{0}$ as a function of the surface pressure ratio $$p_{s} / p_{\text {atm }^{*}}\left(F_{0}\right.$$ is the force required when $\left.p_{s}=p_{\text {atm }} .\right)$

Chai Santi

Numerade Educator

Problem 48

A hydropneumatic elevator consists of a piston-cylinder assembly to lift the elevator cab. Hydraulic oil, stored in an accumulator tank pressurized by air, is valved to the piston as needed to lift the elevator. When the elevator descends, oil is returned to the accumulator. Design the least expensive accumulator that can satisfy the system requirements. Assume the lift is 3 floors, the maximum load is 10 passengers, and the maximum system pressure is $800 \mathrm{kPa}$ (gage). For column bending strength, the piston diameter must be at least $150 \mathrm{mm}$. The elevator cab and piston have a combined mass of $3000 \mathrm{kg}$, and are to be purchased. Perform the analysis needed to define, as a function of system operating pressure, the piston diameter, the accumulator volume and diameter, and the wall thickness. Discuss safety features that your company should specify for the complete elevator system. Would it be preferable to use a completely pneumatic design or a completely hydraulic design? Why?

Chai Santi

Numerade Educator

Problem 49

Find the pressures at points $A, B,$ and $C,$ as shown in the figure, and in the two air cavities.

Chai Santi

Numerade Educator

Problem 50

Semicircular plane gate $A B$ is hinged along $B$ and held by horizontal force $F_{A}$ applied at $A$. The liquid to the left of the gate is water. Calculate the force $F_{A}$ required for equilibrium.

Keshav Singh

Numerade Educator

Problem 51

A triangular access port must be provided in the side of a form containing liquid concrete. Using the coordinates and dimensions shown, determine the resultant force that acts on the port and its point of application.

Chai Santi

Numerade Educator

Problem 52

A plane gate of uniform thickness holds back a depth of water as shown. Find the minimum weight needed to keep the gate closed.

Keshav Singh

Numerade Educator

Problem 53

Consider a semicylindrical trough of radius $R$ and length $L$ Develop general expressions for the magnitude and line of action of the hydrostatic force on one end, if the trough is partially filled with water and open to atmosphere. Plot the results (in nondimensional form) over the range of water depth $0 \leq d / R \leq 1$.

Chai Santi

Numerade Educator

Problem 54

A rectangular gate (width $w=2 \mathrm{m}$ ) is hinged as shown, with a stop on the lower edge. At what depth $H$ will the gate tip?

Keshav Singh

Numerade Educator

Problem 55

For a mug of tea $(65 \mathrm{mm}$ diameter), imagine it cut symmetrically in half by a vertical plane. Find the force that each half experiences due to an $80-\mathrm{mm}$ depth of tea.

Narayan Hari

Numerade Educator

Problem 56

Gates in the Poe Lock at Sault Ste. Marie, Michigan, close a channel $W=34 \mathrm{m}$ wide, $L=360 \mathrm{m}$ long, and $D=10 \mathrm{m}$ decp. The geometry of one pair of gates is shown; each gate is hinged at the channel wall. When closed, the gate edges are forced together at the center of the channel by water pressure. Evaluate the force exerted by the water on gate $A$. Determine the magnitude and direction of the force components exerted by the gate on the hinge. (Neglect the weight of the gate.)

Chai Santi

Numerade Educator

Problem 57

A section of vertical wall is to be constructed from readymix concrete poured between forms. The wall is to be $3 \mathrm{m}$ high, $0.25 \mathrm{m}$ thick, and $5 \mathrm{m}$ wide. Calculate the force exerted by the ready-mix concrete on each form. Determine the line of application of the force.

Keshav Singh

Numerade Educator

Problem 58

A window in the shape of an isosceles triangle and hinged at the top is placed in the vertical wall of a form that contains liquid concrete. Determine the minimum force that must be applied at point $D$ to keep the window closed for the configuration of form and concrete shown. Plot the results over the range of concrete depth $0 \leq c \leq a$.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 59

Solve Example 3.6 again using the two separate pressures method. Consider the distributed force to be the sum of a force $F_{1}$ caused by the uniform gage pressure and a force $F_{2}$ caused by the liquid. Solve for these forces and their lines of action. Then sum moments about the hinge axis to calculate $F_{r}$.

Chai Santi

Numerade Educator

Problem 60

A large open tank contains water and is connected to a 6-ft-diameter conduit as shown. A circular plug is used to seal the conduit. Determine the magnitude, direction, and location of the force of the water on the plug.

Narayan Hari

Numerade Educator

Problem 61

What holds up a car on its rubber tires? Most people would tell you that it is the air pressure inside the tires. However, the air pressure is the same all around the hub (inner wheel), and the air pressure inside the tire therefore pushes down from the top as much as it pushes up from below, having no net effect on the hub. Resolve this paradox by explaining where the force is that keeps the car off the ground.

Keshav Singh

Numerade Educator

Problem 62

The circular access port in the side of a water standpipe has a diameter of $0.6 \mathrm{m}$ and is held in place by eight bolts evenly spaced around the circumference. If the standpipe diameter is $7 \mathrm{m}$ and the center of the port is located $12 \mathrm{m}$ below the free surface of the water, determine (a) the total force on the port and
(b) the appropriate bolt diameter.

Narayan Hari

Numerade Educator

Problem 63

As water rises on the left side of the rectangular gate, the gate will open automatically. At what depth above the hinge will this occur? Neglect the mass of the gate.

Chai Santi

Numerade Educator

Problem 64

The gate $A O C$ shown is 6 ft wide and is hinged along $O$ Neglecting the weight of the gate, determine the force in bar $A B .$ The gate is sealed at $C$.

Chai Santi

Numerade Educator

Problem 65

The gate shown is $3 \mathrm{m}$ wide and for analysis can be considered massless. For what depth of water will this rectangular gate be in equilibrium as shown?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 66

The gate shown is hinged at $H .$ The gate is $3 \mathrm{m}$ wide normal to the plane of the diagram. Calculate the force required at $A$ to hold the gate closed.

Chai Santi

Numerade Educator

Problem 67

A long, square wooden block is pivoted along one edge. The block is in equilibrium when immersed in water to the depth shown. Evaluate the specific gravity of the wood, if friction in the pivot is negligible.

Keshav Singh

Numerade Educator

Problem 68

A solid concrete dam is to be built to hold back a depth $D$ of water. For ease of construction the walls of the dam must be planar. Your supervisor asks you to consider the following dam cross-sections: a rectangle, a right triangle with the hypotenuse in contact with the water, and a right triangle with the vertical in contact with the water. She wishes you to determine which of these would require the least amount of concrete. What will your report say? You decide to look at one more possibility: a nonright triangle, as shown. Develop and plot an expression for the cross-section area $A$ as a function of $a,$ and find the minimum cross-sectional area.

Chai Santi

Numerade Educator

Problem 69

For the geometry shown, what is the vertical force on the dam? The steps are $0.5 \mathrm{m}$ high, $0.5 \mathrm{m}$ deep, and $3 \mathrm{m}$ wide.

Keshav Singh

Numerade Educator

Problem 70

For the dam shown, what is the vertical force of the water on the dam?

Keshav Singh

Numerade Educator

Problem 71

The gate shown is $1.5 \mathrm{m}$ wide and pivoted at $O$ $a=1.0 \mathrm{m}^{-2}, D=1.20 \mathrm{m},$ and $H=1.40 \mathrm{m} .$ Determine (a) the magnitude and moment of the vertical component of the force about $O,$ and (b) the horizontal force that must be applied at point $A$ to hold the gate in position.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 72

The parabolic gate shown is $2 \mathrm{m}$ wide and pivoted at $O$ $c=0.25 \mathrm{m}^{-1}, D=2 \mathrm{m},$ and $H=3 \mathrm{m} .$ Determine
(a) the magnitude and line of action of the vertical force on the gate due to the water, (b) the horizontal force applied at $A$ required to maintain the gate in equilibrium, and
(c) the vertical force applied at $A$ required to maintain the gate in equilibrium.

Chai Santi

Numerade Educator

Problem 73

Liquid concrete is poured into the form $(R=2 \mathrm{ft}) .$ The form is $w=15$ ft wide normal to the diagram. Compute the magnitude of the vertical force exerted on the form by the concrete, and specify its line of action.

Chai Santi

Numerade Educator

Problem 74

An open tank is filled with water to the depth indicated. Atmospheric pressure acts on all outer surfaces of the tank. Determine the magnitude and line of action of the vertical component of the force of the water on the curved part of the tank bottom.

Chai Santi

Numerade Educator

Problem 75

A spillway gate formed in the shape of a circular arc is $w \mathrm{m}$ wide. Find the magnitude and line of action of the vertical component of the force due to all fluids acting on the gate.

Chai Santi

Numerade Educator

Problem 76

A dam is to be constructed using the cross-section shown. Assume the dam width is $w=160 \mathrm{ft}$. For water height $H=9 \mathrm{ft},$ calculate the magnitude and line of action of the vertical force of water on the dam face. Is it possible for water forces to overturn this dam? Under what circumstances will this happen?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 77

A Tainter gate used to control water flow from the Uniontown Dam on the Ohio River is shown; the gate width is $w=35 \mathrm{m} .$ Determine the magnitude, direction, and line of action of the force from the water acting on the gate.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 78

A gate, in the shape of a quarter-cylinder, hinged at $A$ and sealed at $B,$ is $3 \mathrm{m}$ wide. The bottom of the gate is $4.5 \mathrm{m}$ below the water surface. Determine the force on the stop at $B$ if the gate is made of concrete; $R=3 \mathrm{m}$.

Pawan Yadav

Numerade Educator

Problem 79

Consider the cylindrical weir of diameter $3 \mathrm{m}$ and length $6 \mathrm{m} .$ If the fluid on the left has a specific gravity of $1.6,$ and on the right has a specific gravity of $0.8,$ find the magnitude and direction of the resultant force.

Chai Santi

Numerade Educator

Problem 80

A cylindrical weir has a diameter of $3 \mathrm{m}$ and a length of
$6 \mathrm{m} .$ Find the magnitude and direction of the resultant force acting on the weir from the water.

Chai Santi

Numerade Educator

Problem 81

A cylindrical weir has a diameter of $3 \mathrm{m}$ and a length of $6 \mathrm{m} .$ Find the magnitude and direction of the resultant force acting on the weir from the water.

Chai Santi

Numerade Educator

Problem 82

A curved surface is formed as a quarter of a circular cylinder with $R=0.750 \mathrm{m}$ as shown. The surface is $w=3.55$ $\mathrm{m}$ wide. Water stands to the right of the curved surface to depth $H=0.650 \mathrm{m} .$ Calculate the vertical hydrostatic force on the curved surface. Evaluate the line of action of this force. Find the magnitude and line of action of the horizontal force on the surface.

Chai Santi

Numerade Educator

Problem 83

If you throw an anchor out of your canoe but the rope is too short for the anchor to rest on the bottom of the pond, will your canoe float higher, lower, or stay the same? Prove your answer.

Keshav Singh

Numerade Educator

Problem 84

A curved submerged surface, in the shape of a quarter cylinder with radius $R=1.0 \mathrm{ft}$ is shown. The form can withstand a maximum vertical load of 350 lbf before breaking. The width is $w=4$ ft. Find the maximum depth $H$ to which the form may be filled. Find the line of action of the vertical force for this condition. Plot the results over the range of concrete depth $0 \leq H \leq R$.

Chai Santi

Numerade Educator

Problem 85

The cross-sectional shape of a canoe is modeled by the curve $y=a x^{2},$ where $a=1.2 \mathrm{ft}^{-1}$ and the coordinates are in feet. Assume the width of the canoe is constant at $w=2 \mathrm{ft}$ over its entire length $L=18 \mathrm{ft}$. Set up a general algebraic expression relating the total mass of the canoe and its contents to distance $d$ between the water surface and the gunwale of the floating canoe. Calculate the maximum total mass allowable without swamping the canoe.

Chai Santi

Numerade Educator

Problem 86

The cylinder shown is supported by an incompressible liquid of density $\rho,$ and is hinged along its length. The cylinder, of mass $M,$ length $L,$ and radius $R,$ is immersed in liquid to depth $H .$ Obtain a general expression for the cylinder specific gravity versus the ratio of liquid depth to cylinder radius, $\alpha=H / R,$ needed to hold the cylinder in equilibrium for $0 \leq \alpha<1 .$ Plot the results.

Chai Santi

Numerade Educator

Problem 87

A canoe is represented by a right semicircular cylinder, with $R=1.2 \mathrm{ft}$ and $L=17 \mathrm{ft} .$ The canoe floats in water that is $d=1 \mathrm{ft}$ deep. Set up a general algebraic expression for the total mass (canoe and contents) that can be floated, as a function of depth. Evaluate for the given conditions. Plot the results over the range of water depth $0 \leq d \leq R$.

Chai Santi

Numerade Educator

Problem 88

A glass observation room is to be installed at the corner of the bottom of an aquarium. The aquarium is filled with seawater to a depth of 35 ft. The glass is a segment of a sphere, radius $5 \mathrm{ft}$ mounted symmetrically in the corner. Compute the magnitude and direction of the net force on the glass structure.

Chai Santi

Numerade Educator

Problem 89

A hydrometer is a specific gravity indicator, the value being indicated by the level at which the free surface intersects the stem when floating in a liquid. The 1.0 mark is the level when in distilled water. For the unit shown, the immersed volume in distilled water is $15 \mathrm{cm}^{3} .$ The stem is $6 \mathrm{mm}$ in diameter. Find the distance, $h,$ from the 1.0 mark to the surface when the hydrometer is placed in a nitric acid solution of specific gravity 1.5.

Keshav Singh

Numerade Educator

Problem 90

Find the specific weight of the sphere shown if its volume is $0.025 \mathrm{m}^{3}$. State all assumptions. What is the equilibrium position of the sphere if the weight is removed?

Chai Santi

Numerade Educator

Problem 91

The fat-to-muscle ratio of a person may be determined from a specific gravity measurement. The measurement is made by immersing the body in a tank of water and measuring the net weight. Develop an expression for the specific gravity of a person in terms of their weight in air, net weight in water, and $\mathrm{SG}=f(T)$ for water.

Narayan Hari

Numerade Educator

Problem 92

Quantify the statement, "Only the tip of an iceberg shows (in seawater)."

James Kiss

Numerade Educator

Problem 93

An open tank is filled to the top with water. A steel cylindrical container, wall thickness $\delta=1 \mathrm{mm},$ outside diameter $D=100 \mathrm{mm},$ and height $H=1 \mathrm{m},$ with an open top, is gently placed in the water. What is the volume of water that overflows from the tank? How many 1 kg weights must be placed in the container to make it sink? Neglect surface tension effects.

Chai Santi

Numerade Educator

Problem 94

Quantify the experiment performed by Archimedes to identify the material content of King Hiero's crown. Assume you can measure the weight of the king's crown in air, $W_{a}$ and the weight in water, $W_{w^{*}}$ Express the specific gravity of the crown as a function of these measured values.

Keshav Singh

Numerade Educator

Problem 95

Gas bubbles are released from the regulator of a submerged scuba diver. What happens to the bubbles as they rise through the seawater? Explain.

Keshav Singh

Numerade Educator

Problem 96

Hot-air ballooning is a popular sport. According to a recent article, "hot-air volumes must be large because air heated to $150^{\circ} \mathrm{F}$ over ambient lifts only $0.018 \mathrm{lbf} / \mathrm{ft}^{3} \mathrm{com}$ pared to 0.066 and 0.071 for helium and hydrogen, respectively." Check these statements for sea-level conditions. Calculate the effect of increasing the hot-air maximum temperature to $250^{\circ} \mathrm{F}$ above ambient.

Chai Santi

Numerade Educator

Problem 97

Hydrogen bubbles are used to visualize water flow streaklines in the video, Flow Visualization. A typical hydrogen bubble diameter is $d=0.001$ in. The bubbles tend to rise slowly in water because of buoyancy; eventually they reach terminal speed relative to the water. The drag force of the water on a bubble is given by $F_{D}=3 \pi \mu V d,$ where $\mu$ is the viscosity of water and $V$ is the bubble speed relative to the water. Find the buoyancy force that acts on a hydrogen bubble immersed in water. Estimate the terminal speed of a bubble rising in water.

Keshav Singh

Numerade Educator

Problem 98

It is desired to use a hot air balloon with a volume of $320,000 \mathrm{ft}^{3}$ for rides planned in summer morning hours when the air temperature is about $48^{\circ} \mathrm{F}$. The torch will warm the air inside the balloon to a temperature of $160^{\circ} \mathrm{F}$. Both inside and outside pressures will be "standard" (14.7 psia). How much mass can be carried by the balloon (basket, fuel, passengers, personal items, and the component of the balloon itself) if neutral buoyancy is to be assured? What mass can be carried by the balloon to ensure vertical takeoff acceleration of $2.5 \mathrm{ft} / \mathrm{s}^{2} ?$ For this, consider that both balloon and inside air have to be accelerated, as well as some of the surrounding air (to make way for the balloon). The rule of thumb is that the total mass subject to acceleration is the mass of the balloon, all its appurtenances, and twice its volume of air. Given that the volume of hot air is fixed during the flight, what can the balloonists do when they want to go down?

Chai Santi

Numerade Educator

Problem 99

Scientific balloons operating at pressure equilibrium with the surroundings have been used to lift instrument packages to extremely high altitudes. One such balloon, filled with helium, constructed of polyester with a skin thickness of $0.013 \mathrm{mm}$ and a diameter of $120 \mathrm{m},$ lifted a payload of $230 \mathrm{kg}$. The specific gravity of the skin material is 1.28. Determine the altitude to which the balloon would rise. Assume that the helium used in the balloon is in thermal equilibrium with the ambient air, and that the balloon is a perfect sphere.

Keshav Singh

Numerade Educator

Problem 100

A helium balloon is to lift a payload to an altitude of $40 \mathrm{km},$ where the atmospheric pressure and temperature are 3.0 mbar and $-25^{\circ} \mathrm{C},$ respectively. The balloon skin is polyester with specific gravity of 1.28 and thickness of 0.015 $\mathrm{mm} .$ To maintain a spherical shape, the balloon is pressurized to a gage pressure of 0.45 mbar. Determine the maximum balloon diameter if the allowable tensile stress in the skin is limited to $62 \mathrm{MN} / \mathrm{m}^{2}$. What payload can be carried?

Anand Jangid

Numerade Educator

Problem 101

A block of volume $0.025 \mathrm{m}^{3}$ is allowed to sink in water as shown. A circular rod $5 \mathrm{m}$ long and $20 \mathrm{cm}^{2}$ in crosssection is attached to the weight and also to the wall. If the rod mass is $1.25 \mathrm{kg}$ and the rod makes an angle of 12 degrees with the horizontal at equilibrium, what is the mass of the block?

James Kiss

Numerade Educator

Problem 102

The stem of a glass hydrometer used to measure specific gravity is $5 \mathrm{mm}$ in diameter. The distance between marks on the stem is $2 \mathrm{mm}$ per 0.1 increment of specific gravity. Calculate the magnitude and direction of the error introduced by surface tension if the hydrometer floats in kerosene. (Assume the contact angle between kerosene and glass is $0^{\circ}$.)

Chai Santi

Numerade Educator

Problem 103

A sphere, of radius $R,$ is partially immersed, to depth $d,$ in a liquid of specific gravity SG. Obtain an algebraic expression for the buoyancy force acting on the sphere as a function of submersion depth $d$. Plot the results over the range of water depth $0 \leq d \leq 2 R$.

Chai Santi

Numerade Educator

Problem 104

If the mass $M$ in Problem 3.101 is released from the rod, at equilibrium how much of the rod will remain submerged? What will be the minimum required upward force at the tip of the rod to just lift it out of the water?

Chai Santi

Numerade Educator

Problem 105

In a logging operation, timber floats downstream to a lumber mill. It is a dry year, and the river is running low, as low as $60 \mathrm{cm}$ in some locations. What is the largest diameter log that may be transported in this fashion (leaving a minimum $5 \mathrm{cm}$ clearance between the log and the bottom of the river)? For the wood, $\mathrm{SG}=0.8$.

Keshav Singh

Numerade Educator

Problem 106

A sphere of radius 1 in., made from material of specific gravity of $\mathrm{SG}=0.95,$ is submerged in a tank of water. The sphere is placed over a hole of radius 0.075 in., in the tank bottom. When the sphere is released, will it stay on the bottom of the tank or float to the surface?

Chai Santi

Numerade Educator

Problem 107

A cylindrical timber, with $D=1 \mathrm{ft}$ and $L=15 \mathrm{ft}$, is weighted on its lower end so that it floats vertically with $10 \mathrm{ft}$ submerged in seawater. When displaced vertically from its equilibrium position, the timber oscillates or "heaves" in a vertical direction upon release, Estimate the frequency of oscillation in this heave mode. Neglect viscous effects and water motion.

Keshav Singh

Numerade Educator

Problem 108

You are in the Bermuda Triangle when you see a bubble plume eruption (a large mass of air bubbles, similar to a foam ) off to the side of the boat. Do you want to head toward it and be part of the action? What is the effective density of the water and air bubbles in the drawing on the right that will cause the boat to sink? Your boat is 10 ft long, and weight is the same in both cases.

Chai Santi

Numerade Educator

Problem 109

A bowl is inverted symmetrically and held in a dense fluid, $\mathrm{SG}=15.6,$ to a depth of $200 \mathrm{mm}$ measured along the centerline of the bowl from the bowl rim. The bowl height is $80 \mathrm{mm},$ and the fluid rises $20 \mathrm{mm}$ inside the bowl. The bowl is $100 \mathrm{mm}$ inside diameter, and it is made from an old clay recipe, $\mathrm{SG}=6.1 .$ The volume of the bowl itself is about $0.9 \mathrm{L}$ What is the force required to hold it in place?

Prashant Bana

Numerade Educator

Problem 110

In the "Cartesian diver" child's toy, a miniature "diver" is immersed in a column of liquid. When a diaphragm at the top of the column is pushed down, the diver sinks to the bottom. When the diaphragm is released, the diver again rises. Explain how the toy might work.

Keshav Singh

Numerade Educator

Problem 111

Consider a conical funnel held upside down and submerged slowly in a container of water. Discuss the force needed to submerge the funnel if the spout is open to the atmosphere. Compare with the force needed to submerge the funnel when the spout opening is blocked by a rubber stopper.

Keshav Singh

Numerade Educator

Problem 112

Three steel balls (each about half an inch in diameter ) lie at the bottom of a plastic shell floating on the water surface in a partially filled bucket. Someone removes the steel balls from the shell and carefully lets them fall to the bottom of the bucket, leaving the plastic shell to float empty. What happens to the water level in the bucket? Does it rise, go down, or remain unchanged? Explain.

Keshav Singh

Numerade Educator

Problem 113

A proposed ocean salvage scheme involves pumping air into "bags" placed within and around a wrecked vessel on the sea bottom. Comment on the practicality of this plan, supporting your conclusions with analyses.

Keshav Singh

Numerade Educator

Problem 114

A cylindrical container, similar to that analyzed in Example 3.10 (on the $\mathrm{Web}$ ), is rotated at a constant rate of $2 \mathrm{Hz}$ about its axis. The cylinder is $0.5 \mathrm{m}$ in diameter and initially contains water that is $0.3 \mathrm{m}$ deep. Determine the height of the liquid free surface at the center of the container. Does your answer depend on the density of the liquid? Explain.

Keshav Singh

Numerade Educator

Problem 115

A crude accelerometer can be made from a liquidfilled U-tube as shown. Derive an expression for the liquid level difference $h$ caused by an acceleration $\vec{a},$ in terms of the tube geometry and fluid properties.

Keshav Singh

Numerade Educator

Problem 116

A rectangular container of water undergoes constant acceleration down an incline as shown. Determine the slope of the free surface using the coordinate system shown.

Chai Santi

Numerade Educator

Problem 117

The U-tube shown is filled with water at $T=68^{\circ} \mathrm{F}$. It is sealed at $A$ and open to the atmosphere at $D$. The tube is rotated about vertical axis $A B$ at 1600 rpm. For the dimensions shown, would cavitation occur in the tube?

Chai Santi

Numerade Educator

Problem 118

If the U-tube of Problem 3.117 is spun at $300 \mathrm{rpm}$ what will the pressure be at $A$ ? If a small leak appears at $A$ how much water will be lost at $D ?$

Chai Santi

Numerade Educator

Problem 119

A centrifugal micromanometer can be used to create small and accurate differential pressures in air for precise measurement work. The device consists of a pair of parallel disks that rotate to develop a radial pressure difference. There is no flow between the disks. Obtain an expression for pressure difference in terms of rotation speed, radius, and air density. Evaluate the speed of rotation required to develop a differential pressure of $8 \mu \mathrm{m}$ of water using a device with a $50 \mathrm{mm}$ radius.

Keshav Singh

Numerade Educator

Problem 120

A test tube is spun in a centrifuge. The tube support is mounted on a pivot so that the tube swings outward as rotation speed increases. At high speeds, the tube is nearly horizontal. Find (a) an expression for the radial component of acceleration of a liquid clement located at radius $r,$ (b) the radial pressure gradient $d p / d r,$ and (c) the required angular velocity to generate a pressure of $250 \mathrm{MPa}$ in the bottom of a test tube containing water. (The free surface and bottom radii are 50 and $130 \mathrm{mm},$ respectively.

Keshav Singh

Numerade Educator

Problem 121

A rectangular container, of base dimensions $0.4 \mathrm{m} \times$ $0.2 \mathrm{m}$ and height $0.4 \mathrm{m},$ is filled with water to a depth of $0.2 \mathrm{m} ;$ the mass of the empty container is $10 \mathrm{kg} .$ The container is placed on a plane inclined at $30^{\circ}$ to the horizontal. If the coefficient of sliding friction between the container and the plane is $0.3,$ determine the angle of the water surface relative to the horizontal.

Chai Santi

Numerade Educator

Problem 122

If the container of Problem 3.121 slides without friction, determine the angle of the water surface relative to the horizontal. What is the slope of the free surface for the same acceleration up the plane?

Chai Santi

Numerade Educator

Problem 123

A cubical box, $80 \mathrm{cm}$ on a side, half-filled with oil $(\mathrm{SG}=0.80),$ is given a constant horizontal acceleration of 0.25 $g$ parallel to one edge. Determine the slope of the free surface and the pressure along the horizontal bottom of the box.

Chai Santi

Numerade Educator

Problem 124

Gas centrifuges are used in one process to produce enriched uranium for nuclear fuel rods. The maximum peripheral speed of a gas centrifuge is limited by stress considerations to about $950 \mathrm{ft} / \mathrm{s}$. Assume a gas centrifuge containing uranium hexafluoride gas, with molecular gas $M_{m}=$ $352,$ and ideal gas behavior. Develop an expression for the ratio of maximum pressure to pressure at the centrifuge axis. Evaluate the pressure ratio for a gas temperature of $620^{\circ} \mathrm{F}$.

Keshav Singh

Numerade Educator

Problem 125

A pail, $400 \mathrm{mm}$ in diameter and $400 \mathrm{mm}$ deep, weighs $15 \mathrm{N}$ and contains $200 \mathrm{mm}$ of water. The pail is swung in a vertical circle of $1-\mathrm{m}$ radius at a speed of $5 \mathrm{m} / \mathrm{s}$. Assume the water moves as a rigid body. At the instant when the pail is at the top of its trajectory, compute the tension in the string and the pressure on the bottom of the pail from the water.

Chai Santi

Numerade Educator

Problem 126

A partially full can of soda is placed at the outer edge of a child's merry-go-round, located $R=5$ ft from the axis of rotation. The can diameter and height are 2.5 in. and 5 in. respectively. The can is half full, and the soda has specific gravity $\mathrm{SG}=1.05 .$ Evaluate the slope of the liquid surface in the can if the merry-go-round spins at 20 rpm. Calculate the spin rate at which the can would spill, assuming no slippage between the can bottom and the merry-go-round. Would the can most likely spill or slide off the merry-go-round?

Anand Jangid

Numerade Educator

Problem 127

When a water polo ball is submerged below the surface in a swimming pool and released from rest, it is observed to pop out of the water. How would you expect the height to which it rises above the water to vary with depth of submersion below the surface? Would you expect the same results for a beach ball? For a table-tennis ball?

Chai Santi

Numerade Educator

Problem 128

Cast iron or steel molds are used in a horizontalspindle machine to make tubular castings such as liners and tubes. A charge of molten metal is poured into the spinning mold. The radial acceleration permits nearly uniformly thick wall sections to form. A steel liner, of length $L=6 \mathrm{ft}$, outer radius $r_{o}=6$ in. and inner radius $r_{i}=4$ in., is to be formed by this process. To attain nearly uniform thickness, the angular velocity should be at least 300 rpm. Determine (a) the resulting radial acceleration on the inside surface of the liner and (b) the maximum and minimum pressures on the surface of the mold.

Chai Santi

Numerade Educator

Problem 129

The analysis of Problem 3.121 suggests that it may be possible to determine the coefficient of sliding friction between two surfaces by measuring the slope of the free surface in a liquid-filled container sliding down an inclined surface. Investigate the feasibility of this idea.

Keshav Singh

Numerade Educator