For the two-dimensional stress field shown in Fig. P2.1 it is found that

\[

\sigma_{x x}=3000 \text { lbf/ft }^{2} \quad \sigma_{y y}=2000 \text { lbf/ft }^{2} \quad \sigma_{x y}=500 \text { lbf/ft }^{2}

\]

Find the shear and normal stresses (in $\operatorname{lbf} / \mathrm{ft}^{2}$ ) acting on plane $A A$ cutting through the element at a $30^{\circ}$ angle as shown.

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For the two-dimensional stress field shown in Fig. P2.1 suppose that

\[

\sigma_{x x}=2000 \mathrm{lbf} / \mathrm{ft}^{2} \quad \sigma_{y y}=3000 \mathrm{lbf} / \mathrm{ft}^{2} \quad \sigma_{n}(A A)=2500 \mathrm{lbf} / \mathrm{ft}^{2}

\]

Compute $(a)$ the shear stress $\sigma_{x y}$ and $(b)$ the shear stress on plane $A A$

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A vertical, clean, glass piezometer tube has an inside diameter of $1 \mathrm{mm}$. When pressure is applied, water at $20^{\circ} \mathrm{C}$ rises into the tube to a height of $25 \mathrm{cm} .$ After correcting for surface tension, estimate the applied pressure in Pa.

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Pressure gages, such as the Bourdon gage in Fig. P2.4, are calibrated with a deadweight piston. If the Bourdon gage is designed to rotate the pointer 10 degrees for every 2 psig of internal pressure, how many degrees does the pointer rotate if the piston and weight together total 44 newtons?

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Denver, Colorado, has an average altitude of $5300 \mathrm{ft}$ On a standard day (Table A.6), pressure gage $A$ in a laboratory experiment reads $83 \mathrm{kPa}$ and gage $B$ reads 105 kPa. Express these readings in gage pressure or vacuum pressure (Pa), whichever is appropriate.

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Any pressure reading can be expressed as a length or head, $h=$ plpg. What is standard sea-level pressure expressed in $(a)$ ft of glycerin, $(b)$ inHg, $(c) \mathrm{m}$ of water and $(d) \mathrm{mm}$ of ethanol? Assume all fluids are at $20^{\circ} \mathrm{C}$

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La Paz, Bolivia is at an altitude of approximately $12,000 \mathrm{ft}$. Assume a standard atmosphere. How high would the liquid rise in a methanol barometer, assumed at $20^{\circ} \mathrm{C} ?$ Hint: Don't forget the vapor pressure.

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A diamond mine is two miles below sea level. (a) Estimate the air pressure at this depth. ( $b$ ) If a barometer, accurate to $1 \mathrm{mm}$ of mercury, is carried into this mine, how accurately can it estimate the depth of the mine? List your assumptions carefully.

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A storage tank, 26 ft in diameter and 36 ft high, is Hed with SAE $30 \mathrm{W}$ oil at $20^{\circ} \mathrm{C}$. ( $a$ ) What is the gage pressure, in $1 \mathrm{bf} / \mathrm{in}^{2},$ at the bottom of the $\operatorname{tank} ?(b)$ How does your result in $(a)$ change if the tank diameter is reduced to 15 ft? (c) Repeat $(a)$ if leakage has caused a layer of $5 \mathrm{ft}$ of water to rest at the bottom of the (full) tank.

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A closed tank contains 1.5 $\mathrm{m}$ of SAE 30 oil, 1 $\mathrm{m}$ of water, $20 \mathrm{cm}$ of mercury, and an air space on top, all at $20^{\circ} \mathrm{C}$ The absolute pressure at the bottom of the tank is $60 \mathrm{kPa}$ What is the pressure in the air space?

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In Fig. P2.11, pressure gage $A$ reads 1.5 kPa (gage). The fluids are at $20^{\circ} \mathrm{C}$. Determine the elevations $z$, in meters, of the liquid levels in the open piezometer tubes $B$ and $C$

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In Fig. P2.12 the tank contains water and immiscible oil at $20^{\circ} \mathrm{C}$. What is $h$ in $\mathrm{cm}$ if the density of the oil is $898 \mathrm{kg} / \mathrm{m}^{3} ?$

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In Fig. P2.13 the $20^{\circ} \mathrm{C}$ water and gasoline surfaces are open to the atmosphere and at the same elevation. What is the height $h$ of the third liquid in the right leg?

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For the three-liquid system shown, compute $h_{1}$ and $h_{2}$ Neglect the air density.

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The air-oil-water system in Fig. P2.15 is at 20 $^{\circ} \mathrm{C}$. Knowing that gage $A$ reads 15 lbf/in $^{2}$ absolute and gage $B$ reads $1.25 \mathrm{lbf} / \mathrm{in}^{2}$ less than gage $C,$ compute $(a)$ the specific weight of the oil in $16 \mathrm{f} / \mathrm{ft}^{3}$ and $(b)$ the actual reading of gage $C$ in $\mathrm{Ibf} / \mathrm{in}^{2}$ absolute.

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If the absolute pressure at the interface between water and mercury in Fig. $\mathrm{P} 2.16$ is $93 \mathrm{kPa}$, what, in $\mathrm{lbf} / \mathrm{ft}^{2},$ is $(a)$ the pressure at the surface and ( $b$ ) the pressure at the bottom of the container?

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The system in Fig. P2.17 is at $20^{\circ} \mathrm{C}$. If the pressure at point $A$ is 1900 lbf $/ \mathrm{ft}^{2},$ determine the pressures at points $B, C,$ and $D$ in $1 \mathrm{bf} / \mathrm{ft}^{2}$

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The system in Fig. $\mathrm{P} 2.18$ is at $20^{\circ} \mathrm{C}$. If atmospheric pressure is $101.33 \mathrm{kPa}$ and the pressure at the bottom of the $\operatorname{tank}$ is $242 \mathrm{kPa}$, what is the specific gravity of fluid $X ?$

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The U-tube in Fig. P2.19 has a $1-\mathrm{cm}$ ID and contains mercury as shown. If $20 \mathrm{cm}^{3}$ of water is poured into the righthand leg, what will the free-surface height in each leg be after the sloshing has died down?

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The hydraulic jack in Fig. P2.20 is filled with oil at $56 \mathrm{lbf} / \mathrm{ft}^{3} .$ Neglecting the weight of the two pistons, what force $F$ on the handle is required to support the 2000 -lbf weight for this design?

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At $20^{\circ} \mathrm{C}$ gage $A$ reads 350 kPa absolute. What is the height $h$ of the water in $\mathrm{cm} ?$ What should gage $B$ read in kPa absolute? See Fig. P2.21.

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The fuel gage for a gasoline tank in a car reads proportional to the bottom gage pressure as in Fig. P2.22. If the tank is $30 \mathrm{cm}$ deep and accidentally contains $2 \mathrm{cm}$ of water plus gasoline, how many centimeters of air remain at the top when the gage erroneously reads "full"?

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In Prob. 1.2 we made a crude integration of the density distribution $\rho(z)$ in Table $A .6$ and estimated the mass of the earth's atmosphere to be $m \approx 6$ E18 kg. Can this result be used to estimate sea-level pressure on the earth? Conversely, can the actual sea-level pressure of 101.35 $\mathrm{kPa}$ be used to make a more accurate estimate of the atmospheric mass?

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As measured by NASA's Viking landers, the atmosphere of Mars, where $g \approx 3.71 \mathrm{m} / \mathrm{s}^{2},$ is almost entirely carbon dioxide, and the surface pressure averages 700 Pa. The temperature is cold and drops off exponentially: $T \approx T_{\mathrm{g}} \mathrm{e}^{-\mathrm{C}_{\mathrm{Z}}}$ where $C=1.3 \mathrm{E}-5 \mathrm{m}^{-1}$ and $T_{\mathrm{o}}=250 \mathrm{K} .$ For example, at $20,000 \mathrm{m}$ altitude, $T \approx 193 \mathrm{K} .(a)$ Find an analytic formula for the variation of pressure with altitude. (b) Find the altitude where pressure on Mars has dropped to 1 pascal.

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For gases that undergo large changes in height, the linear approximation, Eq. $(2.14),$ is inaccurate. Expand the troposphere power-law, Eq. $(2.20),$ into a power series, and show that the linear approximation $p \approx p_{a}-\rho_{a} g z$ is adequate when

\[

\delta z \leqslant \frac{2 T_{0}}{(n-1) B} \quad \text { where } n=\frac{g}{R B}

\]

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Conduct an experiment to illustrate atmospheric pressure. Note: Do this over a sink or you may get wet! Find a drinking glass with a very smooth, uniform rim at the top. Fill the glass nearly full with water. Place a smooth, light, flat plate on top of the glass such that the entire rim of the glass is covered. A glossy postcard works best. A small index card or one flap of a greeting card will also work. See Fig. P2.27a. (a) Hold the card against the rim of the glass and turn the glass upside down. Slowly release pressure on the card. Does the water fall out of the glass? Record your experimental observations.

(b) Find an expression for the pressure at points 1 and 2 in Fig. $\mathrm{P} 2.27 b$. Note that the glass is now inverted, so the original top rim of the glass is at the bottom of the picture, and the original bottom of the glass is at the top of the picture. The weight of the card can be neglected. $(c)$ Estimate the theoretical maximum glass height at which this experiment could still work, such that the water would not fall out of the glass.

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A correlation of numerical calculations indicates that, all other things being equal, the distance traveled by a well-hit baseball varies inversely as the cube root of the air density. If a home-run ball hit in New York City travels $400 \mathrm{ft},$ estimate the distance it would travel in

(a) Denver, Colorado, and (b) La Paz, Bolivia.

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An airplane flies at a Mach number of 0.82 at a standard altitude of $24,000 \mathrm{ft}$. ( $a$ ) What is the plane's velocity, in $\mathrm{mi} / \mathrm{h} ?$ (b) What is the standard density at that altitude?

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For the traditional equal-level manometer measurement in Fig. $\mathrm{E} 2.3,$ water at $20^{\circ} \mathrm{C}$ flws through the plug device from $a$ to $b .$ The manometer fluid is mercury. If $L=12 \mathrm{cm}$ and $h=24 \mathrm{cm},(a)$ what is the pressure drop through the device? $(b)$ If the water flows through the pipe at a velocity $V=18 \mathrm{ft} / \mathrm{s},$ what is the dimensionless loss coeffient of the device, defined by $K=\Delta p /\left(\rho V^{2}\right) ?$ We will study loss coefficients in Chap. 6

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In Fig. P2.31 all fluids are at $20^{\circ} \mathrm{C}$. Determine the pressure difference (Pa) between points $A$ and $B.$

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For the inverted manometer of Fig. $\mathrm{P} 2.32,$ all fluids are at $20^{\circ} \mathrm{C} .$ If $p_{B}-p_{A}=97 \mathrm{kPa},$ what must the height $\mathrm{H}$ be in $\mathrm{cm} ?$

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Sometimes manometer dimensions have a significant effect. In Fig. $\mathrm{P} 2.34$ containers $(a)$ and $(b)$ are cylindrical and conditions are such that $p_{a}=p_{b} .$ Derive a formula for the pressure difference $p_{a}-p_{b}$ when the oil-water interface on the right rises a distance $\Delta h<h$ for $(a) d \ll D$ and (b) $d=0.15 D .$ What is the percentage change in the value of $\Delta p ?$

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Water flows upward in a pipe slanted at $30^{\circ},$ as in Fig. P2.35. The mercury manometer reads $h=12 \mathrm{cm} .$ Both fluids are at $20^{\circ} \mathrm{C}$. What is the pressure difference $p_{1}-p_{2}$ in the pipe?

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In Fig. $P 2.36$ both the tank and the tube are open to the atmosphere. If $L=2.13 \mathrm{m},$ what is the angle of tilt $\theta$ of the tube?

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The inclined manometer in Fig. P2.37 contains Meriam red manometer oil, $\mathrm{SG}=0.827 .$ Assume that the reservoir is very large. If the inclined arm is fitted with graduations 1 in apart, what should the angle $\theta$ be if each graduation corresponds to 1 lbf/ft $^{2}$ gage pressure for $p_{A} ?$

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If the pressure in container A in Fig. P2.38 is $150 \mathrm{kPa}$ compute the pressure in container $\mathrm{B}$

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In Fig. P2.39 the right leg of the manometer is open to the atmosphere. Find the gage pressure, in $\mathrm{Pa}$, in the air gap in the tank.

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In Fig. $\mathrm{P} 2.40$ the pressures at $A$ and $B$ are the same, 100 kPa. If water is introduced at $A$ to increase $p_{A}$ to $130 \mathrm{kPa}$, find and sketch the new positions of the mercury menisci. The connecting tube is a uniform 1-cm diameter. Assume no change in the liquid densities.

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The system in Fig. P2.41 is at $20^{\circ} \mathrm{C}$. Compute the pressure at point $A$ in $16 f / f t^{2}$ absolute.

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Very small pressure differences $p_{A}-p_{B}$ can be measured accurately by the two-fluid differential manometer in Fig. P2.42. Density $\rho_{2}$ is only slightly larger than that of the upper fluid $\rho_{1}$. Derive an expression for the proportionality between $h$ and $p_{A}-p_{B}$ if the reservoirs are very large.

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The traditional method of measuring blood pressure uses a sphygmomanometer, first recording the highest (systolic) and then the lowest (diastolic) pressure from which flowing "Korotkoff" sounds can be heard. Patients with dangerous hypertension can exhibit systolic pressures as high as 5 lbf/in $^{2}$. Normal levels, however, are 2.7 and $1.7 \mathrm{lbf} / \mathrm{in}^{2},$ respectively, for systolic and diastolic pressures. The manometer uses mercury and air as fluids. (a) How high in $\mathrm{cm}$ should the manometer tube be?

(b) Express normal systolic and diastolic blood pressure in millimeters of mercury.

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Water flows downward in a pipe at $45^{\circ},$ as shown in Fig. P2.44. The pressure drop $p_{1}-p_{2}$ is partly due to gravity and partly due to friction. The mercury manometer reads a 6 -in height difference. What is the total pressure drop $p_{1}-p_{2}$ in $1 \mathrm{bf} / \mathrm{in}^{2} ?$ What is the pressure drop due to friction only between 1 and 2 in Ibf/in $^{2} ?$ Does the manometer reading correspond only to friction drop? Why?

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In Fig. $P 2.45,$ determine the gage pressure at point $A$ in Pa. Is it higher or lower than atmospheric?

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In Fig. P2.46 both ends of the manometer are open to the atmosphere. Estimate the specific gravity of fluid $X$

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The cylindrical tank in Fig. P2.47 is being filled with water at $20^{\circ} \mathrm{C}$ by a pump developing an exit pressure of 175 kPa. At the instant shown, the air pressure is 110 kPa and $H=35 \mathrm{cm} .$ The pump stops when it can no longer raise the water pressure. For isothermal air compression, estimate $H$ at that time.

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The system in Fig. $\mathrm{P} 2.48$ is open to 1 atm on the right side. $(a)$ If $L=120 \mathrm{cm},$ what is the air pressure in container $A ?$ (b) Conversely, if $p_{A}=135 \mathrm{kPa}$, what is the length $L ?$

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Conduct the following experiment to illustrate air pressure. Find a thin wooden ruler (approximately 1 ft in length) or a thin wooden paint stirrer. Place it on the edge of a desk or table with a little less than half of it hanging over the edge lengthwise. Get two full-size sheets of newspaper; open them up and place them on top of the ruler, covering only the portion of the ruler resting on the desk as illustrated in Fig. P2.49.

(a) Estimate the total force on top of the newspaper due to air pressure in the

room.

(b) Careful! To avoid potential injury, make sure nobody is standing directly in front of the desk. Perform a karate chop on the portion of the ruler sticking out over the edge of the desk. Record your results. $(c)$ Explain your results.

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A small submarine, with a hatch door 30 in in diameter, is submerged in seawater. $(a)$ If the water hydrostatic force on the hatch is 69,000 lbf, how deep is the sub? (b) If the sub is 350 ft deep, what is the hydrostatic force on the hatch?

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Gate $A B$ in Fig. P2.51 is $1.2 \mathrm{m}$ long and $0.8 \mathrm{m}$ into the paper. Neglecting atmospheric pressure, compute the force $F$ on the gate and its center-of-pressure position $X$

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Example 2.5 calculated the force on plate $A B$ and its line of action, using the moment-of-inertia approach. Some teachers say it is more instructive to calculate these by direct integration of the pressure forces. Using Figs. P2.52 and E2.5 $a,(a)$ find an expression for the pressure variation $p(\xi)$ along the plate; $(b)$ integrate this expression to find the total force $F ;(c)$ integrate the moments about point $A$ to find the position of the center of pressure.

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Panel $A B C$ in the slanted side of a water tank is an isosceles triangle with the vertex at $A$ and the base $B C=2 \mathrm{m}$ as in Fig. P2.53. Find the water force on the panel and its line of action.

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In Fig. P2.54, the hydrostatic force $F$ is the same on the bottom of all three containers, even though the weights of liquid above are quite different. The three bottom shapes and the fluids are the same. This is called the hydrostatic paradox. Explain why it is true and sketch a free body of each of the liquid columns.

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Gate $A B$ in Fig. $P 2.55$ is 5 ft wide into the paper, hinged at $A,$ and restrained by a stop at $B .$ The water is at $20^{\circ} \mathrm{C}$ Compute $(a)$ the force on stop $B$ and $(b)$ the reactions at $A$ if the water depth $h=9.5 \mathrm{ft}$

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In Fig. P2.55, gate $A B$ is 5 ft wide into the paper, and stop $B$ will break if the water force on it equals 9200 lbf. For what water depth $h$ is this condition reached?

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The tank in Fig. P2.57 is 2 $\mathrm{m}$ wide into the paper. Neglecting atmospheric pressure, find the resultant hydrostatic force on panel $B C(a)$ from a single formula and

(b) by computing horizontal and vertical forces separately, in the spirit of Section 2.6

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In Fig. P2.58, the cover gate $A B$ closes a circular opening $80 \mathrm{cm}$ in diameter. The gate is held closed by a $200-\mathrm{kg}$ mass as shown. Assume standard gravity at $20^{\circ} \mathrm{C}$. At what water level $h$ will the gate be dislodged? Neglect the weight of the gate.

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Gate $A B$ has length $L$ and width $b$ into the paper, is hinged at $B,$ and has negligible weight. The liquid level $h$ remains at the top of the gate for any angle $\theta$. Find an analytic expression for the force $P,$ perpendicular to $A B,$ required to keep the gate in equilibrium in Fig. P2.59.

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Determine the water hydrostatic force on one side of the vertical equilateral triangle panel BCD in Fig. P2.60. Neglect atmospheric pressure.

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Gate $A B$ in Fig. $P 2.61$ is a homogeneous mass of $180 \mathrm{kg}$ $1.2 \mathrm{m}$ wide into the paper, hinged at $A,$ and resting on a smooth bottom at $B$. All fluids are at $20^{\circ} \mathrm{C}$. For what water depth $h$ will the force at point $B$ be zero?

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Gate $A B$ in Fig. $P 2.62$ is $15 \mathrm{ft}$ long and $8 \mathrm{ft}$ wide into the paper and is hinged at $B$ with a stop at $A .$ The water is at $20^{\circ} \mathrm{C} .$ The gate is 1 -in-thick steel, $\mathrm{SG}=7.85 .$ Compute the water level $h$ for which the gate will start to fall.

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The tank in Fig. $\mathrm{P} 2.63$ has a $4-\mathrm{cm}$ -diameter plug at the bottom on the right. All fluids are at $20^{\circ} \mathrm{C}$. The plug will pop out if the hydrostatic force on it is 25 N. For this condition, what will be the reading $h$ on the mercury manometer on the left side?

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Gate $A B C$ in Fig. $\mathrm{P} 2.64$ has a fixed hinge line at $B$ and is $2 \mathrm{m}$ wide into the paper. The gate will open at $A$ to release water if the water depth is high enough. Compute the depth $h$ for which the gate will begin to open.

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Gate $A B$ in Fig. $\mathrm{P} 2.65$ is semicircular, hinged at $B,$ and held by a horizontal force $P$ at $A$. What force $P$ is required for equilibrium?

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Dam $A B C$ in Fig. $\mathrm{P} 2.66$ is $30 \mathrm{m}$ wide into the paper and made of concrete $(\mathrm{SG}=2.4) .$ Find the hydrostatic force on surface $A B$ and its moment about $C .$ Assuming no seepage of water under the dam, could this force tip the dam over? How does your argument change if there is seepage under the dam?

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Generalize Prob. P2.66 as follows. Denote length $A B$ as

$H,$ length $B C$ as $L,$ and angle $A B C$ as $\theta .$ Let the dam material have specific gravity SG. The width of the dam is

$b$. Assume no seepage of water under the dam. Find an analytic relation between $\mathrm{SG}$ and the critical angle $\theta_{c}$ for which the dam will just tip over to the right. Use your relation to compute $\theta_{c}$ for the special case $\mathrm{SG}=$ $2.4(\text { concrete })$

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Isosceles triangle gate $A B$ in Fig. $\mathrm{P} 2.68$ is hinged at $A$ and weighs $1500 \mathrm{N}$. What horizontal force $P$ is required at point $B$ for equilibrium?

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Consider the slanted plate $A B$ of length $L$ in Fig. $\mathrm{P} 2.69$

(a) Is the hydrostatic force $F$ on the plate equal to the weight of the missing water above the plate? If not, correct this hypothesis. Neglect the atmosphere.

(b) Can a "missing water" theory be generalized to curved surfaces of this type?

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The swing-check valve in Fig. P2.70 covers a 22.86 -cm diameter opening in the slanted wall. The hinge is $15 \mathrm{cm}$ from the centerline, as shown. The valve will open when the hinge moment is $50 \mathrm{N} \cdot \mathrm{m}$. Find the value of $h$ for the water to cause this condition.

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The swing-check valve in Fig. $\mathrm{P} 2.70$ covers a $22.86-\mathrm{cm}$ diameter opening in the slanted wall. The hinge is $15 \mathrm{cm}$ from the centerline, as shown. The valve will open when the hinge moment is $50 \mathrm{N} \cdot \mathrm{m}$. Find the value of $h$ for the water to cause this condition.

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In Fig. $\mathrm{P} 2.71$ gate $A B$ is $3 \mathrm{m}$ wide into the paper and is connected by a rod and pulley to a concrete sphere $(\mathrm{SG}=2.40) .$ What diameter of the sphere is just sufficient to keep the gate closed?

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Gate $B$ in Fig. $\mathrm{P} 2.72$ is $30 \mathrm{cm}$ high, $60 \mathrm{cm}$ wide into the paper, and hinged at the top. What water depth $h$ will first cause the gate to open?

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Gate $A B$ is $5 \mathrm{ft}$ wide into the paper and opens to let fresh water out when the ocean tide is dropping. The hinge at $A$ is $2 \mathrm{ft}$ above the freshwater level. At what ocean level $h$ will the gate first open? Neglect the gate weight.

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Find the height $H$ in Fig. $P 2.74$ for which the hydrostatic force on the rectangular panel is the same as the force on the semicircular panel below.

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The cap at point $B$ on the 5 -cm-diameter tube in Fig. P2.75 will be dislodged when the hydrostatic force on its base reaches 22 lbf. For what water depth $h$ does this occur?

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Panel $B C$ in Fig. $P 2.76$ is circular. Compute ( $a$ ) the hydrostatic force of the water on the panel, ( $b$ ) its center of presand $(c)$ the moment of this force about point $B$

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The circular gate $A B C$ in Fig. $\mathrm{P} 2.77$ has a $1-\mathrm{m}$ radius and is hinged at $B$. Compute the force $P$ just sufficient to keep the gate from opening when $h=8 \mathrm{m} .$ Neglect atmospheric pressure.

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Panels $A B$ and $C D$ in Fig. $P 2.78$ are each $120 \mathrm{cm}$ wide into the paper. (a) Can you deduce, by inspection, which panel has the larger water force? ( $b$ ) Even if your deduction is brilliant, calculate the panel forces anyway.

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Gate $A B C$ in Fig. $\mathrm{P} 2.79$ is $1 \mathrm{m}$ square and is hinged at

$B .$ It will open automatically when the water level $h$ becomes high enough. Determine the lowest height for which the gate will open. Neglect atmospheric pressure. Is this result independent of the liquid density?

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A concrete dam $(\mathrm{SG}=2.5)$ is made in the shape of an isosceles triangle, as in Fig. $\mathrm{P} 2.80 .$ Analyze this geometry to find the range of angles $\theta$ for which the hydrostatic force will tend to tip the dam over at point $B$. The width into the paper is $b$

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For the semicircular cylinder $C D E$ in Example $2.9,$ find the vertical hydrostatic force by integrating the vertical component of pressure around the surface from $\theta=0$ to $\boldsymbol{\theta}=\pi$

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The dam in Fig. $\mathrm{P} 2.82$ is a quarter circle $50 \mathrm{m}$ wide into the paper. Determine the horizontal and vertical components of the hydrostatic force against the dam and the point CP where the resultant strikes the dam.

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Gate $A B$ in Fig. $\mathrm{P} 2.83$ is a quarter circle $10 \mathrm{ft}$ wide into the paper and hinged at $B$. Find the force $F$ just sufficient to keep the gate from opening. The gate is uniform and weighs 3000 lbf.

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Panel AB in Fig. P2.84 is a parabola with its maximum at point $A$. It is $150 \mathrm{cm}$ wide into the paper. Neglect atmospheric pressure. Find $(a)$ the vertical and $(b)$ the horizontal water forces on the panel.

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Compute the horizontal and vertical components of the hydrostatic force on the quarter-circle panel at the bottom of the water tank in Fig. $\mathrm{P} 2.85$

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The quarter circle gate $B C$ in Fig. $P 2.86$ in hinged at $C$ Find the horizontal force $P$ required to hold the gate stationary. Neglect the weight of the gate.

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The bottle of champagne $(\mathrm{SG}=0.96)$ in Fig. $\mathrm{P} 2.87$ is under pressure, as shown by the mercury-manometer reading. Compute the net force on the 2 -in-radius hemispherical end cap at the bottom of the bottle.

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Gate $A B C$ is a circular arc, sometimes called a Tainter gate, which can be raised and lowered by pivoting about point $O .$ See Fig. $\mathrm{P} 2.88 .$ For the position shown, deter$\operatorname{mine}$ (a) the hydrostatic force of the water on the gate and $(b)$ its line of action. Does the force pass through point $O ?$

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The tank in Fig. $\mathrm{P} 2.89$ contains benzene and is pressurized to $200 \mathrm{kPa}$ (gage) in the air gap. Determine the vertical hydrostatic force on circular-arc section $A B$ and its line of action.

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The tank in Fig. $\mathrm{P} 2.90$ is $120 \mathrm{cm}$ long into the paper. Determine the horizontal and vertical hydrostatic forces on the quarter-circle panel $A B$. The fluid is water at $20^{\circ} \mathrm{C}$. Neglect atmospheric pressure.

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The hemispherical dome in Fig. P2.91 weighs $30 \mathrm{kN}$ and is filled with water and attached to the floor by six equally spaced bolts. What is the force in each bolt required to hold down the dome?

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A 4 -m-diameter water tank consists of two half cylinders, each weighing $4.5 \mathrm{kN} / \mathrm{m},$ bolted together as shown in Fig. $\mathrm{P} 2.92 .$ If the support of the end caps is neglected, determine the force induced in each bolt.

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In Fig. $\mathrm{P} 2.93,$ a one-quadrant spherical shell of radius $R$ is submerged in liquid of specific weight $\gamma$ and depth $h>R$ Find an analytic expression for the resultant hydrostatic force, and its line of action, on the shell surface.

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Find an analytic formula for the vertical and horizontal forces on each of the semicircular panels $A B$ in Fig. $\mathrm{P} 2.94$ The width into the paper is $b .$ Which force is larger? Why?

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The uniform body $A$ in Fig. $P 2.95$ has width $b$ into the paper and is in static equilibrium when pivoted about hinge $O .$ What is the specific gravity of this body if

(a) $h=0$ and

$(b) h=R ?$

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Curved panel $B C$ in Fig. $\mathrm{P} 2.96$ is a $60^{\circ}$ arc, perpendicular to the bottom at $C .$ If the panel is $4 \mathrm{m}$ wide into the paper, estimate the resultant hydrostatic force of the water on the panel.

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The contractor ran out of gunite mixture and finished the deep corner, of a 5 -m-wide swimming pool, with a quarter-circle piece of PVC pipe, labeled $A B$ in Fig. P2.97. Compute the horizontal and vertical water forces on the curved panel $A B$

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Gate $A B C$ in Fig. $\mathrm{P} 2.98$ is a quarter circle $8 \mathrm{ft}$ wide into the paper. Compute the horizontal and vertical hydrostatic forces on the gate and the line of action of the resultant force

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The mega-magnum cylinder in Fig. P2.99 has a hemispherical bottom and is pressurized with air to $75 \mathrm{kPa}$ (gage) at the top. Determine ( $a$ ) the horizontal and $(b)$ the vertical hydrostatic forces on the hemisphere, in lbf.

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Pressurized water fills the tank in Fig. P2.100. Compute the net hydrostatic force on the conical surface $A B C$.

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The closed layered box in Fig. $\mathrm{P} 2.101$ has square horizontal cross sections everywhere. All fluids are at $20^{\circ} \mathrm{C}$ Estimate the gage pressure of the air if ( $a$ ) the hydrostatic force on panel $A B$ is $48 \mathrm{kN}$ or $(b)$ the hydrostatic force on the bottom panel $B C$ is $97 \mathrm{kN}.$

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A cubical tank is $3 \times 3 \times 3 \mathrm{m}$ and is layered with 1 meter of fluid of specific gravity 1.0,1 meter of fluid with $\mathrm{SG}=0.9,$ and 1 meter of fluid with $\mathrm{SG}=0.8 .$ Neglect atmospheric pressure. Find $(a)$ the hydrostatic force on the bottom and $(b)$ the force on a side panel.

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A solid block, of specific gravity $0.9,$ floats such that 75 percent of its volume is in water and 25 percent of its volume is in fluid $X$, which is layered above the water. What is the specific gravity of fluid $X ?$

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The can in Fig. $\mathrm{P} 2.104$ floats in the position shown. What is its weight in $\mathrm{N} ?$

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It is said that Archimedes discovered the buoyancy laws when asked by King Hiero of Syracuse to determine whether his new crown was pure gold (SG = 19.3 ). Archimedes measured the weight of the crown in air to be $11.8 \mathrm{N}$ and its weight in water to be $10.9 \mathrm{N}$. Was it pure gold?

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A spherical helium balloon is $2.5 \mathrm{m}$ in diameter and has a total mass of 6.7 kg. When released into the U.S. standard atmosphere, at what altitude will it settle?

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Repeat Prob. $2.62,$ assuming that the 10,000 -lbf weight is aluminum (SG $=2.71$ ) and is hanging submerged in the water.

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A 7 -cm-diameter solid aluminum ball $(\mathrm{SG}=2.7)$ and a solid brass ball (SG = 8.5) balance nicely when submerged in a liquid, as in Fig. P2.108. $(a)$ If the fluid is water at $20^{\circ} \mathrm{C}$, what is the diameter of the brass ball? (b) If the brass ball has a diameter of $3.8 \mathrm{cm},$ what is the density of the fluid?

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A hydrometer floats at a level that is a measure of the specific gravity of the liquid. The stem is of constant diameter $D,$ and a weight in the bottom stabilizes the body to float vertically, as shown in Fig. P2.109. If the position $h=0$ is pure water $(\mathrm{SG}=1.0),$ derive a formula for $h$ as a function of total weight $W, D, \mathrm{SG},$ and the specific weight $\gamma_{0}$ of water.

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A solid sphere, of diameter $18 \mathrm{cm},$ floats in $20^{\circ} \mathrm{C}$ water with 1,527 cubic centimeters exposed above the surface.

(a) What are the weight and specific gravity of this sphere?

(b) Will it float in $20^{\circ} \mathrm{C}$ gasoline? If so, how many cubic centimeters will be exposed?

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A hot-air balloon must be designed to support basket, cords, and one person for a total weight of $1300 \mathrm{N}$. The balloon material has a mass of $60 \mathrm{g} / \mathrm{m}^{2} .$ Ambient air is at $25^{\circ} \mathrm{C}$ and 1 atm. The hot air inside the balloon is at $70^{\circ} \mathrm{C}$ and 1 atm. What diameter spherical balloon will just support the total weight? Neglect the size of the hotair inlet vent.

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The uniform 5 -m-long round wooden rod in Fig. $\mathrm{P} 2.112$ is tied to the bottom by a string. Determine ( $a$ ) the tension in the string and ( $b$ ) the specific gravity of the wood. Is it possible for the given information to determine the inclination angle $\theta ?$ Explain.

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A spar buoy is a buoyant rod weighted to float and protrude vertically, as in Fig. P2.113. It can be used for measurements or markers. Suppose that the buoy is maple wood $(\mathrm{SG}=0.6), 2$ in by 2 in by $12 \mathrm{ft},$ floating in seawater $(\mathrm{SG}=1.025) .$ How many pounds of steel $(\mathrm{SG}=7.85)$ should be added to the bottom end so that $h=18$ in?

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The uniform rod in Fig. P2.114 is hinged at point $B$ on the waterline and is in static equilibrium as shown when $2 \mathrm{kg}$ of lead $(\mathrm{SG}=11.4)$ are attached to its end. What is the specific gravity of the rod material? What is peculiar about the rest angle $\theta=30^{\circ} ?$

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The 2 -in by 2 -in by 12 -ft spar buoy from Fig. $\mathrm{P} 2.113$ has $5 \mathrm{lbm}$ of steel attached and has gone aground on a rock, as in Fig. P2.115. Compute the angle $\theta$ at which the buoy will lean, assuming that the rock exerts no moments on the spar.

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The deep submersible vehicle ALVIN in the chapteropener photo has a hollow titanium (SG = 4.50) spherical passenger compartment with an inside diameter of 78.08 in and a wall thickness of 1.93 in.

(a) Would the empty sphere float in seawater?

(b) Would it float if it contained $1000 \mathrm{lbm}$ of people and equipment inside?

(c) What wall thickness would cause the empty sphere to be neutrally buoyant?

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The balloon in Fig. $\mathrm{P} 2.117$ is filled with helium and pressurized to $135 \mathrm{kPa}$ and $20^{\circ} \mathrm{C}$. The balloon material has a mass of $85 \mathrm{g} / \mathrm{m}^{2}$. Estimate $(a)$ the tension in the mooring line and $(b)$ the height in the standard atmosphere to which the balloon will rise if the mooring line is cut.

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An intrepid treasure-salvage group has discovered a steel box, containing gold doubloons and other valuables, resting in $80 \mathrm{ft}$ of seawater. They estimate the weight of the box and treasure (in air) at 7000 lbf. Their plan is to attach the box to a sturdy balloon, inflated with air to 3 atm pressure. The empty balloon weighs 250 lbf. The box is $2 \mathrm{ft}$ wide, $5 \mathrm{ft}$ long, and 18 in high. What is the proper diameter of the balloon to ensure an upward lift force on the box that is 20 percent more than required?

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When a 5 -lbf weight is placed on the end of the uniform floating wooden beam in Fig. $\mathrm{P} 2.119$, the beam tilts at an angle $\theta$ with its upper right corner at the surface, as shown. Determine ( $a$ ) the angle $\theta$ and (b) the specific gravity of the wood. Hint: Both the vertical forces and the moments about the beam centroid must be balanced.

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A uniform wooden beam (SG $=0.65$ ) is $10 \mathrm{cm}$ by $10 \mathrm{cm}$ by $3 \mathrm{m}$ and is hinged at $A,$ as in Fig. $\mathrm{P} 2.120 .$ At what angle $\theta$ will the beam float in the $20^{\circ} \mathrm{C}$ water?

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The uniform beam in Fig. $\mathrm{P} 2.121$, of size $L$ by $h$ by $b$ and with specific weight $\gamma_{b},$ floats exactly on its diagonal when a heavy uniform sphere is tied to the left corner, as shown. Show that this can happen only $(a)$ when $\gamma_{b}=\gamma / 3$ and $(b)$ when the sphere has size

\[

D=\left[\frac{L h b}{\pi(\mathrm{SG}-1)}\right]^{1 / 3}

\]

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A uniform block of steel $(\mathrm{SG}=7.85)$ will "float" at a mercury-water interface as in Fig. P2.122. What is the ratio of the distances $a$ and $b$ for this condition?

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A barge has the trapezoidal shape shown in Fig. $\mathrm{P} 2.123$ and is $22 \mathrm{m}$ long into the paper. If the total weight of barge and cargo is 350 tons, what is the draft $H$ of the barge when floating in seawater?

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A balloon weighing 3.5 lbf is $6 \mathrm{ft}$ in diameter. It is filled with hydrogen at 18 lbf/in absolute and $60^{\circ} \mathrm{F}$ and is released. At what altitude in the U.S. standard atmosphere will this balloon be neutrally buoyant?

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A solid sphere, of diameter $20 \mathrm{cm},$ has a specific gravity of $0.7 .(a)$ Will this sphere float in $20^{\circ} \mathrm{C}$ SAE $10 \mathrm{W}$ oil? If so,

(b) how many cubic centimeters are exposed, and

$(c)$ how high will a spherical cap protrude above the surface? Note: If your knowledge of offbeat sphere formulas is lacking, you can "Ask Dr. Math" at Drexel University, <http://mathforum.org/dr.math/> EES is recommended for the solution.

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A block of wood $(\mathrm{SG}=0.6)$ floats in fluid $X$ in Fig. $\mathrm{P} 2.126$ such that 75 percent of its volume is submerged in fluid $X$. Estimate the vacuum pressure of the air in the tank.

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Consider a cylinder of specific gravity $S<1$ floating vertically in water $(S=1),$ as in Fig. $\mathrm{P} 2.127 .$ Derive a formula for the stable values of $D / L$ as a function of $S$ and apply it to the case $D / L=1.2$

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An iceberg can be idealized as a cube of side length $L$, as in Fig. $\mathrm{P} 2.128 .$ If seawater is denoted by $S=1.0,$ then glacier ice (which forms icebergs) has $S=0.88$. Determine if this "cubic" iceberg is stable for the position shown in Fig. $\mathrm{P} 2.128$

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The iceberg idealization in Prob. $\mathrm{P} 2.128$ may become unstable if its sides melt and its height exceeds its width. In Fig. $\mathrm{P} 2.128$ suppose that the height is $L$ and the depth into the paper is $L,$ but the width in the plane of the paper is $H<L$ Assuming $S=0.88$ for the iceberg, find the ratio $H / L$ for which it becomes neutrally stable (about to overturn)

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Consider a wooden cylinder $(\mathrm{SG}=0.6) 1 \mathrm{m}$ in diameter and $0.8 \mathrm{m}$ long. Would this cylinder be stable if placed to float with its axis vertical in oil $(\mathrm{SG}=0.8) ?$

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A barge is $15 \mathrm{ft}$ wide and $40 \mathrm{ft}$ long and floats with a draft of $4 \mathrm{ft}$. It is piled so high with gravel that its center of gravity is $3 \mathrm{ft}$ above the waterline. Is it stable?

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A solid right circular cone has $\mathrm{SG}=0.99$ and floats vertically as in Fig. P2.132. Is this a stable position for the cone?

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Consider a uniform right circular cone of specific gravity $S<1$, floating with its vertex down in water $(S=1)$ The base radius is $R$ and the cone height is $H$. Calculate and plot the stability $M G$ of this cone, in dimensionless form, versus $H / R$ for a range of $S<1$

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Consider a homogeneous right circular cylinder of length $L,$ radius $R,$ and specific gravity $\mathrm{SG},$ floating in water $(\mathrm{SG}=1) .$ Show that the body will be stable with its axis vertical if

\[

\frac{R}{L}>[2 \mathrm{SG}(1-\mathrm{SG})]^{1 / 2}

\]

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Consider a homogeneous right circular cylinder of length $L,$ radius $R,$ and specific gravity $\mathrm{SG}=0.5,$ floating in water $(\mathrm{SG}=1) .$ Show that the body will be stable with its axis horizontal if $L / R>2.0$

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A tank of water $4 \mathrm{m}$ deep receives a constant upward acceleration $a_{z}$. Determine $(a)$ the gage pressure at the tank bottom if $a_{z}=5 \mathrm{m}^{2} / \mathrm{s}$ and $(b)$ the value of $a_{z}$ that causes the gage pressure at the tank bottom to be 1 atm.

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A 12 -fl-oz glass, of 3 -in diameter, partly full of water, is attached to the edge of an 8 -ft-diameter merrygo-round, which is rotated at 12 r/min. How full can the glass be before water spills? Hint: Assume that the glass is much smaller than the radius of the merry-go-round.

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Suppose an elliptical-end fuel tank that is $10 \mathrm{m}$ long and has a 3 -m horizontal major axis and $2-\mathrm{m}$ vertical major axis is filled completely with fuel oil $\left(\rho=890 \mathrm{kg} / \mathrm{m}^{3}\right) .$ Let the tank be pulled along a horizontal road. For rigid-body motion, find the acceleration, and its direction, for which $(a)$ a constant-pressure surface extends from the top of the front end wall to the bottom of the back end and $(b)$ the top of the back end is at a pressure 0.5 atm lower than the top of the front end.

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The same tank from Prob. P2.139 is now moving with constant acceleration up a $30^{\circ}$ inclined plane, as in Fig. P2.141. Assuming rigid-body motion, compute (a) the value of the acceleration $a,(b)$ whether the acceleration is up or down, and (c) the gage pressure at point $A$ if the fluid is mercury at $20^{\circ} \mathrm{C}.$

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The tank of water in Fig. $\mathrm{P} 2.142$ is $12 \mathrm{cm}$ wide into the paper. If the tank is accelerated to the right in rigid-body motion at $6.0 \mathrm{m} / \mathrm{s}^{2},$ compute $(a)$ the water depth on side $A B$ and $(b)$ the water-pressure force on panel $A B$. Assume no spilling.

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The tank of water in Fig. $\mathrm{P} 2.143$ is full and open to the atmosphere at point $A .$ For what acceleration $a_{x}$ in $\mathrm{ft} / \mathrm{s}^{2}$ will the pressure at point $B$ be

(a) atmospheric and

(b) zero absolute?

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Consider a hollow cube of side length $22 \mathrm{cm},$ filled completely with water at $20^{\circ} \mathrm{C}$. The top surface of the cube is horizontal. One top corner, point $A$, is open through a small hole to a pressure of 1 atm. Diagonally opposite to point $A$ is top corner $B$. Determine and discuss the various rigidbody accelerations for which the water at point $B$ begins to cavitate, for (a) horizontal motion and (b) vertical motion.

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A fish tank 14 in deep by 16 by 27 in is to be carried in a car that may experience accelerations as high as $6 \mathrm{m} / \mathrm{s}^{2} .$ What is the maximum water depth that will avoid spilling in rigid-body motion? What is the proper alignment of the tank with respect to the car motion?

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The tank in Fig. $\mathrm{P} 2.146$ is filled with water and has a vent hole at point $A$. The tank is $1 \mathrm{m}$ wide into the paper. Inside the tank, a 10 -cm balloon, filled with helium at $130 \mathrm{kPa}$ is tethered centrally by a string. If the tank accelerates to the right at $5 \mathrm{m} / \mathrm{s}^{2}$ in rigid-body motion, at what angle will the balloon lean? Will it lean to the right or to the left?

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The tank of water in Fig. $\mathrm{P} 2.147$ accelerates uniformly by freely rolling down a $30^{\circ}$ incline. If the wheels are frictionless, what is the angle $\theta ?$ Can you explain this interesting result?

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A child is holding a string onto which is attached a heliumfilled balloon. (a) The child is standing still and suddenly accelerates forward. In a frame of reference moving with the child, which way will the balloon tilt, forward or backward? Explain. (b) The child is now sitting in a car that is stopped at a red light. The helium-filled balloon is not in contact with any part of the car (seats, ceiling, etc.) but is held in place by the string, which is in turn held by the child. All the windows in the car are closed. When the traffic light turns green, the car accelerates forward. In a frame of reference moving with the car and child, which way will the balloon tilt, forward or backward? Explain. (c) Purchase or borrow a helium-filled balloon. Conduct a scientific experiment to see if your predictions in parts $(a)$ and (b) above are correct. If not, explain.

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The 6 -ft-radius waterwheel in Fig. $\mathrm{P} 2.149$ is being used to lift water with its 1 -ft-diameter half-cylinder blades. If the wheel rotates at $10 \mathrm{r} / \mathrm{min}$ and rigid-body motion is assumed, what is the water surface angle $\theta$ at position $A$ ?

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A cheap accelerometer, probably worth the price, can be made from a U-tube as in Fig. P2.150. If $L=18 \mathrm{cm}$ and $D=5 \mathrm{mm},$ what will $h$ be if $a_{x}=6 \mathrm{m} / \mathrm{s}^{2} ?$ Can the scale markings on the tube be linear multiples of $a_{x} ?$

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The U-tube in Fig. $\mathrm{P} 2.151$ is open at $A$ and closed at $D$. If accelerated to the right at uniform $a_{x},$ what acceleration will cause the pressure at point $C$ to be atmospheric? The fluid is water $(\mathrm{SG}=1.0)$

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A 16 -cm-diameter open cylinder $27 \mathrm{cm}$ high is full of water. Compute the rigid-body rotation rate about its central axis, in $\mathrm{r} / \mathrm{min},(a)$ for which one-third of the water will spill out and $(b)$ for which the bottom will be barely exposed.

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A tall cylindrical container, 14 in in diameter, is used to make a mold for forming 14 -in salad bowls. The bowls are to be 8 in deep. The cylinder is half-filled with molten plastic, $\mu=1.6 \mathrm{kg} /(\mathrm{m}-\mathrm{s}),$ rotated steadily about the central axis, then cooled while rotating. What is the appropriate rotation rate, in $\mathrm{r} / \mathrm{min}$ ?

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A very tall 10 -cm-diameter vase contains $1178 \mathrm{cm}^{3}$ of water. When spun steadily to achieve rigid-body rotation, a $4-\mathrm{cm}$ -diameter dry spot appears at the bottom of the vase. What is the rotation rate, in r/min, for this condition?

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For what uniform rotation rate in $\mathrm{r} / \mathrm{min}$ about axis $C$ will the U-tube in Fig. $\mathrm{P} 2.155$ take the configuration shown? The fluid is mercury at $20^{\circ} \mathrm{C}$

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Suppose that the U-tube of Fig. $\mathrm{P} 2.151$ is rotated about axis $D C .$ If the fluid is water at $122^{\circ} \mathrm{F}$ and atmospheric pressure is 2116 lbf/ft $^{2}$ absolute, at what rotation rate will the fluid within the tube begin to vaporize? At what point will this occur?

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The $45^{\circ}$ V-tube in Fig. $P 2.157$ contains water and is open at $A$ and closed at $C .$ What uniform rotation rate in $\mathrm{r} / \mathrm{min}$ about axis $A B$ will cause the pressure to be equal at points $B$ and $C ?$ For this condition, at what point in $\operatorname{leg} B C$ will the pressure be a minimum?

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It is desired to make a 3 -m-diameter parabolic telescope mirror by rotating molten glass in rigid-body motion until the desired shape is achieved and then cooling the glass to a solid. The focus of the mirror is to be $4 \mathrm{m}$ from the mirror, measured along the centerline. What is the proper mirror rotation rate, in $\mathrm{r} / \mathrm{min}$, for this task?

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The three-legged manometer in Fig. $\mathrm{P} 2.159$ is filled with water to a depth of $20 \mathrm{cm}$. All tubes are long and have equal small diameters. If the system spins at angular velocity $\Omega$ about the central tube, $(a)$ derive a formula to find the change of height in the tubes;

(b) find the height in $\mathrm{cm}$ in each tube if $\Omega=120 \mathrm{r} / \mathrm{min}$. Hint: The central tube must supply water to both the outer legs.

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