Fluid Mechanics for Engineers in SI Units

David A Chin

Chapter 2

Fluid Statics - all with Video Answers

Educators

Chapter Questions

Problem 1

A glycerin storage tank has been contaminated with crude oil as shown in Figure 2.43. The tank is open to the atmosphere, where the atmospheric pressure is $101 \mathrm{kPa}$. Within the tank, a 0.7 -m-thick layer of crude oil floats on the glycerin that is $2.25 \mathrm{~m}$ deep. Both liquids are at $20^{\circ} \mathrm{C}$. Determine the pressure on the bottom of the tank, both as an absolute pressure and as a gauge pressure.

Abid Hussain

Abid Hussain

Numerade Educator

Problem 2

(a) At what depth below the surface of a water body will the (gauge) pressure be equal to $200 \mathrm{kPa}$ ? (b) If a 1.55-m-tall person orients himself vertically underwater in a pool, what pressure difference does he feel between his head and his toes? Assume water at $20^{\circ} \mathrm{C}$.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 3

A liquid is stratified such that the specific gravity of the fluid at the surface is 0.98 and the specific gravity at a depth of $12 \mathrm{~m}$ is equal to 1.07 . Assuming that the specific gravity varies linearly between the liquid surface and a depth of $12 \mathrm{~m}$, determine the pressure at a depth of $12 \mathrm{~m}$. State whether this is a gauge pressure or an absolute pressure.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 4

(a) A large tank contains water at $20^{\circ} \mathrm{C},$ and the absolute pressure $12 \mathrm{~m}$ below the surface of the water is measured as $200 \mathrm{kPa}$. Estimate the atmospheric pressure above the tank. (b) If the water in the tank was replaced by a liquid with a specific gravity of $0.85,$ what absolute pressure and what gauge pressure would be measured $6 \mathrm{~m}$ below the surface of the liquid?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 5

A pipeline leads from a reservoir to a closed valve as illustrated in Figure $2.44 .$ Calculate the gauge and absolute pressures at the valve. Assume water at $20^{\circ} \mathrm{C}$ and standard atmospheric pressure.

Abid Hussain

Numerade Educator

Problem 6

The pressure in the airspace above an oil $(\mathrm{SG}=0.80)$ surface in a tank is $14 \mathrm{kPa}$. Find the pressure $1.5 \mathrm{~m}$ below the surface of the oil.

Abid Hussain

Numerade Educator

Problem 7

Three tanks contain water at $20^{\circ} \mathrm{C}$ and air as shown in Figure 2.45 . The three tanks have open connections to each other, and the air (gauge) pressure in tank B is $6 \mathrm{kPa}$. What are the air pressures in tanks $\mathrm{A}$ and $\mathrm{C}$ ?

Abid Hussain

Numerade Educator

Problem 8

A 6-mm-diameter air bubble is released at a location $25 \mathrm{~m}$ below the surface of a lake, and the temperature of the lake is approximately uniform at $20^{\circ} \mathrm{C}$. Estimate the diameter of the bubble as it approaches the water surface. Assume standard sea-level atmospheric conditions above the surface of the lake.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 9

A bubble is released from an ocean vent located $20 \mathrm{~m}$ below the ocean surface. Atmospheric pressure above the ocean surface is equal to $101.3 \mathrm{kPa}$, and the ocean temperature down to a depth of $20 \mathrm{~m}$ is approximately constant at $20^{\circ} \mathrm{C}$. Estimate the ratio of the density of the air in the bubble at a depth of $20 \mathrm{~m}$ to the density of the air in the bubble just as it reaches the ocean surface.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 10

A container that is open to the atmosphere stores two immiscible liquids. The liquid on top has a thickness of $7 \mathrm{~m}$ and specific weight of $9 \mathrm{kN} / \mathrm{m}^{3},$ and the liquid on the bottom has a thickness of $2.3 \mathrm{~m}$. If a transducer measures a (gauge) pressure of $92 \mathrm{kPa}$ on the bottom of the tank, estimate the specific gravity of the liquid on the bottom. Must the liquid on the bottom necessarily be denser than the liquid on the top? Explain.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 11

Water pressure at a pipeline junction measures $450 \mathrm{kPa}$. What is the corresponding pressure head (expressed as a height of water)?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 12

What is the pressure head (of water) corresponding to a pressure of $810 \mathrm{kPa}$ ? What depth of mercury at $20^{\circ} \mathrm{C}$ will be required to produce a pressure of $810 \mathrm{kPa}$ ?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 13

Find the pressure head in millimeters of mercury (Hg) equivalent to $80 \mathrm{~mm}$ of water plus $60 \mathrm{~mm}$ of a fluid whose specific gravity is $2.90 .$ The specific weight of mercury can be taken as $133 \mathrm{kN} / \mathrm{m}^{3}$. Assume a temperature of $20^{\circ} \mathrm{C}$.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 14

Find the pressure head corresponding to an atmospheric pressure of $101.3 \mathrm{kPa}$. Give your answer in terms of millimeters of mercury.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 15

A 8 -mm-diameter glass tube is attached to a pressurized storage tank containing water at $25^{\circ} \mathrm{C}$. The water in the tube is observed to rise to a height of $85 \mathrm{~mm}$ above the attachment point. (a) What correction to the rise height must be made to account for surface tension? (b) What is the pressure head in the reservoir where the tube is attached?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 16

It is common practice to use elevated reservoirs to maintain the pressures in municipal water supply systems. Such a system is illustrated in Figure $2.46 .$ If the pressure in the pipeline is to be maintained in the range of $350-500 \mathrm{kPa}$ and the reservoir is not to be less than half full, estimate the height of the midpoint of the reservoir above the pipeline and the minimum required space between the midpoint and the top of the reservoir. The pressure in the air above the tank is maintained at atmospheric pressure.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 17

Blood pressure in humans is normally expressed as a ratio $x / y,$ where $x$ is the maximum arterial pressure in $\mathrm{mm} \mathrm{Hg},$ called the systolic pressure, and $y$ is the minimum arterial pressure in $\mathrm{mm} \mathrm{Hg},$ called the diastolic pressure. A typical blood pressure is $120 / 70$. Blood pressure readings are normally taken as the same level as the heart, and blood at $37^{\circ} \mathrm{C}$ has a density of around $1060 \mathrm{~kg} / \mathrm{m}^{3}$. (a) Consider a (tall) person whose head is $0.48 \mathrm{~m}$ above her heart and whose toes are $1.46 \mathrm{~m}$ below her heart. Assuming static conditions, compare the blood pressure in her head to the blood pressure in her toes. (b) If a tube were connected to the artery in which the blood pressure was being measured, what would be the maximum height that blood would rise in the tube?

Abid Hussain

Numerade Educator

Problem 18

Fluid is to be infused intravenously into the arm of a patient by placing the fluid in a bottle and suspending the bottle at a sufficient height above the patient's arm that the fluid can be fed by gravity into the body. Under typical conditions, an arm-level (gauge) pressure of $150 \mathrm{~mm} \mathrm{Hg}$ is required to provide a sufficient flow rate of fluid. If the intravenous fluid has a density of $1025 \mathrm{~kg} / \mathrm{m}^{3}$, how far above arm level must the fluid level in the bottle be held?

Abid Hussain

Numerade Educator

Problem 19

The head of an adult male giraffe is typically $6 \mathrm{~m}$ above the ground. The normal blood pressure of a giraffe (at heart level) is typically stated as $280 / 180,$ where 280 the maximum arterial pressure in $\mathrm{mm} \mathrm{Hg}$ and 180 is the minimum arterial pressure in $\mathrm{mm} \mathrm{Hg}$. The density of a giraffe's blood is approximately $1060 \mathrm{~kg} / \mathrm{m}^{3}$. (a) What is the change in the blood pressure in the giraffe's head, in millimeters of mercury, as it moves from grazing on a tall tree to drinking from a pond at ground level? (b) Assuming that there is a static distribution of blood pressure in the giraffe's body and that its heart is at approximately the mid-elevation of its body, estimate the maximum blood pressure in its head.

Abid Hussain

Numerade Educator

Problem 20

The hydraulic system shown in Figure 2.47 uses compressed air at a pressure of $310 \mathrm{kPa}$ to lift a load on a platform. The area of the piston connected to the chamber of compressed air is $8 \mathrm{~cm}^{2},$ the weight of the piston is $55 \mathrm{~N},$ the area of the platform is $600 \mathrm{~cm}^{2},$ the weight of the platform is $820 \mathrm{~N},$ the bottom of the platform is located $1.2 \mathrm{~m}$ above the bottom of the piston, and the density of the hydraulic fluid is $900 \mathrm{~kg} / \mathrm{m}^{3}$. (a) What force is exerted by the compressed air on the piston? (b) What weight mounted on the platform can be lifted by the compressed air? (c) If the piston is displaced by $12 \mathrm{~cm}$, what is the displacement of the platform?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 21

Two piston diameters are being considered for use in a hydraulic system: a 25-mmdiameter piston and a 100 -mm-diameter piston. If an applied force of $500 \mathrm{~N}$ to the 25 -mm piston is found to be satisfactory, what force on the 100 -mm piston at the same location would be required so as not to compromise the performance of the hydraulic system?

Abid Hussain

Numerade Educator

Problem 22

The summit of Mount Rainier (in Washington) is at an elevation of $4342 \mathrm{~m}$ and is reported to experience an average annual temperature of $-11^{\circ} \mathrm{C}$ and a typical atmospheric pressure of $58 \mathrm{kPa}$. Compare these measured conditions with the temperature and pressure expected using the standard atmosphere given by Equations 2.25 and $2.26 .$

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 23

Within the standard atmosphere, the stratosphere exists between elevations $11-20 \mathrm{~km}$ and is characterized by a constant temperature of $-56.5^{\circ} \mathrm{C}$. The standard-atmosphere pressure at the bottom of the stratosphere (at elevation $11 \mathrm{~km}$ ) is equal to $22.63 \mathrm{kPa}$, and the acceleration due to gravity, $g$, within the stratosphere varies within the narrow range of $9.776-9.761 \mathrm{~m} / \mathrm{s}^{2}$. Assuming that air within the stratosphere behaves like an ideal gas, use the given data to estimate the theoretical pressure at the top of the stratosphere. Compare your result with the standard-atmosphere pressure at $20 \mathrm{~km}$ (given in Appendix B.3).

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 24

Assume standard conditions of temperature and pressure at sea level, a lapse rate of $6.5^{\circ} \mathrm{C} / \mathrm{km},$ and a hydrostatic distribution of pressure in the atmosphere. Using these assumptions, calculate the theoretical temperature and pressure at $1-\mathrm{km}$ elevation intervals between elevations $0 \mathrm{~km}$ and $11 \mathrm{~km}$. Compare your result with the U.S. standard atmosphere given in Appendix B.3. What is the maximum temperature difference in ${ }^{\circ} \mathrm{C}$ and the maximum pressure difference in $\mathrm{kPa}$ ?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 25

Atmospheric pressure at ground level is due to the weight of the overlying atmosphere. The pressure and temperature at ground level are $101.3 \mathrm{kPa}$ and $20^{\circ} \mathrm{C},$ respectively, and the lapse rate is estimated as $6.3^{\circ} \mathrm{C} / \mathrm{km}$. If the top of the atmosphere is estimated as the elevation at which the pressure is $1 \mathrm{~Pa}$, estimate the thickness of the atmosphere.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 26

Blue Mountain Peak is the highest point in Jamaica with an elevation of $2256 \mathrm{~m}$. The average temperature at Blue Mountain Peak is $5^{\circ} \mathrm{C}$, and average climatic conditions at sea level in Jamaica are a temperature of $27^{\circ} \mathrm{C}$ and an atmospheric pressure of $101 \mathrm{kPa}$. (a) Estimate the atmospheric pressure at Blue Mountain Peak. (b) At what temperature will water boil at Blue Mountain Peak?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 27

Show that the parameter $g / R b$ that is used to calculate the pressure distribution in the troposphere can be taken as 5.26 for the standard atmosphere. Use the standard atmosphere to estimate the temperature at which water boils in the capital city of La Paz (Bolivia), which is $3640 \mathrm{~m}$ above sea level.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 28

A gold mine extends to a level $4 \mathrm{~km}$ below land surface. The land surface is approximately at sea level, and a vertical air shaft connects the atmosphere at the surface to the lower elevations of the mine. Assuming standard atmospheric conditions at the land surface, estimate the air pressure at the bottom of the mine. State any assumptions.

Abid Hussain

Numerade Educator

Problem 29

A barometer located at the entrance to the ground floor of the Burj Khalifa building in Dubai estimates a pressure of $100.8 \mathrm{kPa}$ on an average day in August, when the temperature is $37^{\circ} \mathrm{C}$. The height of the Burj Khalifa is reported to be $829.8 \mathrm{~m}$. Estimate the barometric pressure at the top of the building.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 30

A light airplane uses an aneroid barometer to measure changes in altitude. At one elevation, the pressure is $96 \mathrm{kPa}$, and at a higher elevation, the pressure is $85.3 \mathrm{kPa}$. Estimate the difference in altitude corresponding to these measurements. Assume that the airplane is operating in a standard atmosphere.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 31

Consider the case of a manometer attached to a tank containing kerosene at $20^{\circ} \mathrm{C}$ as shown in Figure 2.48 . Atmospheric pressure is $101.3 \mathrm{kPa}$. For the given conditions, what would be the pressure reading on the Bourdon gauge mounted at Point $\mathrm{P}$ ?

Abid Hussain

Numerade Educator

Problem 32

A manometer is attached to a tank containing two immiscible liquids as shown in Figure $2.49 .$ The top (light) liquid has a specific gravity of 0.8 and a thickness of $0.3 \mathrm{~m},$ and the bottom (dense) liquid has a specific gravity of 2.8 and a thickness of $0.3 \mathrm{~m}$. The manometer is connected to the bottom fluid at an elevation $0.1 \mathrm{~m}$ below the fluid interface. (a) Determine the elevation, $\Delta z,$ of the liquid in the manometer above the interface. (b) What is the gauge pressure on the bottom of the tank?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 33

The U-tube manometer shown in Figure 2.50 is used to measure the pressure in the pipeline at $\mathrm{A}$. The water is at $30^{\circ} \mathrm{C}$, and the specific weight of the gauge fluid is 45 $\mathrm{kN} / \mathrm{m}^{3}$. For the given fluid heights, determine the pressure in the pipeline.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 34

The U-tube manometer shown in Figure 2.51 is used to measure the pressure exerted by the compressed air in a storage reservoir containing SAE 10 oil. For the measurements shown, what is the gauge pressure exerted by the compressed air? Assume that all liquids are at a temperature of $20^{\circ} \mathrm{C}$.

Abid Hussain

Numerade Educator

Problem 35

Consider the differential manometer shown in Figure $2.52 .$ Express the pressure difference between Points A and B in terms of $\gamma_{\mathrm{w}}, \gamma_{\mathrm{f}}, h_{1}, h_{2},$ and $h_{3}$.

Abid Hussain

Numerade Educator

Problem 36

The differential manometer shown in Figure 2.53 is used to measure the pressure drop across an orifice plate in a pipe. Because the pressure drop across an orifice plate is proportional to the flow rate through the opening in the plate, such an arrangement is commonly used to measure the volume flow rate in pipes. In the present case, the fluid flowing in the pipe is water at $30^{\circ} \mathrm{C}$ and the specific weight of the gauge fluid is $20.5 \mathrm{kN} / \mathrm{m}^{3} .$ For the given fluid heights, what is the difference in pressure between Points 1 and $2 ?$

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 37

Determine the pressure difference between the water pipe and the oil pipe shown in Figure 2.54

Abid Hussain

Numerade Educator

Problem 38

An inclined manometer is attached to a cylindrical storage tank as shown in Figure $2.55,$ where the tank has a diameter of $1 \mathrm{~m}$ and the manometer has a diameter of $10 \mathrm{~mm}$. The tank contains SAE 30 oil at $20^{\circ} \mathrm{C}$, and initially the air above the oil in the tank is at atmospheric pressure, as is the air above the liquid surface in the inclined manometer. If the storage tank is sealed and the pressure in the tank increases by $200 \mathrm{~Pa}$, what angle, $\theta,$ of the inclined manometer is required such that the liquid in the manometer moves $200 \mathrm{~mm}$ from its original location?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 39

The pressure of water flowing through a pipe is measured by the inclined differential manometer shown in Figure $2.56 .$ For the values given, calculate the pressure in the pipe.

Abid Hussain

Numerade Educator

Problem 40

A $3.2 \mathrm{~m} \times 4.1 \mathrm{~m}$ rectangular gate is mounted vertically in the side of a water storage reservoir such that the 3.2 -m side of the gate is parallel to the water surface. The gate weighs $20 \mathrm{kN}$ and is mounted in vertical guides such that the gate can be opened by pulling upward with cables attached to the top of the gate. The coefficient of friction between the guides and the gate is $0.05 .$ When the water level is $2 \mathrm{~m}$ above the top of the gate, what force is required to lift the gate?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 41

A section of a wall is to be constructed by pouring liquid concrete into a form in the shape of a wall as shown in Figure 2.57 . The wall section is $4 \mathrm{~m}$ high, $3.5 \mathrm{~m}$ long, and $0.3 \mathrm{~m}$ thick. The specific gravity of liquid concrete can be estimated as $2.5,$ and before the concrete dries, the pressure distribution in the liquid concrete can be assumed to be hydrostatic. The support beams on each side of the formwork should be located such that they are aligned with the resultant hydrostatic force on the formwork. How far above the bottom of the wall should the support beams be located? What should be the lateral location of the support beams?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 42

A $2 \mathrm{~m} \times 3 \mathrm{~m}$ rectangular gate is located on the sloping side of a water reservoir such that the $2-\mathrm{m}$ side of the gate is parallel to the water surface. The side of the reservoir (and the gate) slopes at an angle of $60^{\circ}$ to the horizontal, and the top of the gate is $2.5 \mathrm{~m}$ vertically below the water surface. Estimate the resultant hydrostatic force on the gate and the effective location of this resultant force, as measured vertically downward from the water surface.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 43

The 2.2 -m-diameter pipe illustrated in Figure 2.58 drains water from a reservoir, where the crown of the pipe is $3 \mathrm{~m}$ below the surface of the reservoir. If the pipe entrance is covered with a 2.2 -m-diameter gate hinged at A, determine the magnitude and location of the net hydrostatic force on the gate. What torque would need to be applied at $A$ to open the gate?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 44

A vertical rectangular gate of height $h$ and width $w$ is installed such that the top of the gate is a distance $d$ below the water surface in a reservoir. Calculate the hydrostatic force on the gate and show that the location of the center of pressure is given by
$$
y_{\mathrm{cp}}=d+\frac{L(3 d+2 h)}{6 d+3 h}
$$

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 45

A reservoir is to be built to contain a slurry as shown in Figure $2.59 .$ It is proposed that the retaining wall of the reservoir be $3 \mathrm{~m}$ high and $1 \mathrm{~m}$ thick and be made of concrete with a density of $2800 \mathrm{~kg} / \mathrm{m}^{3}$. A review of site conditions indicates that the coefficient of friction between the concrete wall and the underlying soil is equal to $0.35 .$ It is expected that the slurry will have a density of $1500 \mathrm{~kg} / \mathrm{m}^{3}$. (a) Estimate the height, $h,$ of slurry that will cause the wall to fail by sliding over the soil, (i.e., shear failure). (b) Estimate the height, $h$, of slurry that will cause the wall to topple over by rotating about the point $\mathrm{P}$. Which is the more likely failure mode?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 46

The proposed concrete dam shown in Figure 2.60 has a bottom width of $25 \mathrm{~m}$, it has a top width of $5 \mathrm{~m}$, and the water level behind the dam is to be $4 \mathrm{~m}$ below the top of the dam. The specific gravity of the concrete used in the dam is $2.4 .$ The water depth downstream of the dam is $3 \mathrm{~m}$. Water seepage under the dam causes an uplift pressure distribution that varies linearly from the upstream hydrostatic pressure to the downstream hydrostatic pressure. Estimate the height, $h,$ of water behind the dam that will cause the dam to overturn. Assume water at $20^{\circ} \mathrm{C}$.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 47

A vertical circular gate of diameter $D$ is installed such that the top of the gate is a distance $d$ below the water surface in a reservoir. Calculate the hydrostatic force on the gate and show that the location of the center of pressure is given by
$$
y_{\mathrm{cp}}=D+\frac{D(8 d+5 D)}{16 d+8 D}
$$

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 48

A $2.5-\mathrm{m}$ -diameter gate is connected to a water reservoir as shown in Figure 2.61 . The gate is oriented at $35^{\circ}$ to the horizontal, the centroid of the gate is $1.5 \mathrm{~m}$ vertically below the water surface, and the weight of the gate is $500 \mathrm{kN}$. The gate is pin-connected to the reservoir at the top of the gate (at $\mathrm{P}$ ) and is opened by applying a vertical force at the bottom of the gate (at Q). (a) What is the magnitude of the resultant hydrostatic force and its location relative to the top of the gate?
(b) What is the magnitude of the force required to open the gate?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 49

A semicircular steel gate is installed in the side of a water storage reservoir that has a side slope of $50^{\circ}$ as shown in Figure 2.62 . The gate has a radius of $3 \mathrm{~m}$, the top of the gate is supported by a horizontal shaft attached to the reservoir, and the bottom of the gate is supported at the point $\mathrm{P}$ with a force $F_{\mathrm{P}}$. When the depth of water in the tank is $15 \mathrm{~m}$, what support force is necessary to keep the gate from opening?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 50

A circular hatch is located in a sloping wall of a water storage reservoir, where the wall slopes at $35^{\circ}$ to the horizontal, the radius of the hatch is $420 \mathrm{~mm}$, and the center of the hatch is $3 \mathrm{~m}$ below the water surface (measured along the sloping wall). Find the magnitude and location of the resultant hydrostatic force on the hatch.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 51

Determine the force at $P$ needed to hold the 4 -m-wide gate in position as shown in Figure 2.63. You may assume that the support surface at $\mathrm{P}$ is smooth (i.e., frictionless).

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 52

The $500-\mathrm{kg}$, 5 -m-wide rectangular gate shown in Figure 2.64 is hinged at $\mathrm{B}$ and makes an angle of $45^{\circ}$ with the floor at A. (a) If the gate is opened by applying a normal force at the center of the gate, determine the force required to open the gate. (b) If you could choose any location to apply the force to open the gate, what location would you choose and what force would need to be applied?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 53

The rectangular gate shown in Figure 2.65 is intended to control the flow of seawater into a freshwater reservoir. If the fresh water has a density of $998 \mathrm{~kg} / \mathrm{m}^{3},$ the seawater has a density of $1025 \mathrm{~kg} / \mathrm{m}^{3},$ and the gate weighs $0.448 \mathrm{kN} / \mathrm{m}$ (into the page), find the depth of seawater when the gate opens.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 54

The water in a reservoir is contained by an elliptical gate illustrated in Figure 2.66 , where $D$ is the diameter of the pipe leading to the gate and $D$ and $D / \sin \theta$ are the lengths of the minor and major principal axes of the elliptical gate, respectively. If $D=1.2 \mathrm{~m}, \theta=30^{\circ},$ and the water surface in the reservoir is $9 \mathrm{~m}$ above the centerline of the gate, determine the resultant hydrostatic force on the gate and the location of the center of pressure. What moment at Point $\mathrm{P}$ would be required to keep the gate closed? Neglect the weight of the gate.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 55

A square flap gate is shown in Figure $2.67,$ where the side dimensions of the gate $(b)$ are $1.5 \mathrm{~m},$ the gate weighs $8 \mathrm{kN},$ the gate is hinged $1 \mathrm{~m}$ above its center $(a=0.25$ $\mathrm{m}$ ), and the face of the gate is sloped $4^{\circ}$ from the vertical. To what depth will water rise behind the gate before it opens?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 56

The 2 -m-wide gate shown in Figure 2.68 is pinned at $\mathrm{P}$ and is connected to a restraining spring that keeps the gate closed until the hydrostatic force is sufficient to open it. If the gate is to open when the depth of water, $y,$ is $4 \mathrm{~m}$, determine the required restraining force that should be exerted by the spring.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 57

Gate $\mathrm{AB}$ in Figure 2.69 is $1.52 \mathrm{~m}$ wide and opens to let fresh water out when the ocean tide is dropping. The hinge at $\mathrm{A}$ is $0.61 \mathrm{~m}$ above the freshwater level. At what ocean level $h$ will the gate open? Neglect the weight of the gate.

Hunza Gilgit

Numerade Educator

Problem 58

A hinged gate is located under an elevated dock in a freshwater reservoir as shown in Figure $2.70 .$ The hinge is located $0.5 \mathrm{~m}$ from the bottom of the reservoir, and the top of the gate is $0.7 \mathrm{~m}$ above the hinge. The width of the gate is $3 \mathrm{~m}$.
(a) What depth of water, $h$, will cause the gate to open? Assume water at $20^{\circ} \mathrm{C}$.
(b) Show that the depth of liquid required to open the gate is independent of the density of the liquid. Note that the gate consists of a single $1.2 \mathrm{~m} \times 3 \mathrm{~m}$ rectangular piece and that the dock offers no resistance to the rotation of the gate.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 59

The two sides of a V-shaped water trough are hinged together at the bottom where they meet, as shown in Figure 2.71 , making an angle of $45^{\circ}$ with the ground from both sides. Each side is $0.75 \mathrm{~m}$ long, and the two sides are held together by cables placed every $6 \mathrm{~m}$ along the length of the trough. Calculate the tension in each cable when the trough is filled to the level of the cable.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 60

Consider the pressurized storage tank shown in Figure $2.72,$ where water at $20^{\circ} \mathrm{C}$ is contained by a rectangular gate that is $2 \mathrm{~m}$ long by $3 \mathrm{~m}$ wide and is open to the atmosphere on one side. The side of the tank in contact with water makes an angle of $30^{\circ}$ with the horizontal, and above the gate support is $2 \mathrm{~m}$ of inclined surface and $1 \mathrm{~m}$ of vertical surface below the water surface. The air above the water surface is at a gauge pressure of $300 \mathrm{kPa}$. What is the magnitude of the net hydrostatic force on the gate? At what depth below the water surface does the resultant hydrostatic force act?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 61

Find the magnitude and location of the resultant hydrostatic force on the gate shown in Figure 2.73 . Also find the magnitude of the moment about axis XX that would be required to open the gate. The pressure above the water surface is equal to atmospheric pressure, as is the pressure on the other side of the gate.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 62

Consider the semi-elliptical gate shown in Figure 2.74. Determine the magnitude and location of the resultant hydrostatic force on the gate and find the magnitude of the moment about axis XX that would be required to open the gate. The pressure above the water surface is equal to atmospheric pressure, as is the pressure on the other side of the gate.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 63

Consider the submerged gate in a stratified liquid as shown in Figure $2.75 .$ Find the magnitude and location of the resultant hydrostatic force on the gate.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 64

Determine the magnitude and location of the resultant hydrostatic force on the elliptical gate shown in Figure $2.76 .$ The center of the ellipse coincides with the interface between the two fluids.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 65

Pressurized water fills the tank in Figure 2.77 . Compute the net hydrostatic force on the conical surface $\mathrm{ABC}$.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 66

The step dam shown in Figure 2.78 is to be constructed on an earth foundation that can support a maximum shear force of $2500 \mathrm{kN}$. The length of the dam is $10 \mathrm{~m}$, and the step height and depth are to be equal. What maximum step height can be used without exceeding the support capacity of the foundation? Under this limiting condition, what is the vertical hydrostatic force on the dam? Assume water at $20^{\circ} \mathrm{C}$.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 67

A 5-m-long seawall separates a freshwater body from a saltwater body as shown in Figure $2.79 .$ The wall is a total of $4 \mathrm{~m}$ high, with the top half of the wall being semicircular. Under design conditions, the surface of the freshwater body is at the top of the wall and the surface of the saltwater body is at the midheight of the wall. What is the net hydrostatic force on the wall under the design condition? Assume that both the fresh water and salt water are at $20^{\circ} \mathrm{C}$.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 68

A gate in the shape of a quarter of a circle divides a freshwater body from a saltwater body as shown in Figure $2.80 .$ The gate has a radius of $3.5 \mathrm{~m}$ and a length (perpendicular to the page) of $4.8 \mathrm{~m}$. (a) Determine the height, $h,$ of salt water that will cause the horizontal hydrostatic forces on both sides of the gate to be equal. (b) Determine the height of salt water that will cause the vertical hydrostatic forces on the gate to be equal. Assume that both the fresh water and salt water are at $20^{\circ} \mathrm{C}$.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 69

The flow control gate shown in Figure 2.81 has the shape of a quadrant of a circle of radius $1 \mathrm{~m}$ and a weight of $40 \mathrm{kN}$ per meter of length perpendicular to the page. The gate is pin-connected at P. Determine the height, $h,$ of water in the reservoir behind the gate that will cause the gate to open. Assume water at $20^{\circ} \mathrm{C}$.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 70

An 8 -m-deep by 20 -m-long aquarium is to be designed with a glass viewing area at the bottom. If the viewing section is shaped like the quadrant of a circle (Figure 2.82), calculate the hydrostatic force on the viewing glass.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 71

The gate shown in Figure 2.83 is used to contain water in an aquarium. The gate extends from A to C, consists of an arc of a circle between A and B (the viewing window), and is planar between $\mathrm{B}$ and $\mathrm{C}$. The gate is $2 \mathrm{~m}$ wide, is pinned at $\mathrm{A}$, and rests on a frictionless surface at $\mathrm{C}$. For the water-surface elevation shown in Figure $2.83,$ determine the hydrostatic force on the gate.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 72

Water is contained in a reservoir by the Tainter gate illustrated in Figure $2.84 .$ The gate has the shape of a quadrant of a circle, it is $5 \mathrm{~m}$ wide, it weighs $10 \mathrm{kN},$ and its center of gravity is at Point $\mathrm{G}$. Determine the net hydrostatic force on the gate and the magnitude and direction of the moment that must be applied at $\mathrm{P}$ to open the gate.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 73

The face of the dam shown in Figure 2.85 is curved according to the relation $y=$ $x^{2} / 2.4,$ where $y$ and $x$ are in meters. The height of the water surface above the horizontal plane through $\mathrm{A}$ is $15.25 \mathrm{~m} .$ Calculate the resultant force due to the water acting on a unit breadth of the dam and determine the position of the point $\mathrm{B}$ at which the line of action of this force cuts the horizontal plane through $\mathrm{A}$.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 74

An aquarium proposes using a 1 -m-diameter hemispherical viewing glass in its piranha exhibit as shown in Figure $2.86 .$ The center of the viewing glass is proposed to be $5 \mathrm{~m}$ below the water surface. The viewing glass is to be fixed to the wall using rivets, each capable of supporting $100 \mathrm{~N}$ of normal force and $5 \mathrm{~N}$ of shear force. What is the minimum number of rivets required? If a flat circular viewing glass is used instead, what is the minimum number of rivets required? What is the ratio of the hydrostatic force on the top half of the viewing glass to the hydrostatic force on the bottom half of the viewing glass? Comment on how you would distribute the rivets around the viewing glass.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 75

A water trough of a semicircular cross section of radius $0.5 \mathrm{~m}$ consists of two symmetric parts hinged together at the bottom as shown in Figure 2.87 . The two parts are held together by cables placed every $3 \mathrm{~m}$ along the length of the trough. Calculate the tension in each cable when the trough is filled to the rim.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 76

Gate $\mathrm{AB}$ in Figure 2.88 is a three-eighths circle, is $3 \mathrm{~m}$ wide into the page, is hinged at $\mathrm{B}$, and is resting against a smooth wall at $\mathrm{A}$. The specific weight of the seawater being retained by the gate is $10.05 \mathrm{kN} / \mathrm{m}^{3}$. Compute the reaction forces at Points $\mathrm{A}$ and $\mathrm{B}$.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 77

A natural gas pipeline has a diameter of $410 \mathrm{~mm}$ and a wall thickness of $4.2 \mathrm{~mm}$. If the pressure of the gas in the pipeline is equal to $800 \mathrm{kPa}$, estimate the average circumferential stress in the pipe wall.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 78

The weight of a solid object in air is measured as $40 \mathrm{~N},$ and the weight of this same object in water at $20^{\circ} \mathrm{C}$ is measured as $25 \mathrm{~N}$. Estimate the specific weight and volume of the object.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 79

Archimedes' principle was developed as a by-product of a royal project undertaken by Archimedes to determine whether the crown of King Hiero II of Syracuse, Sicily, was made of pure gold. Knowing the specific gravity of gold and being able to weigh the king's crown in both air and water, Archimedes, in effect, developed a relationship between the specific gravity of an object and the ratio of the weight of an object in water to the weight of that same object in air. (a) Determine this relationship. (b) Apply this relationship to determine the specific gravity of the object described in Problem 2.78

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 80

A hot air balloon has a diameter of $15 \mathrm{~m}$, and the weight of the balloon plus attached load is $2 \mathrm{kN}$. To what temperature must the air in the balloon be heated to achieve liftoff? Assume standard sea-level conditions in the surrounding air.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 81

A 3 -m-diameter balloon with a hard (nonexpandable) shell is filled with helium and released into the atmosphere. The balloon plus the helium has a mass of $8 \mathrm{~kg} .$ At what elevation will the balloon stabilize in a standard atmosphere?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 82

A hot air balloon is to carry a weight of $1.5 \mathrm{kN}$ over an area that has a typical atmospheric pressure of $101 \mathrm{kPa}$ and a typical summer temperature of $20^{\circ} \mathrm{C}$. The balloon is made of a material that has a mass of $80 \mathrm{~g} / \mathrm{m}^{2}$, and the air in the balloon is to be heated to a temperature of $80^{\circ} \mathrm{C}$. What will be the diameter of the balloon under stable conditions?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 83

The drag force, $F_{\mathrm{D}}$, on a spherical particle settling with velocity, $v$, in a fluid can be approximated by
$$
F_{\mathrm{D}}=3 \pi \mu v D
$$
where $\mu$ is the viscosity of the fluid and $D$ is the particle diameter. If a 2 -mmdiameter particle with a specific gravity of 2.65 is stirred up from the bottom of a river, estimate the sedimentation velocity, $v$. (Hint: A particle settles with a constant velocity when the sum of the forces on the particle is equal to zero.)

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 84

A cabin used by recreational scuba divers in a lake is $10 \mathrm{~m}$ long by $15 \mathrm{~m}$ wide by $4 \mathrm{~m}$ high as shown in Figure $2.89 .$ The lake contains fresh water at $20^{\circ} \mathrm{C}$. The lighter section of the cabin has an effective specific gravity of $1.5,$ and a heavier section of the cabin has an effective specific gravity of $3.0 .$ The cabin is kept horizontal by a linear support wall located at a distance $x$ from the centroidal axis of the cabin. Determine the required support force and the distance of the support wall from the centroidal axis.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 85

A tank contains a layer of SAE 30 oil floating on a layer of water, where both liquids are at $20^{\circ} \mathrm{C}$. A cube of dimensions $0.15 \mathrm{~m} \times 0.15 \mathrm{~m} \times 0.15 \mathrm{~m}$ is placed in the tank, and it is observed that the cube locates itself such that its top and bottom faces are parallel to the interface between fluids and $15 \%$ of the height of the cube is located in the oil layer. Estimate the density of the cube.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 86

The volume of seawater displaced by ships is usually controlled by the intake and discharge of ballast water, which increases and decreases the weight of the ship, respectively. Show that the percentage change in displaced seawater is equal to the percentage change in the weight of the ship.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 87

The hollow steel sphere shown in Figure 2.90 is to be used to house instruments for oceanic research and is to be sized such that $90 \%$ of its volume is below the surface of the ocean. The wall thickness is to be $25 \mathrm{~mm}$, the weight of the instrumentation to be contained in the sphere is $500 \mathrm{~N},$ and the density of the steel to be used in manufacturing the sphere is $8000 \mathrm{~kg} / \mathrm{m}^{3}$. What should be the diameter of the sphere?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 88

The cylindrical object shown in Figure 2.91 is to be sized such that it floats in a $1-$ m-deep pool of water with a minimum clearance of $100 \mathrm{~mm}$ from the bottom of the pool. The specific gravity of the object is estimated as $0.85 .$ What is the maximum allowable diameter of the object?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 89

The standard canoe model from a manufacturer weighs $810 \mathrm{~N},$ and an ultralight (and ultrastrong) model of the same size and shape weighs $220 \mathrm{~N}$. Both models are capable of carrying the same load. For any given load, determine the additional volume of water displaced by the heavier model.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 90

If a body of specific gravity $\mathrm{SG}_{1}$ is placed in a liquid of specific gravity $\mathrm{SG}_{2}$, what fraction of the total volume of the body will be above the surface of the liquid? If an iceberg has a specific gravity of 0.95 and is floating in seawater with a specific gravity of $1.25,$ what fraction of the iceberg is above water?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 91

A new tactic in naval warfare is being proposed in which ships are sunk by a highdensity release of bubbles from below the ship to reduce the effective density of the water surrounding the ship. This tactic is illustrated in Figure 2.92. If a ship has a specific gravity of 0.8 and bubbles with a diameter of $10 \mathrm{~mm}$ are released, what is the number of bubbles per $\mathrm{m}^{3}$ required to sink the ship? Does your answer depend on the shape of the ship? Explain.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 92

It is conventional wisdom that in areas where the groundwater is close to the land surface, swimming pools should not be completely emptied. The dimensions of a swimming pool are $10 \mathrm{~m}$ long by $5 \mathrm{~m}$ wide by $2.5 \mathrm{~m}$ deep, the weight of the pool is $500 \mathrm{kN},$ and the groundwater is $1.25 \mathrm{~m}$ below the top of the pool. Determine the minimum depth of water that must be maintained in the pool. (Hint: The net force on the pool structure must remain downward.)

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 93

The 3 -m-wide barge shown in Figure 2.93 weighs $20 \mathrm{kN}$ when it is empty. It is proposed that the barge carry a $250-\mathrm{kN}$ load. Predict the draft in salt water $(\mathrm{SG}=1.03)$.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 94

Observations of a floating body indicate that $75 \%$ of the body is submerged below the water surface. It is known that $90 \%$ of the volume of the body consists of open (air) space. Estimate the average density of the whole body and the average density of the solid material that constitutes the body.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 95

A rectangular prism of height $2 \mathrm{~m}$, width $W,$ and length $L$ is placed in a stratified liquid as shown in Figure 2.94 . The top layer of the liquid has a specific gravity of 1.2 and a thickness of $1.2 \mathrm{~m}$, and the bottom layer has a specific gravity of 1.6 .
(a) Determine the minimum specific gravity of the body for which the body will fully penetrate the top layer. (b) Determine the depth of penetration, if any, into the bottom layer when the body has a specific gravity equal to 1.0 .

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 96

Consider a buoy attached to a supporting cable as shown in Figure $2.95 .$ The buoy is $3 \mathrm{~m}$ long, it has a diameter of $200 \mathrm{~mm}$, and its specific gravity relative to seawater is $0.60 .$ In a coastal (seawater) environment at low tide, the buoy is not fully submerged and inclines as shown in Figure $2.95 .$ At high tide, the buoy is fully submerged and is oriented vertically. (a) What is the tension in the support cable at low tide? (b) What is the tension in the support cable at high tide?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 97

A hydrometer with a stem diameter of $12 \mathrm{~mm}$ weighs $0.246 \mathrm{~N}$. An engineer places the hydrometer in pure water at $20^{\circ} \mathrm{C}$ and marks the place on the stem corresponding to the water surface. When the engineer places the hydrometer in a test liquid, the mark is $2 \mathrm{~cm}$ above the surface of the liquid. Estimate the specific gravity of the test liquid.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 98

A hydrometer with a stem diameter of $9 \mathrm{~mm}$ is placed in distilled water, and the volume of the hydrometer below the water surface is estimated to be $20 \mathrm{~cm}^{3}$. If the hydrometer is placed in a liquid with a specific gravity of 1.2 , how far above the liquid surface will the distilled water mark be located?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 99

A cylinder of diameter $0.70 \mathrm{~m}$ and height $0.60 \mathrm{~m}$ is placed in a liquid as shown in Figure 2.96. The cylinder has a specific gravity of $0.65,$ and the liquid has a specific gravity of 0.90 . Determine whether the cylinder is stable at its initial orientation.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 100

The canoe shown in Figure 2.97 has a length of $4 \mathrm{~m},$ and with one rower in the canoe, the dimensions of the submerged portion of the canoe are as shown as well. Estimate the maximum height of the center of gravity of the canoe plus the rower (relative to the bottom of the canoe) for the canoe to be stable.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 101

A cylindrical container has a diameter of $0.4 \mathrm{~m}$ and contains kerosene to a depth of $0.6 \mathrm{~m}$. The temperature of the kerosene is $20^{\circ} \mathrm{C}$. If the container, with its long axis oriented vertically, is placed on the floor of a delivery elevator that ascends with an acceleration of $1.5 \mathrm{~m} / \mathrm{s}^{2}$, what is the pressure in the fluid on the bottom of the container? What force does the container exert on the floor of the elevator? Assume that the mass of the container is negligible compared with that of the kerosene.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 102

A water truck is mounted with a cylindrical tank that has a diameter of $2 \mathrm{~m}$ and a length of $10 \mathrm{~m}$. The long axis of the tank is oriented with the direction of truck motion. If the tank is filled with water at $20^{\circ} \mathrm{C}$ and the truck accelerates at a rate of $2 \mathrm{~m} / \mathrm{s}^{2}$, estimate the difference in magnitude between the resultant hydrostatic force on the front and back of the tank.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 103

A 4-m-diameter, 2 -m-deep cylindrical tank containing a caustic liquid with a density of $1040 \mathrm{~kg} / \mathrm{m}^{3}$ is placed on a moving platform as shown in Figure 2.98 . The tank is filled to a depth of $1.2 \mathrm{~m}$. (a) What is the maximum allowable acceleration of the moving platform that will prevent the liquid from spilling out of the tank? (b) At this limiting condition, what is the (gauge) pressure at the bottom front of the tank and what is the gauge pressure at the bottom back of the tank?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 104

A rectangular tank $3 \mathrm{~m}$ long, $0.8 \mathrm{~m}$ wide, and $1.6 \mathrm{~m}$ deep is filled with kerosene to a depth of $1.2 \mathrm{~m}$ and placed on a truck. Consider two orientations of the tank on the truck: (a) the long side aligned with the direction of truck motion and (b) the short side aligned with the direction of truck motion. Which orientation would allow the greatest truck acceleration without spillage? What is that acceleration?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 105

Sensors in a decelerating tanker truck indicate that the liquid in the tank has a $10^{\circ}$ slope. At what rate is the truck decelerating?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 106

If a tanker truck accelerates from rest to a highway speed of $90 \mathrm{~km} / \mathrm{h}$ in 10 seconds, what slope of (unconfined) liquid would be expected to occur in the tank? Assume that the truck accelerates at a constant rate.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 107

The tanker truck shown in Figure 2.99 accelerates at a rate of $5 \mathrm{~m} / \mathrm{s}^{2}$ down a $25^{\circ}$ incline. Determine the slope of the liquid surface in the tank. Is the slope of the liquid surface greater than or less than the slope of the incline?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 108

Consider the case in which the tanker truck illustrated in Figure 2.99 completely loses power, the road is iced over, and the truck slides down the incline without any frictional resistance from the road. Under this condition, what is the slope of the liquid surface in the tank?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 109

Water at $20^{\circ} \mathrm{C}$ is contained in a U-tube as shown in Figure 2.100 . Atmospheric pressure is $101 \mathrm{kPa}$. (a) What rate of acceleration toward the right would cause the liquid level in the left-hand arm of the U-tube to be $40 \mathrm{~mm}$ higher than the liquid level in the right-hand arm of the U-tube? (b) What rate of rotation around the $z$ -axis would cause cavitation to occur at the center of the U-tube (i.e., at Point P)?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 110

If the $U$ -tube shown in Figure 2.100 is rotated about a vertical axis that is $50 \mathrm{~mm}$ to the right of Point $\mathrm{P}$ and parallel to the $z$ -axis, what rate of rotation would cause the water levels in the arms of the $U$ -tube to differ by $40 \mathrm{~mm}$ ?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 111

The U-tube shown in Figure 2.101 is in the shape of a semicircle between A and $\mathrm{B}$, where the semicircle has a radius of $0.2 \mathrm{~m}$ and is centered at $\mathrm{P}$. The $\mathrm{U}$ -tube is filled with a liquid such that the levels are the same at $\mathrm{A}$ and $\mathrm{B}$, and then the tube is sealed at A. The liquid remains open to the atmosphere at $\mathrm{B}$. The U-tube is then rotated at 450 rpm about a vertical axis that is $0.1 \mathrm{~m}$ from $\mathrm{P}$. (a) Identify the location within the U-tube where the pressure is a minimum. (b) Plot the gauge pressure in the U-tube as a function of the radial distance from the axis of rotation. (c) If the liquid is water at $20^{\circ} \mathrm{C}$ and atmospheric pressure is $101.3 \mathrm{kPa}$, determine the minimum absolute pressure in the U-tube and assess whether cavitation will occur under the given conditions.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 112

A barrel is filled with SAE 10 oil at $20^{\circ} \mathrm{C}$ and then sealed. The barrel has a diameter of $0.6 \mathrm{~m}$, has a height of $2 \mathrm{~m},$ and is rotated about its central axis at a rate of $40 \mathrm{rad} / \mathrm{s}$. At any given height above the bottom of the barrel, what is the difference between the pressure at the center of the barrel and the pressure on the perimeter of the barrel?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 113

A sealed cylinder of diameter $0.50 \mathrm{~m}$ is rotated at $400 \mathrm{rpm}$ as shown in Figure 2.102 . The cylinder contains water at $20^{\circ} \mathrm{C},$ and the gauge pressure on the periphery of the top surface of the cylinder is measured as $200 \mathrm{kPa}$. Determine the force on the top of the cylinder.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 114

A cylindrical reservoir containing a liquid can be used to determine the rotational speed of a platform as shown in Figure $2.103 .$ In this particular case, a 4 -cm-diameter cylindrical reservoir is placed at the center of a rotating platform and the range of the liquid surface in the reservoir is observed to be $1.5 \mathrm{~cm}$. Determine the rate at which the platform is rotating in revolutions per minute.

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator

Problem 115

A 1.6-m-diameter cylinder is filled with a liquid to a depth of $1.1 \mathrm{~m}$ and rotated about its center axis. (a) Assuming that the cylinder is tall enough for the liquid not to spill, at what rotational speed will the liquid surface intersect the bottom of the cylinder? (b) If the cylinder is rotated at 60 rpm, what is the minimum height of the cylinder that prevents spillage?

Kudakwashe Mapiki

Kudakwashe Mapiki

Numerade Educator