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Chapter 2 FLUID STATICS

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The science of fluid statics : the study of pressure and its variation throughout a fluid the study of pressure forces on finite surfaces Special cases of fluids moving as solids are included in the treatment of statics because of the similarity of forces involved. Since there is no motion of a fluid layer relative to an adjacent layer, there are no shear stresses in the fluid all free bodies in fluid statics have only normal pressure forces acting on their surfaces

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2.1 PRESSURE AT A POINT Average pressure: dividing the normal force pushing against a plane area by the area. Pressure at a point: the limit of the ratio of normal force to area as the area approaches zero size at the point. At a point: a fluid at rest has the same pressure in all directions an element δA of very small area, free to rotate about its center when submerged in a fluid at rest, will have a force of constant magnitude acting on either side of it, regardless of its orientation. To demonstrate this, a small wedge-shaped free body of unit width is taken at the point (x, y) in a fluid at rest (Fig.2.1)

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Figure 2.1 Free-body diagram of wedge-shaped particle

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There can be no shear forces the only forces are the normal surface forces and gravity the equations of motion in the x and y directions p x, p y, p s are the average pressures on the three faces, γ is the unit gravity force of the fluid, ρ is its density, and a x, a y are the accelerations When the limit is taken as the free body is reduced to zero size by allowing the inclined face to approach (x, y) while maintaining the same angle θ, and using the equations simplify to Last term of the second equation – infinitestimal of higher of smallness, may be neglected

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When divided by δy and δx, respectively, the equations can be combined θ is any arbitrary angle this equation proves that the pressure is the same in all directions at a point in a static fluid Although the proof was carried out for a two-dimensional case, it may be demonstrated for the three-dimensional case with the equilibrium equations for a small tetrahedron of fluid with three faces in the coordinate planes and the fourth face inclined arbitrarily. If the fluid is in motion (one layer moves relative lo an adjacent layer), shear stresses occur and the normal stresses are no longer the same in all directions at a point the pressure is defined as the average of any three mutually perpendicular normal compressive stresses at a point, Fictitious fluid of zero viscosity (frictionless fluid): no shear stresses can occur at a point the pressure is the same in all directions

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2.2 BASIC EQUATION OF FLUID STATICS Pressure Variation in a Static Fluid Force balance: The forces acting on an element of fluid at rest (Fig. 2.2): surface forces and body forces. With gravity the only body force acting, and by taking the y axis vertically upward, it is -γ δx δy δz in the y direction With pressure p at its center (x, y, z) the approximate force exerted on the side normal to the y axis closest to the origin and the opposite e side are approximately δy/2 – the distance from center to a face normal to y

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Figure 2.2 Rectangular parallelepiped element of fluid at rest

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Summing the forces acting on the element in the y direction For the x and z directions, since no body forces act, The elemental force vector δF If the element is reduced to zero size, alter dividing through by δx δy δz = δV, the expression becomes exact. This is the resultant force per unit volume at a point, which must be equated to zero for a fluid at rest. The gradient ∇ is

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- ∇ p is the vector field f or the surface pressure force per unit volume The fluid static law or variation of pressure is then For an inviscid fluid in motion, or a fluid so moving that the shear stress is everywhere zero, Newton's second law takes the form a is the acceleration of the fluid element, f - jγ is the resultant fluid force when gravity is the only body force acting

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In component form, Eq. (2.2.4) becomes The partials, for variation in horizontal directions, are one form of Pascal's law; they state that two points at the same elevation in the same continuous mass or fluid at rest have the same pressure. Since p is a function of y only, relates the change of pressure to unit gravity force and change of elevation and holds for both compressible and incompressible fluids For fluids that may be considered homogeneous and incompressible, γ is constant, and the above equation, when integrated, becomes in which c is the constant of integration. The hydrostatic law of variation of pressure is frequently written in the form h = -y, p is the increase in pressure from that at the free surface

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Example 2.1 An oceanographer is to design a sea lab 5 m high to withstand submersion to 100 m, measured from sea level to the top of the sea lab. Find the pressure variation on a side of the container and the pressure on the top if the relative density of salt water is 1.020. At the top, h = 100 m, and If y is measured from the top of the sea lab downward, the pressure variation is #

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Pressure Variation in a Compressible Fluid When the fluid is a perfect gas at rest at constant temperature When the value of γ in Eq. (2.2.7) is replaced by ρ g and ρ is eliminated between Eqs. (2.2.7) and (2.2.9), If P = P 0 when ρ = ρ 0, integration between limits - the equation for variation of pressure with elevation in an isothermal gas - constant temperature gradient of atmosphere

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Example 2.2 Assuming isothermal conditions to prevail in the atmosphere, compute the pressure and density at 2000 m elevation if P = 105Pa, ρ = 1.24 kg/m 3 at sea level. From Eq. (2.2.12) Then, from Eq. (2.2.9) #

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2.3 UNITS AND SCALES OF PRESSURE MEASUREMENT Pressure may be expressed with reference to any arbitrary datum absolute zero local atmospheric pressure Absolute pressure: difference between its value and a complete vacuum Gage pressure: difference between its value and the local atmospheric pressure

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Figure 2.3 Bourdon gage.

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The bourdon gage (Fig. 2.3): typical of the devices used for measuring gage pressures pressure element is a hollow, curved, flat metallic tube closed at one end; the other end is connected to the pressure to be measured when the internal pressure is increased, the tube tends to straighten, pulling on a linkage to which is attached a pointer and causing the pointer to move the dial reads zero when the inside and outside of the tube are at the same pressure, regardless of its particular value the gage measures pressure relative to the pressure of the medium surrounding the tube, which is the local atmosphere

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Figure 2.4 Units and scales for pressure measurement

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Figure 2.4: the data and the relations of the common units of pressure measurement Standard atmospheric pressure is the mean pressure at sea level, 760 mm Hg. A pressure expressed in terms of the length of a column of liquid is equivalent to the force per unit area at the base of the column. The relation for variation of pressure with altitude in a liquid p = γh [Eq. (2.2.8)] (p is in pascals, γ in newtons per cubic metre, and h in metres) With the unit gravity force of any liquid expressed as its relative density S times the unit gravity force of water: Water: γ may be taken as 9806 N/m 3.

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Figure 2.5 Mercury barometer Local atmospheric pressure is measured by mercury barometer aneroid barometer (measures the difference in pressure between the atmosphere and an evacuated box or tube in a manner analogous to the bourdon gage except that the tube is evacuated and sealed) Mercury barometer: glass tube closed at one end, filled with mercury, and inverted so that the open end is submerged in mercury. It has a scale: the height of column R can be determined The space above the mercury contains mercury vapor. If the pressure of the mercury vapor hv is given in millimetres of mercury and R is measured in the same units, the pressure at A may be expressed as (mm Hg)

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Figure 2.4: a pressure may be located vertically on the chart, which indicates its relation to absolute zero and to local atmospheric pressure. If the point is below the local-atmospheric-pressure line and is referred to gage datum, it is called negative, suction, or vacuum. Example: the pressure 460 mm Hg abs, as at 1, with barometer reading 720 mm, may be expressed as -260 mm Hg, 260 mm Hg suction, or 260 mm Hg vacuum. Note: P abs = p bar + p gage Absolute pressures : P, gage pressures : p.

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Example 2.3 The rate of temperature change in the atmosphere with change in elevation is called its lapse rate. The motion of a parcel of air depends on the density of the parcel relative to the density of the surrounding (ambient) air. However, as the parcel ascends through the atmosphere, the air pressure decreases, the parcel expands, and its temperature decreases at a rate known as the dry adiabatic lapse rate. A firm wants lo burn a large quantity of refuse. It is estimated that the temperature of the smoke plume at 10 m above the ground will be 11 o C greater than that of the ambient air. For the following conditions determine what will happen to the smoke. (a) At standard atmospheric lapse rate β = -0.00651 o C per meter and t 0 =20 o C. (b) At an inverted lapse rate β = 0.00365 o C per meter.

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By combining Eqs. (2.2.7) and (2.2.14), The relation between pressure and temperature for a mass of gas expanding without heat transfer (isentropic relation, Sec. 6.1) is in which T 1 is the initial smoke absolute temperature and P 0 the initial absolute pressure; k is the specific heat ratio, 1.4 for air and other diatomic gases. Eliminating P/P 0 in the last two equations Since the gas will rise until its temperature is equal to the ambient temperature, the last two equations may be solved for y. Let Then For β = -0.00651 o C per metre, R = 287 m·N/(kg·K), a = 2.002, and y = 3201 m. For the atmospheric temperature inversion β = -0.00365 o C per metre, a = -0.2721, and y = 809.2 m. #

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2.4 MANOMETERS Manometers are devices that employ liquid columns for determining differences in pressure. Figure 2.6a: the most elementary manometer – piezometer It measures the pressure in a liquid when it is above zero gage Glass tube is mounted vertically so that it is connected to the space within the container Liquid rises in the tube until equilibrium is reached The pressure is then given by the vertical distance h from the meniscus (liquid surface) to the point where the pressure is to be measured, expressed in units of length of the liquid in the container. Piezometer would not work for negative gage pressures, because air would flow into the container through the tube

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Figure 2.6 Simple manometers.

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Figure 2.6b: for small negative or positive gage pressures in a liquid With this arrangement the meniscus may come to rest below A, as shown. Since the pressure at the meniscus is zero gage and since pressure decreases with elevation, units of length H 2 O Figure 2.6c: for greater negative or positive gage pressures (a second liquid of greater relative density employed) It must be immiscible in the first fluid, which may now be a gas If the relative density of the fluid at A is S 1 (based on water) and the relative density of the manometer liquid is S 2, the equation for pressure at A h A - the unknown pressure, expressed in length units of water, h 1, h 2 - in length units

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A general procedure in working all manometer problems : 1. Start at one end (or any meniscus if the circuit is continuous) and write the pressure there in an appropriate unit (say pascals) or in an appropriate symbol if it is unknown. 2. Add to this the change in pressure, in the same unit, from one meniscus to the next (plus if the next meniscus is lower, minus if higher). (For pascals this is the product of the difference in elevation in metres and the unit gravity force of the fluid in newtons per cubic metre.) 3. Continue until the other end of the gage (or the starting meniscus) is reached and equate the expression to the pressure at that point, known or unknown. The expression will contain one unknown for a simple manometer or will give a difference in pressures for the differential manometer. In equation form,

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Figure 2.7 Differential manometers

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A differential manometer (Fig. 2.7) determines the difference in pressures at two points A and B when the actual pressure at any point in the system cannot be determined Application of the procedure outlined above to Fig. 2.7a produces For Fig. 2.7b: If the pressures at A and B are expressed in length of the water column, the above results can be written, for Fig. 2.7a, For Fig 2.7b:

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Example 2.4 In Fig. 2.7a the liquids at A and B are water and the manometer liquid is oil. S = 0.80; h 1 = 300 mm; h 2 = 200 mm; and h 3 = 600 mm. (a) Determine p A - p B, in pacals. (b) If p B = 50kPa and the barometer reading is 730 mm Hg, find the pressure at A, in meters of water absolute. (a) (b) (a)

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Micromanometers For determining very small differences in pressure or determining large pressure differences precisely – several types of manometers One type very accurately measures the differences in elevation of two menisci of a manometer. By means of small telescopes with horizontal cross hairs mounted along the tubes on a rack which is raised and lowered by a pinion and slow motion screw so that the cross hairs can be set accurately, the difference in elevation of menisci (the gage difference) can be read with verniers.

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Figure 2.8 Micromanometer using two gage liquids

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Fig. 2.8: two gage liquids, immiscible in each other and in the fluid to be measured a large gage difference R can be produced for a small pressure difference. The heavier gage liquid fills the lower U tube up to 0-0; then the lighter gage liquid is added to both sides, filling the larger reservoirs up to 1-1. The gas or liquid in the system fills the space above 1-1. When the pressure at C is slightly greater than at D, the menisci move as indicated in Fig. 2.8. The volume of liquid displaced in each reservoir equals the displacement in the U tube Manometer equation γ 1, γ 2 and γ 3 are the unit gravity force

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Example 2.5 In the micromanometer of Fig 2.8 the pressure difference is wanted, in pascals, when air is in the system, S 2 = 1.0, S 3 = 1.10, a/A = 0.01, R = 5 mm, t = 20 o C, and the barometer reads 760 mm Hg. The term γ 1 (a/A) may be neglected. Substituting into Eq. (2.4.1) gives #

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Figure 2.9 Inclined manometer The inclined manometer: frequently used for measuring small differences in gas pressures. Adjusted to read zero, by moving the inclined scale, when A and B are open. Since the inclined tube requires a greater displacement of the meniscus for given pressure difference than a vertical tube, it affords greater accuracy in reading the scale. Surface tension causes a capillary rise in small tubes. If a U tube is used with a meniscus in each leg, the surface-tension effects cancel.

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2.5 FORCES ON PLANE AREAS In the preceding sections variations oF pressure throughout a fluid have been considered. The distributed forces resulting from the action of fluid on a finite area may be conveniently replaced by a resultant force, insofar as external reactions to the force system are concerned. In this section the magnitude of resultant force and its line of action (pressure center) are determined by integration, by formula, and by use of the concept of the pressure prism.

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Horizontal Surfaces A plane surface in a horizontal position in a fluid at rest is subjected to a constant pressure. The magnitude of the force acting on one side of the surface is The elemental forces pdA acting on A are all parallel and in the same sense a scalar summation of all such elements yields the magnitude of the resultant force. Its direction is normal to the surface and toward the surface if p is positive. Fig. 2.10: arbitrary xy axes - to find the line of action of the resultant, i.e., the point in the area where the moment of the distributed force about any axis through the point is zero, Then, since the moment of the resultant must equal the moment of the distributed force system about any axis, say the y axis, x’ – the distance from the y axis to the resultant

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Figure 2.10 Notation for determining the line of action of a force

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Momentum (1) First moment The moment of an area A about the y axis The moment about a parallel axis, for example, x = k, the moment Centroidal axis Volume center Mass center: center of gravity of a body

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Figure A.1 Notation for first and second moments

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(2) Second moment The second moment of an area A (the moment of inertia of the area) The moment about a parallel axis, for example, x = k, the moment Figure A.2 Moments of inertia of simple areas about centroidal axes

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The product of inertia I xy of an area - the product of inertia about centroidal axes parallel to the xy axes.

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Inclined Surfaces Fig. 2.11: a plane surface is indicated by its trace A'B‘;it is inclined θo from the horizontal. x axis: intersection of the plane of the area and the free surface. y axis: taken in the plane of the area, with origin O in the free surface. The xy plane portrays the arbitrary inclined area. The magnitude, direction, and line of action of the resultant force due to the liquid, acting on one side of the area, are sought. For δA: Since all such elemental forces are parallel, the integral over the area yields the magnitude of force F, acting on one side of the area, Magnitude of force exerted on one side of a plane area submerged in a liquid is the product of the area and the pressure at its centroid The presence of a free surface is unnecessary

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Figure 2.11 Notation for force of liquid on one side of a plane inclined area.

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Center of Pressure Fig. 2.11: the line of action of the resultant force has its piercing point in the surface at a point called the pressure center, with coordinates (xp, yp). Center of pressure of an inclined surface is not at the centroid. To find the pressure center, the moments of the resultant xpF, ypF are equated to the moment of the distributed forces about the y axis and x axis, respectively - may be evaluated conveniently through graphical integration, for simple areas they may be transformed into general formulas:

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When either of the centroidal axes is an axis of symmetry for the surface, vanishes and the pressure center lies on x = x -. Since may be either positive or negative, the pressure center may lie on either side of the line x = x -. To determine y p by formula, with Eqs. (2.5.2) and (2.5.6) In the parallel-axis theorem for moments of inertia in which I G is the second moment or the area about its horizontal centroidal axis. If I G is eliminated from Eq. (2.5.9)

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Figure 2.12 Triangular gate Example 2.6 The triangular gate CDE (Fig. 2.12) is hinged along CD and is opened by a normal force P applied at E. It holds oil, relative density 0.80, above it and is open to the atmosphere on its lower side Neglecting the weight of the gate, find (a) the magnitude of force exerted on the gate by integration and by Eq. (2.5.2); (b) the location of pressure center; (c) the force P needed to open the gate.

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(a) By integration with reference to Fig. 2.12, When y = 4, x = 0, and when y = 6.5, x = 3, with x varying linearly with y; thus in which the coordinates have been substituted to find x in terms of y. Solving for a and b gives Similarly, y = 6.5, x = 3; y = 9, x = 0; and x = 6/5(9 - y). Hence, Integrating and substituting for γsinθ leads to By Eq. (2.5.2)

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(b) With the axes as shown, In Eq. (2.5.8) I - xy is zero owing to symmetry about the centroidal axis parallel to the x axis; hence In Eq. (2.5.11), i.e., the pressure center is 0.16 m below the centroid, measured in the plane of the area. (c) When moments about CD are taken and the action of the oil is replaced by the resultant,

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The Pressure Prism Pressure prism: another approach to determine the resultant force and line of action of the force on a plane surface - prismatic volume with its base the given surface area and with altitude at any point of the base given by p = γh. h is the vertical distance to the free surface, Fig. 2.13. (An imaginary free surface may be used to define h if no real free surface exists.) (in the figure, γh may be laid off to any convenient scale such that its trace is OM) The force acting on an elemental area δA is (2.5.12) - an element of volume of the pressure prism. After integrating, F = ϑ From Eqs. (2.5.5) and (2.5.6), (2.5.13) x p, y p are distances to the centroid of the pressure prism the line of action of the resultant passes through the centroid of the pressure prism

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Figure 2.13 Pressure prism

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Effects of Atmospheric Pressure on Forces on Plane Areas In the discussion of pressure forces the pressure datum was not mentioned : p = γh the datum taken was gage pressure zero, or the local atmospheric pressure When the opposite side of the surface is open to the atmosphere, a force is exerted on it by the atmosphere equal to the product of the atmospheric pressure P 0 and the area, or P 0 A, based on absolute zero as datum. On the liquid side the force is The effect P 0 A of the atmosphere acts equally on both sides and in no way contributes to the resultant force or its location So long as the same pressure datum is selected for all sides of a free body, the resultant and moment can be determined by constructing a free surface at pressure zero on this datum and using the above methods

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Example 2.8 An application of pressure forces on plane areas is given in the design of a gravity dam. The maximum and minimum compressive stresses in the base of the dam are computed from the forces which act on the dam. Figure 2.15 shows a cross section through a concrete dam where the unit gravity force of concrete has been taken as 2.5γ and γ is the unit gravity force of water. A 1 m section of dam is considered as a free body; the forces are due to the concrete, the water, the foundation pressure, and the hydrostatic uplift. Determining amount of hydrostatic uplift is beyond the scope of this treatment. but it will be assumed to be one-half the hydrostatic head at the upstream edge, decreasing linearly to zero at the downstream edge of the dam. Enough friction or shear stress must be developed at the base of the dam to balance the thrust due to the water that is R x = 5000γ. The resultant upward force on the base equals the gravity force of the dam less the hydrostatic uplift R y = 6750γ + 2625γ - 1750γ = 7625γ N. The position of Ry is such that the free body is in equilibrium. For moments around O,

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Figure 2.15 Concrete gravity dam

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It is customary to assume that the foundation pressure varies linearly over the base of the dam, i.e., that the pressure prism is a trapezoid with a volume equal to R y ; thus in which σmax, σmin are the maximum and minimum compressive stresses in pascals. The centroid of the pressure prism is at the point where x = 44.8 m. By taking moments about 0 to express the position of the centroid in terms of σ max and σ min, Simplifying gives When the resultant falls within the middle third of the base of the dam, σ min will always be a compressive stress. Owing to the poor tensile properties of concrete, good design requires the resultant to fall within the middle third of the base.

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2.6 FORCE COMPONENTS ON CURVED SURFACES When the elemental forces p δA vary in direction, as in the case of a curved surface, they must be added as vector quantities their components in three mutually perpendicular directions are added as scalars, and then the three components are added vectorially. With two horizontal components at right angles and with the vertical component - all easily computed for a curved surface - the resultant can be determined. The lines of action of the components also are readily determined.

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Horizontal Component of Force on a Curved Surface The horizontal component pressure force on a curved surface is equal to the pressure force exerted on a projection of the curved surface. The vertical plane of projection is normal to the direction of the component. Fig. 2.16: the surface represents any three-dimensional surface, and δA an element of its area, its normal making the angle θ with the negative x direction. Then Projecting each element on a plane perpendicular to x is equivalent to projecting the curved surface as a whole onto the vertical plane force acting on this projection of the curved surface is the horizontal component of force exerted on the curved surface in the direction normal to the plane of projection. To find the horizontal component at right angles to the x direction, the curved surface is projected onto a vertical plane parallel to x and the force on the projection is determined.

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Figure 2.16 Horizontal component of force on a curved surface Figure 2.17 Projections of area elements on opposite sides of a body

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When looking for the horizontal component of pressure force on a closed body, the projection of the curved surface on a vertical plane is always zero, since on opposite sides of the body the area-element projections have opposite signs (Fig. 2.17). Let a small cylinder of cross section δA with axis parallel to x intersect the closed body at B and C. If the element of area of the body cut by the prism at B is δAB and at C is δAC, then and similarly for all other area elements To find the line of action of a horizontal component of force on a curved surface, the resultant of the parallel force system composed of the force components from each area element is required. This is exactly the resultant of the force on the projected area, since the two force systems have an identical distribution of elemental horizontal force components. Hence, the pressure center is located on the projected area by the methods of Sec. 2.5.

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Example 2.9 The equation of an ellipsoid of revolution submerged in water is x 2 /4 + y 2 /4 + z 2 /9 = 1. The center of the body is located 2 m below the free surface. Find the horizontal force components acting on the curved surface that is located in the first octant. Consider the xz plane to be horizontal and y to be positive upward. Projection of the surface on the yz plane has an area of Its centroid is located m below the free surface

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Vertical Component of Force on a Curved Surface The vertical component of pressure force on a curved surface is equal to the weight surface and extending up to the free surface Can be determined by summing up the vertical components of pressure force on elemental areas δA of the surface In Fig.2.18 an area element is shown with the force p δA acting normal to it. Let θ be the angle the normal to the area element makes with the vertical. Then the vertical component of force acting on the area element is p cos θ δA, and the vertical component of force on the curved surface is given by (2.6.2) p replaced by its equivalent γh; cos θ δA is the projection of δA on a horizontal plane Eq. (2.6.2): (2.5.3-4) in which δ ϑ is the volume of the prism of height h and base cos θ δA, or the volume of liquid vertically above the area element

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Figure 2.18 Vertical component of force on a curved surface Figure 2.19 Liquid with equivalent free surface

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Fig. 2.19: the liquid is below the curved surface and the pressure magnitude is known at some point (e.g., O), an imaginary or equivalent free surface s-s can be constructed p/γ above O, so that the product of unit gravity force and vertical distance to any point in the tank is the pressure at the point. The weight of the imaginary volume of liquid vertically above the curved surface is then the vertical component of pressure force on the curved surface. In constructing an imaginary free surface, the imaginary liquid must be of the same unit gravity force as the liquid in contact with the curved surface; otherwise, the pressure distribution over the surface will not be correctly represented. With an imaginary liquid above a surface, the pressure at a point on the curved surface is equal on both sides, but the elemental force components in the vertical direction are opposite in sign the direction of the vertical force component is reversed when an imaginary fluid is above the surface. In some cases a confined liquid may be above the curved surface, and an imaginary liquid must be added (or subtracted) to determine the free surface.

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The line of action of the vertical component is determined by equating moments of the elemental vertical components about a convenient axis with the moment of the resultant force. With the axis at O (Fig.2.18), in which is the distance from 0 to the line of action Since Fv = γ ϑ the distance to the centroid of the volume the line of action of the vertical force passes through the centroid of the volume, real or imaginary, that extends above the curved surface up to the real or imaginary free surface

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Example 2.10 A cylindrical barrier (Fig. 2.20) holds water as shown. The contact between cylinder and wall is smooth. Considering a 1-m length of cylinder, determine (a) its gravity force and (b) the force exerted against the wall. (a) For equilibrium the weight of the cylinder must equal the vertical component of force exerted on it by the water. (The imaginary free surface for CD is at elevation A.) The vertical force on BCD is The vertical force on AB is Hence, the gravity force per metre of length is (b) The force exerted against the wall is the horizontal force on ABC minus the horizontal force on CD. The horizontal components of force on BC and CD cancel; the projection of BCD on a vertical plane is zero , since the projected area is 2 m2 and the pressure at the centroid of the projected area is 9806 Pa.

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Figure 2.20 Semifloating body

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Tensile Stress in a Pipe and Spherical Shell Fig. 2.21: a circular pipe under the action of an internal pressure is in tension around its periphery; assuming that no longitudinal stress occurs, the walls are in tension Consider a section of pipe of unit length (the ring between two planes normal to the axis and unit length apart). If one-half of this ring is taken as a free body, the tensions per unit length at top and bottom are respectively T 1 and T 2 The horizontal component of force acts through the pressure center of the projected area and is 2pr, in which p is the pressure at the centerline and r is the internal pipe radius. For high pressures the pressure center may be taken at the pipe center; then T 1 = T 2, and T is the tensile force per unit length. For wall thickness e, the tensile stress in the pipe wall is

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Figure 2.21 Tensile stress in pipe

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Example 2.11 A 100 mm-ID steel pipe has a 6 mm wall thickness. For an allowable tensile stress of 70 MPa, what is the maximum pressure? From Eq. (2.6.6)

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2.7 BUOYANT FORCE Buoyant force: the resultant force exerted on a body by a static fluid in which it is submerged or floating Always acts vertically upward (there can be no horizontal component of the resultant because the projection of the submerged body or submerged portion of the floating body on a vertical plane is always zero) The buoyant force on a submerged body is the difference between the vertical component of pressure force on its underside and the vertical component of pressure force on its upper side Figure 2.22

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Figure 2.22 Buoyant force on floating and submerged bodies

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Fig. 2.22: the upward force on the bottom is equal to the gravity force of liquid, real or imaginary, which is vertically above the surface ABC, indicated by the gravity force of liquid within ABCEFA. The downward force on the upper surface equals the gravity force of liquid ADCEFA. The difference between the two forces is a force, vertically upward, due to the gravity force of fluid ABCD that is displaced by the solid. In equation form F B is buoyant force, V is the volume of fluid displaced, and γ is the unit gravity force of fluid The same formula holds for floating bodies when V is taken as the volume of liquid displaced

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Fig.2.23: the vertical force exerted on an element of the body in the form of a vertical prism of cross section δA is δV is the volume of the prism. Integrating over the complete body gives γ is considered constant throughout the volume To find the line of action of the buoyant force, moments are taken about a convenient axis O and are equated to the moment of the resultant, thus, i is the distance from the axis to the line of action. This equation yields the distance to the centroid of the volume; the buoyant force acts through the centroid of the displaced volume of fluid; this holds for both submerged and floating bodies. The centroid of the displaced volume of fluid is called the center of buoyancy.

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Figure 2.23 Vertical force components on element of body

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Determining gravity force on an odd-shaped object suspended in two different fluids yields sufficient data to determine its gravity force, volume, unit gravity force, and relative density. Figure 2.24: two free-body diagrams for the same object suspended and gravity force determined in two fluids, F 1 and F 2 ; γ 1 and γ 2 are the unit gravity forces of the fluids. W and V, the gravity force and volume of the object, are to be found. The equations of equilibrium are written and solved:

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Figure 2.24 Free body diagrams for body suspended in a fluid

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A hydrometer uses the principle of buoyant force to determine relative densities of liquids Figure 2.25: a hydrometer in two liquids with a stem of prismatic cross section a Considering the liquid on the left to be distilled water (unit relative density S = 1.00), the hydrometer floats in equilibrium when V 0 is the volume submerged, γ is the unit gravity force of water, and W is the gravity force of hydrometer The position of the liquid surface is marked 1.00 on the stem to indicate unit relative density S. When the hydrometer is floated in another 1iquid, the equation of equilibrium becomes where ΔV = aΔh. Solving for Δh with Eqs. (2.7.2) and (2.7.3)

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Figure 2.25 Hydrometer in water and in liquid of relative density

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Example 2.12 A piece of ore having a gravity force of 1.5 N in air is found to have a gravity force 1.1 N when submerged in water. What is its volume, in cubic centimetres, and what is its relative density? The buoyant force due to air may be neglected. From Fig. 2.24

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2.8 STABILITY OF FLOATING AND SUBMERGED BODIES A body floating in a static liquid has vertical stability. A small upward displacement decreases the volume of liquid displaced an unbalanced downward force which tends to return the body to its original position. Similarly, a small downward displacement results in a greater buoyant force, which causes an unbalanced upward force. A body has linear stability when a small linear displacement in any direction sets up restoring forces tending to return it to its original position. A body has rotational stability when a restoring couple is set up by any small angular displacement.

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Methods for determining rotational stability are developed in the following discussion A body may float in stable equilibrium unstable equilibrium (any small angular displacement sets up a couple that tends to increase the angular displacement) neutral equilibrium (any small angular displacement sets up no couple whatever) Figure 2.26: three cases of equilibrium a. a light piece of wood with a metal mass at its bottom is stable b. when the metal mass is at the top, the body is in equilibrium but any slight angular displacement causes it to assume the position in a c. a homogeneous sphere or right-circular cylinder is in equilibrium for any angular rotation; i.e., no couple results from an angular displacement

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Figure 2.26 Examples of stable, unstable, and neutral equilibrium

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A completely submerged object is rotationally stable only when its center of gravity is below the center of buoyancy (Fig. 2.27a) When the object is rotated counterclockwise, the buoyant force and gravity force produce a couple in the clockwise direction (Fig. 2.27b) Figure 2.27 Rotationally stable submerged body

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Normally, when a body is too heavy to float, it submerges and goes down until it rests on the bottom. Although the unit gravity force of a liquid increases slightly with depth, the higher pressure tends to cause the liquid to compress the body or to penetrate into pores of solid substances and thus decrease the buoyancy of the body Example: a ship is sure to go to the bottom once it is completely submerged, owing to compression of air trapped in its various parts

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Determination of Rotational Stability of Floating Objects Any floating object with center of gravity below its center of buoyancy (centroid of displaced volume) floats in stable equilibrium (Fig. 2.26a). Certain floating objects, however, are in stable equilibrium when their center of gravity is above the center of buoyancy. Figure 2.28a: a cross section of a body with all other parallel cross sections identical. The center of buoyancy is always at the centroid of the displaced volume, which is at the centroid of the cross-sectional area below liquid surface in this case.

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Figure 2.28 Stability of a prismatic body

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when the body is tipped (Fig. 2.28b), the center of buoyancy is at the centroid B' of the trapezoid ABCD, the buoyant force acts upward through B', and the gravity force acts downward through G, the center of gravity of the body When the vertical through B' intersects the original centerline above C, as at M, a restoring couple is produced and the body is in stable equilibrium The intersection of the buoyant force and the centerline is called the metacenter (M) When M is above G, the body is stable; when below G, it is unstable; and when at G, it is in neutral equilibrium The distance MG is called the metacentric height and is a direct measure of the stability of the body. The restoring couple is in which θ is the angular displacement and W the gravity force of the body

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Example 2.13 In Fig. 2.28 a scow 6 m wide and 20 m long has a gross mass of 200 Mg. Its center of gravity is 30 cm above the water surface. Find the metacentric height and restoring couple when Δy = 30 cm. The depth of submergence h in the water is The centroid in the tipped position is located with moments about AB and BC, By similar triangles AEO and B'PM,

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G is 1.97 m from the bottom; hence The scow is stable, since is positive; the righting moment is #

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Nonprismatic Cross Sections For a floating object of variable cross section (e.g., a ship) (Fig. 2.39a), a convenient formula can be developed for determination of metacentric height for very small angles of rotation θ The horizontal shift in center of buoyancy r (Fig. 2.29b) is determined by the change in buoyant forces due to the wedge being submerged, which causes an upward force on the left, and by the other wedge decreasing the buoyant force by an equal amount ΔFB on the right. The force system, consisting of the original buoyant force at B and the couple ΔFB x s due to the wedges, must have as resultant the equal buoyant force at B'. With moments about B to determine the shirt r,

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Figure 2.29 Stability relations in a body of variable cross section

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The amount of the couple can be determined with moments about O, the centerline of the body at the liquid surface For an element of area δA on the horizontal section through the body at the liquid surface, an element of volume of the wedge is xθ δA. The buoyant force due to this element is γxθ δA, and its moment about O is γx2θ δA, in which θ is the small angle of tip in radians. By integrating over the complete original horizontal area at the liquid surface, the couple is determined to be I is the moment of inertia of the area about the axis y-y (Fig.2.29a) Substitution into the above equation produces V is the total volume of liquid displaced Since θ is very small and

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Example 2.14 A barge displacing 1 Gg has the horizontal cross section at the waterline shown in Fig. 2.30. Its center of buoyancy is 2.0 m below the water surface, and its center of gravity is 0.5 m below the water surface. Determine its metacentric height for rolling (about y-y axis) and for pitching (about x-x axis). GB = 2 – 0.5 = 1.5 m For rolling For pitching #

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Figure 2.30 Horizontal cross section of a ship at the waterline

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2.9 RELATIVE EQUILIBRIUM Fluid statics: no shear stresses the variation of pressure is simple to compute For fluid motion such that no layer moves relative to an adjacent layer, the shear stress is also zero throughout the fluid A fluid with a translation at uniform velocity still follows the laws of static variation of pressure. When a fluid is being accelerated so that no layer moves relative to an adjacent one (when the fluid moves as if it were a solid), no shear stresses occur and variation in pressure can be determined by writing the equation of motion for an appropriate free body Two cases are of interest, a uniform linear acceleration and a uniform rotation about a vertical axis When moving thus, the fluid is said to be in relative equilibrium

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Uniform Linear Acceleration Fig. 2.31: a liquid in an open vessel is given a uniform linear acceleration a After some time the liquid adjusts to the acceleration so that it moves as a solid, i.e., the distance between any two fluid particles remains fixed no shear stresses occur By selecting a cartesian coordinate system with y vertical and x such that the acceleration vector a is in the xy plane (Fig. 2.31a), the z axis is normal to a and there is no acceleration component in that direction Fig. 2.31b: the pressure gradient ∇ p is then the vector sum of -ρa and - jγ Since ∇ p is in the direction of maximum change in p (the gradient), at right angles to ∇ p there is no change in p. Surfaces of constant pressure, including the free surface, must therefore be normal to ∇ p

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Figure 2.31 Acceleration with free surface

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To obtain a convenient algebraic expression for variation of p with x, y, and z, that is, p = p(x, y, z), Eq. (2.2.5) is written in component form : Since p is a function of position (x, y, z), its total differential is Substituting for the partial differentials gives which can be integrated for an incompressible fluid,

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To evaluate the constant of integration c: let x = 0, y = 0, p = p 0 ; then c = p 0 and When the accelerated incompressible fluid has a free surface, its equation is given by setting p = 0 in the above eq. Solving it for y gives The lines of constant pressure, p = const, have the slope and are parallel to the free surface. The y intercept of the free surface is

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Example 2.15 The tank in Fig. 2.32 is filled with oil, relative density 0.8, and accelerated as shown. There is a small opening in the rank at A. Determine the pressure at B and C; and the acceleration a x required to make the pressure at B zero. By selecting point A as origin and by applying Eq. (2.9.2) for a y = 0 At B, x = 1.8 m, y = - 1.2 m, and p = 2.35 kPa. Ft C, x = -0.15 m, y = - 1.35 m, and p = 11.18 kPa. For zero pressure at B, from Eq. (2.9.2) with origin at A,

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Figure 2.32 Tank completely filled with liquid

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Example 2.16 A closed box with horizontal base 6 by 6 units and a height of 2 units is half-filled with liquid (Fig. 2.33). It is given a constant linear acceleration a x = g/2, a y = -g/4. Develop an equation for variation of pressure along its base. The free surface has the slope: hence, the free surface is located s shown in the figure. When the origin is taken at 0, Eq. (2.9.2) becomes Then, for y = 0, along the bottom,

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Figure 2.33 Uniform linear acceleration of container

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Uniform Rotation about a Vertical Axis Forced-vortex motion: rotation of a fluid, moving as a solid, about an axis Every particle of fluid has the same angular velocity This motion is to be distinguished from free-vortex motion, in which each particle moves in a circular path with a speed varying inversely as the distance from the center A liquid in a container, when rotated about a vertical axis at constant angular velocity, moves like a solid alter some time interval. No shear stresses exist in the liquid, and the only acceleration that occurs is directed radially inward toward the axis of rotation. By selecting a coordinate system (Fig. 2.34a) with the unit vector i in the r direction and j in the vertical upward direction with y the axis of rotation, the following equation may be applied to determine pressure variation throughout the fluid: (2.2.5)

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Figure 2.34 Rotation of a fluid about a vertical axis

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For constant angular velocity w, any particle of fluid P has an acceleration w 2 r directed radially inward (a = -iw 2 r) Vector addition of -jγ and -ρa (Fig. 2.34b) yields ∇ p, the pressure gradient. The pressure does not vary normal to this line at a point if P is taken at the surface, the free surface is normal to ∇ p Expanding Eq. (2.2.5) k is the unit vector along the z axis (or tangential direction). Then p is a function of y and r only: For a liquid (γ ≈ const) integration yields c is the constant of integration

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If the value of pressure at the origin (r = 0, y = 0) is p 0, then c = p 0 and When the particular horizontal plane (y = 0) for which p 0 = 0 is selected and the above eq. is divided by γ, : the head, or vertical depth, varies as the square of the radius. The surfaces of equal pressure are paraboloids of revolution.

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When a free surface occurs in a container that is being rotated, the fluid volume underneath the paraboloid of revolution is the original fluid volume The shape of the paraboloid depends only upon the angular velocity with respect to the axis (Fig. 2.35). The rise of liquid from its vertex to the wall of the cylinder is w 2 r 0 2 /rg (Eq. (2.9.6)), for a circular cylinder rotating about its axis. Since a paraboloid of revolution has a volume equal to one-half its circumscribing cylinder, the volume of the liquid above the horizontal plane through the vertex is When the liquid is at rest, this liquid is also above the plane through the vertex to a uniform depth of Hence, the liquid rises along the walls the same amount as the center drops, thereby permitting the vertex to be located when w, r 0, and depth before rotation are given

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Figure 2.35 Rotation of circular cylinder about its axis

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Example 2.17 A liquid, relative density 1.2, is rotated at 200 rpm about a vertical axis. At one point A in the fluid 1 m from the axis, the pressure is 70 kPa. What is the pressure at a point B which is 2 m higher than A and 1.5 m from the axis? When Eq. (2.9.5) is written for the two points, Then w = 200 x 2π/60 = 20.95 rad/s, γ = 1.2 x 9806 = 11.767 N/m3, rA = 1 m, and rB = 1.5 m. When the second equation is subtracted from the first and the values are substituted, Hence

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Example 2.18 A straight tube 2 m long, closed at the bottom and filled with water, is inclined 30 o with the vertical and rotated about a vortical axis through its midpoint 6.73 rad/s. Draw the paraboloid of zero pressure, and determine the pressure at the bottom and midpoint of the tube. In Fig. 2.36, the zero-pressure paraboloid passes through point A. If the origin is taken at the vertex, that is, p 0 = 0, Eq. (2.9.6) becomes which locates the vertex at O, 0.577 m below A. The pressure at the bottom of the tube is or At the midpoint, =.289 m and

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Figure 2.36 Rotation of inclined tube of liquid about a vertical axis

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Fluid Pressure Forces in Relative Equilibrium The magnitude of the force acting on a plane area in contact with a liquid accelerating as a rigid body can be obtained by integrating over the surface The nature of the acceleration and orientation of the surface governs the particular variation of p over the surface When the pressure varies linearly over the plane surface (linear acceleration), the magnitude of force is given by the product of pressure at the centroid and area, since the volume of the pressure prism is given by p G A For nonlinear distributions the magnitude and line of action can be found by integration.

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과제물 스캐닝된 파일을 텍스트 파일로 만들 것. (2,3,4,5 장, 자습서 ) 교과서의 모든 예제는 반드시 제출하여야 함. 정지유체의 한점에 작용하는 압력은 모든 방향에 대해서 같음을 증명하라. 힘의 평형으로부터 오일러 방정식과 압력의 식을 유도하라. 기체에 대하여 온도가 변하지 않는 경우와 변하는 경우에 대하여 압력의 식을 유도하라. 액주계에 있어서 일반적으로 압력을 계산하는 절차를 설명하라. ( 단순 액주계 와 차분 액주계의 예를 포함하여 ). 면적이 다른 경우와 경사진 경우의 미차액주계에서 압력을 계산하는 식을 유 도하라. 1 차 2 차 모멘트에 대하여 설명하라. 부록에 나와있는 도형에 대하여 도심축과 x 축에 대한 2 차모멘트에 대한 식을 유도하라. 부록 B 의 수치해석 기법을 설명하라. 프로그램밍 포함. ( 교과서와 다른 책의 내 용 참조 ). 예제 2.7 의 문제를 Bisection 과 Newton-Raphson Method 를 사용하여 컴퓨터 프 로그램을 작성하여 풀어라.

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과제물 Runge-Kutta Method 를 설명하고 수치해석 책에 나와있는 문제를 풀어라 ( 프로 그램밍 포함 : C 및 Fortran).

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