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O R Then from Eq. Bulk Modulus of Elasticity. In the preceding section the compressibility of a perfect gas is described by the perfect-gas law. For most purposes a liquid may be considered as incompressible, but for situations involving either sudden or great changes in pressure, its compressibility becomes important. Liquid compressibility and gas also becomes important also when temperature changes arc involved e. The compressibility of a liquid is expressed by its bulk modulus 07 elasticity.

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The magnitude of the force F per meter of width to keep the gate closed is most nearly R is one-third from the bottom centroid of a triangle from the NCEES Handbook. These topics are listed below for clarification. File Format : PDF or Word A rotating viscometer consists of two concentric cylinders - an inner cylinder of radius Ri rotating at angular velocity rotation rate? Lectures by Walter Lewin. HC Verma Solutions for Part 1 and 2. Scribd is the world's largest social reading and publishing site.

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Techet Introduction to basic principles of fluid mechanics I. Elger, Barbara C. If you have any questions, or would like a receive a sample chapter before your download, please contact us at info testbankteam. Fluids and Buoyant Force A. Hibbeler Fluid Mechanics Chapter 2 Professor Fred Stern Fall 1 Chapter 2: Pressure and Fluid Statics Pressure For a static fluid, the only stress is the normal stress since by definition a fluid subjected to a shear stress must deform and undergo motion.

Sample Problems. Determine the final density of water a if it is heated to 50oC at a constant pressure of 1 atm, and b if it is compressed to atm pressure at constant temperature of 20oC.

All laws in continuum mechanics depart from a CV analysis i. The work is protected by local and international copyright laws and is provided solely for the use of Solution Manual for Fluid Mechanics 1st Edition by Russell C. The second type of stress, shown in Fig. John Finnemore, Joseph B. Fluid Mechanics Fundamentals and Applications Solution Manual The author balances three separate approachesintegral, differential and experimentalto provide a foundation for fluid mechanics concepts and applications.

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Analysis Mass m is the actual mass in grams or kilograms This is completed downloadable version of Fundamentals of Fluid Mechanics 7th solution manual by Bruce R. Introduction to Fluid Mechanics - 6 ed - Fox - Solution Solution Manual "Fluid Mechanics 7th Edition Chapter 2" Stromingsleer "Fundamentals of Fluid Mechanics" offers comprehensive topical coverage, with varied examples and problems, application of visual component of fluid mechanics, and strong focus on effective learning.

Click on date to download homework solutions. Crowe, Donald F. Introduction to Fluid Mechanics 2. This gives the freebody shown at right. With two horizontal corn- ponents at right angles and with the vertical component.

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The horizontal component of pressure force on a curved surface i s equal to the pressure force 3. Pressure prisrn for horizontal component of pressure the force components from each area element is required. The vertical component of force on a curved surface can be determined by summing u p the vertical componrnts of pressure force elemental. T o find the horizontal cxnponent at right angles to thc. Let a small cylinder of cross section 6A with axis parallel to s intersect the closed body a t B and r.

When the horizontal component o f pressure force on a closed body is to be found. Sac 2. This is exactly the resultant of the force on the projected area. Vertical conlponent of force on FIG. IAct 8 I c! When the liquid is below t. With an imaginary liquid tthovt. In the construc. Then the vertical cornpoilent of force acting on t.

I n some cases a confined liquid may be above the curved surface.. Example The contact hcttvcrn cylinder and wall is smooth. With the axis at O Fig For equilibrium the weight of the cylindrtr rnust equal the vertical com- ponent of force exerted on it by the water.

The horizontal ronlponents of forccl on BC and C D c: The force excrteti against the wall is the horizontal force on ABC minus the horizontal forchc on CI. For high pressures the pressure center may be taken a t the pipe center.

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Tensile Stress in a Pipe. Taking one-half of this ring as a free body. A I-in. Tensile stress in pipe. The horizon- tal component of force acts through 1 I the pressure center of the projected I area and is 2pr. Assuming that no longi- tudinal stress occurs. For wall thickness t in. A circular pipe under the action of an internal pressure is in tension around its periphery.

For an allow- able tensile stress of I n equation form in which FB is the buoyant force. The difference between the two forces is a force. The buoyant force always acts vertically upward. Integrating over the complete body. 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.

Buoyant force on floating and submerged bodies. To find the line of action of the buoyant force. The same formula holds for floating bodies when f is taken as the volume of liquid displaced.

There can be no horizontal component of the resultant because the vertical projection of the sub- merged body or submerged portion of the floating body is always zero. Buoyant Force. This is evident from inspection of the floating body in Fig. When the body floats at the interface of a static two-fluid system Fig. To locate the line of act. Integrating over t.

This holds for a b FIG. Vertical foroo components on eIcrnent of body. This equa- tion yields the distitnce t. In solving a statics problem involving submerged or floating objects. Weighing an odd-shaped object when suspended in two different fluids yields sufficient data to determine its weight. W and V..

The' weight of t. YI A hydrometer uses the principle of buoyant force to determine specific gravities of liquids. Free-body diagram for body suspended in a fluid. The action of the fluid is replaced by the buoyant force. Figure A body has linear stability when a small linear displacement in any direction sets up restoring forces tending to return the body to its original position.

Sim- ilarly. It has rotational stability when a restoring couple is set up by any small angular displacement. A body floating in a static liquid has vertical stability. A small upward displacement decreases the volume of liquid displaced.

Examples of a stable. UThenthe hydrometer is floated in another liquid. The positioil of the liquid surface js 1. Considering the liquid on the left to be distilled water. Solving for Ah. What is its volume and specific gravity? The buoyant force due to air may be neglected. Stability of Floating and Submerged Bodies. From Fig. A ship.

When the object is FIG. A body may float in stable. The stability of prismatic bodies 's first considered. Any floating object with center of gravity below its center of buoyancy centroid of displaced volume floats in stable equilibrium. Determination of Rotational Stability of Floating Objects. Although the specific weight of a liquid increases s'ightly with depth. Certsin floating objects. The center of Euoyancy is always at the centroid of.

Rotationally stable submerged body. A submerged object is rotationally stable only when its center of gravity is below the center of buoyancy. With the body in neutral equilibrium. Stability of prismatic.

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When M is above G. When the vertical through R' intersects the original center line above G. The intersection of the buoyant.. The depth of submergence h in the water is The centroid in the tipped position is located with moments about AB and BC.

Its center of gravity is The distance MG is called the metacentric height and is a direct measure of the stability of the body. Zxample 2. The restoring couple is WM G sin 0 in which 0 is the angular displacement and W the weight of the body.

Stability relations in body of variable cross section. For s floating object of variable cross section. Thc horizontal shift in center of buoyancy r Fig. By integrating over the complete original horizontal area at the liquid surface.

With moments about B to determine the shift r.

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Substitution into Eq. Since 8 is very small. Horizontal cross FIG. For an element of area 6A on the horizontal section through the body at. The force system. Cube floating in liquid. Its center of buoyancy is 6.

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Determine its metacentric height for rolling about y-y-axis and for pitching about x-x-axis. A ship displacing tons has the horizontal cross section at water line shown in Fig. Hence After applying Eq. Find the range of specific-gravity ratios S. Substitution shows that is. Solving for dcpth of submergence. The depth of submergence z is determined by application of the buoyant-force equation. For rolling: For pitching: A homogeneous cube of specific gravity S. Ezample 2. Determine the pressure at A and B in feet of water.

The container of Fig. Prove that the pressure is the same in all direction8 at a point in a static fluid or the three-dimensional case. What is the pressure a t A.

Plot of S. The tube in Fig. Determine the reading h-in Fig. Derive the Ian. Express a pressure of 5 psi in: Rourdon gage A inside a pressure tank reads 10 psi. Find PA in. The liquid has a specific gravity of 1. By use of the results of Prob. Determine the heights of colurnns of water. In isothermal air a t G O O F what is thc vertical distance for reduction of density by 10 per cent? Another bourdon gage B outside the pressure tank.

Express the pressure in six other customary ways. Calculate the pressure a t A. Derive the equations that. The diameter of reservoir is 2. At the top of the building f t higher. The free surface of a liquid makes an angle of 20" with the horizontal when accelerated uniformly in a horizontal direction. Gas is contained in vessel A of Fig. A vertical gas pipe in a building contains gas. Barometric pressure 34 ft water. Determine the acceleration.

Find the gage difference hz in Yrob. If the barometer reading is In Prob. A contains water and the manometer fluid has a specific gravity of 2. What is the acceleration? The inclined manometer of Fig.. When the left meniscus is a t zero on the scale.. Find the imaginary free liquid surface and the pressure a t B. Find the imaginary free surface and the pressure at B. The tube of J? Derive an expression far pressure variation in a constant-temperature gas undergoing an acceleration a.

A cubical box 3 ft on an cdgc. Find the pressure a t B. Find the pressure a t A. The box alone weighs lb and has a coefficientof friction with the plane of 0. If one-third of the liquid spills out. When accelerated to the right 8. Find the pressure at A. A circular cross-sectioned tank of 6-ft depth and 4 f t diameter is filled with liquid and accelerated uniformly in a horizontal direction. A right circular cylinder of radius ro and height ho with axis vertical is open at the top and fillet] with liquid.

Show that the pressure is the same in all directions rtt a point in a liquid moving as a solid. Knoiving the pressure at one point in the fluid.. A vessel containing liquid. Locate the circular line of maximum pressure. The pressure a t one point 2 f t radially from the axis is the same as a t another point 4 ft from the axis and with elevation 2 ft higher.

Determine the weight W that can be sustained by the lb force acting on the piston of Fig. When accelerated uni- formly in the x-direction.

A t what speed must it rotate so that half the area of the bottom is exposed? Derive an expression for pressure in a radial direction for speed w. Prove that the interface has the same shape as the a e k pressure surface. Determine the pressure variation along a vertical line through the axis for density p and speed o. The U-tube of Fig. What is the rotational speed? A closed box contains two immiscible liquids. Locate the vertical axis of rotation and the speed of rotation of the U-tube of Fig.

Calculate the rotational speed. A liquid rotating about a horizontal axis as a solid has a pressure of 10 psi at the axis. Prove by integration that a paraboloid of revolution has a volume equal to half its circumscribing cylinder.

A tank containing two immiscible liquids is rotated about a vertical axis. Neglecting compressibility. A right-angled triangular surface has a vertex in the free surface of a liquid Fig. Find the force on one side a by integration and b by formula. The cylindrical container of Fig. What force is exerted on the upper m d of the cylinder'! If an additional lb weight were placed on the cylinder.

It is filled with water and supported on the piston.

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Keglecting the weight of the container Fig. Find the force of oil on the top surface CD of Fig. Determine the magnitude of the force acting on triangle of Fig. A barrel 2 f t in diameter filled with water has a vertical pipe of 0. If there is water on the other side of thc gate Pig. Calculate the force exerted by water on one side of the vertical annular area shown in Fig.

A cubical box 4 ft on an edge is open a t the top and 'filled with water. Determine the forcc acting on one side of the vertical surface of Fig. When accelerated upward 8. Locate a horizontal line below AB of Fig. Water 1 2. Determine the moment a t A required to hold the gate as shown in Fig.

Locate the pressure center of Fig. The dam of Fig. Locate the pressure center for the annular area of Fig. Locate the pressure centcr for the vertical area of Fig. Locate the distance. A vertical square area 4 by 4 ft is submerged in water with upper edge 2 f t below the surface. Determine the maximum value of liquid depth h. Locate the pressure center for the gate of Fig. An equilateral triangle with one edge in a water surfaue extends down- ward a t a 45O angle.

By integration locate the pressure center horizontally. By integration. Locate the pressure center in terms of the length of a side b. The shaft of the gate in Fig. Locate a horizontal line on the surface of the square such that a the force on the upper portion equals the force on the lower portion and b the moment of force about the line due to the upper portion equals the moment due to the lower portion.

By using the pressure prism. Determine the com- pressive force in the strut. Determine the value of h and the force against the stop when this force is a maximum for the gate of Prob. Determine the water-surface position for the gate just to start to come: Determine y of Fig. The gate of Fig. Its ceder of gravity is 1. It is hinbcd at 0. Find h of Prob. Water surface is below the hinge.

Determine the hinge location y of the rectangular gate of Fig. Xeglect hydrostatic uplift. A cylinder 2 ft in diameter and 6 ft long is accelerated uniformly dong its axis in a horizontal direction I t is filled with liquid. For linear stress variation over the-base of the dam of Fig. Find the resultant force on one side of. By use of the pressure prism. Gate 6 ft wide 2. Work Prob. Find the net force exerted against the liquid in the cylinder. Find the moment M a t 0 Fig.

A cube 1 ff on an edge is filled with liquid. When it is filled with water and rotated uniformly about a vertical axis. A closed cube. What force is required to lift the ball from the opening?

A ft-diameter pressure pipe carries liquid a t psi.

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It is in equi- librium as shown. Compute W. When filled with water. For allowable stress of psi determine the minimum wall thickness. I-4 fi-4 2. The plane gate Fig. A 1-in. To obtain the same flowarea. The gate shown in Fig. The weight is made of concrete. A thin-walled hoilow sphere 8 f t in diameter holds gas at psi.

Neglecting the weight of the arm and brace supporting the counterweight. The maximum allowable pipe wall stress is the same in each case. What thickness pipe wall is required for maximum stress of Calculate the force F required to hold the gate of Fig.

Gate 6 ft wide. Its centcr is 3 ft below the water surface. I t is in equilibrium when empty in the position. What is R of Fig. Find a the horizontal component of force on AB per foot of length. Find the resultant force. The cylinder gate of Fig. The cylinder of Fig. Find the vertical component of force on the curved gate of Fig. Gate 5 ft wide 2. The log holds the water as shown in Fig. Determine a the force per foot pushing i t against the dam. The gate position is controlled by pumping water into or out of the cylinder.

Determine the moment M to hold the gate of Fig. The center of gravity of the empty gate is on the line of symmetry 4 ft from the hinge. Will a beam 10 ft long with square cross section. Both liquids are water. Find the weight of prism per foot of length and x in terms of y for equilibrium. Determine the lengths of wooden cylinder for the system to float in stable equilibrium with axis vertical. A sphere 1 ft in diameter. How many cubic feet of water must be added per foot of cylinder to hold the gate in its position when the water surface is raised 3 f t?

A wooden r: Compute the distance between specific gravity markings 1. Two beams. Determine the height of the top of the cube above the interface. Determine the density. I t is submerged into a two-layered fluid. Design a hydrometer to read specific gravities in the range from 0. Two cubes. A cube. What portion of the lighter cube is above the water surface. A hydrometer weighs 0.

How many pounds of concrete. The pressure. Thr differential equation for pressure variation in n static fluid may be written y measured vertically upward. The pressure 5. Determine the metacentric height of the torus shown in Fig.

A spherical balloon 40 ft in diameter is open a t the bottom and filled with hydrogen. For barometer reading of 28 in. The pressure in the air space above an oil sp gr 0. Determine whether the thick-walled cylinder of Fig. The normal stress is the same in all directions at a point in a fluid a only when the fluid is frictionless b only whcn the fluid is frictionless and incompressible c only when the fluid has zero viscosity and is a t rest d when there is no motion of one fluid Iaycr relative to an adjacent layer e regardloss of the motion of one fluid layer relative to an adjacent layer 2.

With the barometer reading 29 in. In an isothermal atmosphere. The pressure a t A is I a 2. Select the three pressures that are equivalent. Select the correct statement.

The pressure a t A. In the inclined manometer of Fig. Then hB. A mercxry-water manometer has a gage difference of 2. When the box of Prob. The difference in pntssurc. When a liquid rotates a t constant angular velocity about a vertical axis as a rigid body.

Then p. When the minimum pressure in the box of Prob. A closed cubical box. When acceleratal vrrtically upward The pressure center of the surface of Prob. The surface is inclined 30" with the horizontal.

A forced vortex a turns in an opposite direction to a free vortex b always occurs in conjunction with a free vortex c has the velocity decreasing with the radius d occurs when fluid rotates as a solid e has the velocity decreasing inversely with the radius 2. The force on one side of the surface is a 3 8. The pressure a t the center of the bottom is a zero h one-fourth its value when cylinder was full c indeterminable. A rectangular surface 3 ft by 4 f t has the lower 3-ft edge horizontal and 6 ft below a free oil surface.

When a liquid rotates about a vertical axis as a rigid body so that points on the axis have the same pressure as points 2 ft higher and 2 ft from the axis. The pressure center is a a t the centroid of the submerged area b the centroid of the pressure prism. A right-circular cylinder. The magnitude of force on one side of a circular surface of unit area.

A vertical triangular arca has one side in a free surface.. Gate 12 ft wide The moment about the bottom of the gate is a Its altitude is h.

The magnitude of the resultant force acting on both sides of the gate Fig. What is the force exerted on the vertical annular area enclosed by concentric circles of radii A vertical gate 4 ft by 4 f t holds water with free surface a t its top..

The center is The line of action of the resultant force on both sides of the gate in Fig.. The pressure center for the annular area of Prob.. For an allowable tensile stress of psi.