Fluid Mechanics Past Paper (University of Mines and Technology)

UNIVERSITY OF MINES AND TECHNOLOGY (UMaT), TARKWA, GHANA SCHOOL OF RAILWAYS AND INFRAUSTRUCTURE DEVELOPMENT (SRID), ESSIKADO -TAKORADI COURSE: Fluid Mechanics...

UNIVERSITY OF MINES AND TECHNOLOGY (UMaT), TARKWA, GHANA SCHOOL OF RAILWAYS AND INFRAUSTRUCTURE DEVELOPMENT (SRID), ESSIKADO -TAKORADI COURSE: Fluid Mechanics ES,GL 264: FLUID MECHANICS COURSE CODE: ES, EL, MC, GL 264 COURSE INSTRUCTOR: EVANS K.S MBRAH EMAIL: [email protected] 1 Mode of Delivery and Assessment Select this paragraph to edit Select this paragraph to edit Class Delivery Assessment methods Lectures (Face to Face/ Quizzes (In Class) Online) Take Home Assignments Group Presentations Research Assignments ES,GL 264: FLUID MECHANICS Group Presentations End of Semester Examination 2 NOTE Attendance to lectures is compulsory Punctuality is key In Class assignments/tests will be unannounced ES,GL 264: FLUID MECHANICS 3 Objectives Engineering fluid mechanics is aimed at introducing students to the fundamentals governing the behavior of static fluids and fluids in motion and how this affects flow in pipes and ducts. Also, solutions to dimensional analysis, energy losses as well as flow equations ES,GL 264: FLUID MECHANICS 4 Objectives Engineering fluid mechanics is aimed at introducing students to the fundamentals governing the behavior of static fluids and fluids in motion and how this affects flow in pipes and ducts. Also, solutions to dimensional analysis, energy losses as well as flow equations ES,GL 264: FLUID MECHANICS 5  QUIZ 2 1. Write the formula for determining the dynamic viscosity of a fluid.(1) 2. What is the relation between dynamic and kinematic viscosity.(1) 3. State the formula for determining the bulk modulus of elasticity.(1) 4. When is a concave meniscus formed?(1) 5. Why does water wet glass? 6. Water at 30oC is able to climb up a clean glass of 0.2-mm-diameter tube due to surface tension. The water-glass angle is 0o with the vertical. How far up the tube does the water climb? Take surface tension at 30oC = 0.0718. (3) State Whether the following are True or False 7. Newtonian fluids are fluids for which shear stress is directly proportional to the rate of angular deformation or a fluid for which the viscosity,  is a constant for a fixed temperature and pressure. (1) ES,GL 264: FLUID MECHANICS 8. An incompressible fluid is a fluid whose density is changed by external forces acting on the fluid.(1) 9. Compressibility of a fluid is a measure of the change in volume of the fluid when it is subjected to outside force.(1) 6  FLUID STATICS Focus Determine the pressure at various locations in a fluid at rest. Explain the concept of manometers and apply appropriate equations to determine pressures. Calculatethe hydrostatic pressure force on a plane or ES,GL 264: FLUID MECHANICS curved submerged surface. Calculate the buoyant force and discuss the stability of floating or submerged objects. 7  Hydrostatics Hydrostatic deals with fluid at rest. Hydrostaticsstudies the laws governing the behaviour of fluid at equilibrium when it is subjected to external and internal forces and bodies at equilibrium when they are immersed in the fluid. Shear stress in a fluid at rest is always zero. Therefore, in fluids ES,GL 264: FLUID MECHANICS at rest, the only stress we shall be dealing with is normal stresses. 8  Hydrostatic Pressure 9 ES,GL 264: FLUID MECHANICS  Properties of Hydrostatic Pressure a) Hydrostatic pressure is a compressive stress and always acts along the inside normal to the element of area. b) The magnitude of the hydrostatic pressure p at a given point in a fluid does not depend on the orientation of the surface i.e. on the incline of the surface ES,GL 264: FLUID MECHANICS 10  PRESSURE ES,GL 264: FLUID MECHANICS 11 Pressure Cont. The actual pressure at a given position is absolute pressure pressure, measured relative to absolute vacuum (i.e., absolute zero pressure). Most pressure measuring devices are calibrated to read zero in the atmosphere and so they indicate the difference between the absolute pressure and the local atmospheric pressure. This difference is called the gage pressure. Pressures below atmospheric pressure are called vacuum pressures and are measured ES,GL 264: FLUID MECHANICS by vacuum gages that indicate the difference between the atmospheric pressure and the absolute pressure. Absolute, gage, and vacum 12 Differential Equation of a Fluid at Rest ES,GL 264: FLUID MECHANICS 13 Equation of a Fluid at Rest Cont. ES,GL 264: FLUID MECHANICS 14 Equation of a Fluid at Rest Cont. Projection of surface forces on the x-axis Force on side ABCD dFx=pdydx Force on side A1B1C1D1 dF1=p1dydz Equation of a Fluid at Rest Cont. Projection of body forces on the x-axis The projection of body forces on the x-axis is the product of the mass of fluid and the projection of acceleration on the x-axis. i.e. dRx=dxdydz. where X is the projection of acceleration of body forces in the x-axis Equation of a Fluid at Rest Cont. Applying Newton’s law in the x-axis Fx=0---sum of surface and body forces in the x-axis equals zero Dividing through by ρdxdydz, we shall obtain Equation of a Fluid at Rest Cont. By analogy, we can write similar equations in the y-axis and z- axis Fy=0; Fz=0 Adding left hand side and the right hand side; Equation of a Fluid at Rest Cont. Since hydrostatic pressure is a function of independent coordinates x, y, z, then the first three functions on the left side of the above equation being the sum of three partial differential equals the exact (total) differential Basic differential equation of hydrostatic Equation of a Fluid at Rest Cont. Since the left hand side of equation is an exact (total) differential, then the right hand side must also be an exact differential of a certain function say U (x, y, z) U(x, y,z) = Xdx + Ydy + Zdx We can write the exact differential dU(x, y, z) into partial differential  Equation of a Fluid at Rest Cont. Therefore; and we can write Equation of a Fluid at Rest Cont. Since U is a function of only coordinates (x, y, z) and its partial differential gives the corresponding projection of body forces per unit mass (X, Y, Z) on the respective axes, then the function U is a Potential Function Conclusion Fluids can be in a state of equilibrium (rest) when and only when it is acted upon by potential forces  Integrating The Basic Differential Equation Of Hydrostatics The basic equation is: Integrating; p=ρU + C where C is the constant of integration To find C, we consider a point in a fluid with p and U known. Assuming at this point when p=p0 when U=U0, then po=ρU0+C C= po – ρUo Integrating The Basic Differential Equation Of Hydrostatics Therefore; po=U0+C p = po +ρ(U-Uo) General equation of hydrostatics in the integral form Hydrostatic Pressure At A Point In A Fluid When Gravity Is The Only Body Force ES,GL 264: FLUID MECHANICS 25 Hydrostatic Pressure At A Point In A Fluid When Gravity Is The Only Body Force The basic differential equation is: Since force of gravity is the only body force acting, we shall have the following: X=0; Y=0; Z=-g and dp = -ρg.dz  Hydrostatic Pressure At A Point In A Fluid When Gravity Is The Only Body Force. Integrating the above equation, we have p=-g.z + C or p = -γ.z +C To find C, let us consider a point at the surface of the fluid. At that point O, z=0; p=po po =C The above equation becomes: p=-z + po  Hydrostatic Pressure At A Point In A Fluid When Gravity Is The Only Body Force Now let h be the depth of immersion of the point M.h=-z Therefore, the above equation becomes: p = po +γh fundamental equation of hydrostatics  Hydrostatic Pressure At A Point In A Fluid When Gravity Is The Only Body Force. P ---- is known as the absolute hydrostatic pressure at the point M h --- is the body pressure i.e. pressure due to the body of column of fluid above M Conclusion The absolute pressure at a point is the sum of the external surface pressure and the body pressure (pressure created by the column of fluid on point).  Hydrostatic Pressure At A Point In A Fluid When Gravity Is The Only Body Force If the external pressure po is atmospheric, ie container is opened, then po =pa p=pa + h pa = atmospheric pressure or barometric pressure p-pa = h -------Gauge or manometric pressure  EXAMPLE If instrument air gauge pressure is 580kPa (g),what is its absolute value? p(a) = p(g) + p(atm) p(a) = 580 kPa (g) + 101.3 kPa = 681.3 kPa (a)  Manometric Gauge Pressure Gauge pressure: is the differential (excess) pressure above atmospheric pressure at a point in a fluid. In practice we often use the manometric pressure instead of the absolute pressure. So from now we shall denote; PA = absolute pressure P= γh -- excess or manometric pressure PA =Po +P Where PA –absolute pressure; Po – external pressure and P– gauge pressure Pascal’s Law: Hydraulic Press Select this paragraph to edit  Pascal’s Law: Hydraulic Press Pascal’s Law states: pressure (external) which arises (or which is applied) at the surface of a liquid at rest is transmitted throughout the liquid in all direction without any change Hydraulic Press Select this paragraph to edit Hydraulic Press Select this paragraph to edit Hydraulic Press From Pascal’s law, this pressure is transmitted to the piston 5. The result is a useful force F2 under whose action the material is pressed. Select this paragraph to edit Select this paragraph to edit Piezometric Height Select this paragraph to edit Piezometric Height Select this paragraph to edit Potential Energy of Fluid at Rest Select this paragraph to edit Potential Energy of Fluid at Rest Select this paragraph to edit Specific Potential Energy Select this paragraph to edit Potential Head Select this paragraph to edit Variation of Pressure in the Earth’s Atmosphere Gases are highly compressible and are characterized by changes in density. The change in density is achieved by both change in pressure and temperature. In the treatment of gases, we shall consider the perfect gas. It must be recognized that there is no such thing as a perfect gas, however, air and other real gases that are far removed from the liquid phase may be so considered.  Equations Of State For Gases The absolute pressure p, the specific volume v, and the absolute temperature are related by the equation of state. For a perfect gas, the equation of state per unit weight is Equations Of State For Gases Select this paragraph to edit Isothermal Process The compression and expansion of a gas may take place according to various laws of thermodynamics. If the temperature is kept constant, the process is called isothermal and the value of n in eq. (3) is unity; i.e. n = 1.  Isentropic Process If a processes is such that there is no heat added to or withdrawn from the gas (i.e. zero heat transfer), it is said to be adiabatic process. An isentropic process is an adiabatic process in which there is no friction and hence is a reversible process. The value of the exponent, n in equation (3) is then denoted by k which is the ratio of the specific heats at constant pressure and constant volume. Pressure Variation In The Atmosphere The atmosphere may be considered as a static fluid and as such can be subjected to the basic differential equation when gravity is the only body force acting. dp/dz = -γ To evaluate the pressure variation in a fluid at rest, one must integrate the above equation. For compressible fluids, however, γ must be expressed algebraically as a function of z and p Pressure Variation In The Atmosphere Let us illustrate some of the problems dealing with pressure variation in the atmosphere. Let us compute the atmospheric pressure at an elevation of H considering the atmosphere as a static fluid. Assume standard atmosphere at sea level. Use:  air at constant density  constant temperature between sea level and H Isentropic conditions Air temperature decreasing linearly with elevation at standard lapse rate of X⁰C/m Pressure Variation In The Atmosphere Select this paragraph to edit PRESSURE MEASURING DEVICES There are generally two types of pressure measuring devices: Tube gauges: those instruments that work on the principle that a particular pressure can support a definite weight of a fluid and this weight is defined by definite column of fluid. Mechanical gauges: - work on the principle that the applied pressure will create a deformation in either a spring or a diaphragm.  Tube Gauges (Piezometer) Piezometer: is the simplest pressure measuring tube device and it consists of a narrow tube so chosen that the effect of surface tension is negligible When connected to the pipe whose pressure is to be measured, the liquid rises up to a height, h, which is an indicative of the pressure in the pipe; p=h Advantages And Disadvantages Of Piezometric Tubes Advantages: Cheap, easy to install and read Disadvantages:  Requires unusually long tube to measure even moderate pressures  Cannot measure gas pressures (gases cannot form free surface)  Cannot measure negative pressures (atmospheric air will enter the pipe through the tube) Tube Gauges (Manometers)) To overcome the above-mentioned limitations of the piezometer, an improved form of the piezometer consisting of a bent tube containing one or more fluids of different specific gravities is used. Such a tube is called a manometer. Types of manometers  Simple manometer  Inclined manometer  Micro manometer  Differential manometer  Inverted differential manometer  Tube Gauges (Manometers) Cont. A simple manometer: consists of a tube bent in U-shape, one end of which is attached to the gauge point and the other is opened to the atmosphere The fluid used in the bent tube is called the manometric fluid (usually mercury) and the fluid whose pressure is to be measured and therefore exerts pressure on the manometric fluid is referred to as the working fluid  SIMPLE MANOMETERS (B) Simple manometer measuring vacuum pressure (A) Simple manometer measuring gauge pressure Inclined manometer SIMPLE MANOMETERS CONT. Inclined Tube Manometer This type is more sensitive than the vertical tube type. Due to the inclination the distance moved by the manometric fluid in the narrow tube will be comparatively more and thus give a higher reading for a given pressure Other Types of Manometers Differential Manometer: consists of a U-tube containing the manometric fluid. The two ends of the tubes are connected to the points, whose differential pressure is to be measured. Inverted U-tube Differential Manometer An inverted U-tube differential manometer is used for measuring difference of low pressures, where accuracy is the prime consideration. It consists of an inverted U-tube containing a light liquid.  Mechanical Gauges Whenever very high fluid pressures are to be measured mechanical gauges are best suited for these purposes. A mechanical gauge is also used for the measurement of pressures in boilers or other pipes, where tube gauges cannot be conveniently used Bourdon's Tube Pressure Gauge It can be used to measure both negative (vacuum) and positive (gauge) pressure. It consists of an elliptical tube ABC, bent into an arc of a circle. When the gauge tube is connected to the fluid (whose pressure is to be found) at C, the fluid under pressure flows into the tube. The Bourdon’s tube as a result of the increased pressure tends to straighten out. With an arrangement of pinion and sector, the elastic de formation of the Bourdon’s tube rotate s a pointer, which moves over a calibrated scale to read directly the pressure of the fluid.  Diaphragm Pressure Gauge The principle of work of the diaphragm pressure gauge is similar to that of the Bourdon tube. However instead of the tube, this gauges consists of a corrugated diaphragm When the gauge is connected to the fluid whose pressure is to be measured at C, the pressure in the fluid causes some deformation of the diaphragm. With the help of pinion arrangement, the elastic deformation of the diaphragm rotates the pointer  Dead Weight Pressure Gauge It is an accurate pressure-measuring instrument and is generally used for the calibration of other pressure gauge A dead weight pressure gauge consists of a piston and a cylinder of known area and connected to a fluid by a tube. The pressure on the fluid in the pipe is calculated by: p=weight/Area of piston A pressure gauge to be calibrated is fitted on the other end of the tube. By changing the weight on the piston the pressure on the fluid is calculated and marked on the gauge Relative Equilibrium Of Liquid ((Liquid under constant acceleration or constant angular speed) When fluid masses move without relative motion between particles, they behave just as much as solid body and are said to be in relative equilibrium Relative equilibrium of a liquid is that situation in which a liquid being in motion, stay together as one mass as a solid body i.e. there is no sliding (displacement) of some particles over others. Relative Equilibrium Of Liquid ((Liquid under constant acceleration or constant angular speed) Since the fluid, though moving is in equilibrium means it could be described by the basic equation of hydrostatics as: To be able to solve any problem, we should be able to identify all the body forces acting on the fluid in relative equilibrium Let us consider some examples Liquid Mass Subjected To Uniform Linear Horizontal Acceleration Consider a tank partially filled and placed on a tanker truck and given a uniform acceleration ax in the x-direction As a result of the acceleration, within the fluid will emerge an inertia acceleration in opposition to the imposed acceleration The inertia acceleration has the same magnitude but in the opposite direction Liquid Mass Subjected To Uniform Linear Horizontal Acceleration Cont. Liquid Mass Subjected To Uniform Linear Horizontal Acceleration Cont. Since this is a static situation, then we can use the general differential equation of statics, i.e On the accelerating fluid, there are two body forces acting, namely gravity force and inertia force. From the above equation, we recognise that Substituting (**) into (*), we shall have Integrating, Liquid Mass Subjected To Uniform Linear Horizontal Acceleration Cont. The pressure distribution within the accelerating fluid is: P = ρ(-ax –gz) The angle the surface of the fluid makes with the horizontal can be obtained by finding the tangent of the angle θ Therefore, in a uniform accelerating fluid, the angle of inclination of the fluid surface to the horizontal is the ratio of the horizontal body force acceleration to that of the vertical body force acceleration Motion In The Vertical Plane With Constant Acceleration Motion In The Vertical Plane With Constant Acceleration Cont. Motion In The Vertical Plane With Constant Acceleration Cont. Equilibrium Of A Rotating Container Equilibrium Of A Rotating Container Consider a cylindrical container filled with a liquid and rotating with a constant angular velocity ω about the vertical axis As a result of the liquid rotating with the same angular velocity as the container the liquid is considered to be at rest relative to the container. Frictional force (both internal, and external i.e. friction between particles of liquid walls) is zero Equilibrium Of A Rotating Container If the coordinate axis shown on the diagram is considered fixed to the container, then relative to the rotating vessel, the liquid will also be at rest Therefore, the basic differential equation of hydrostatic of Euler is applicable in the case of a rotating fluid with the above conditions The body forces acting on the fluid are: Equilibrium Of A Rotating Container 1.Gravity dF G = gdM or Z = -g 2.Centripetal force Equilibrium Of A Rotating Container Equilibrium Of A Rotating Container To find C we can look at the conditions at the point x=0; y=0, z=0; and p=Po. Therefore C= Po, Then Forces Of Hydrostatic Pressure On Plane Surfaces Immersed In Fluids Where are these applied: 1. Irrigation Engineering for water distribution on the field 2. Dam engineering for all types of gates 3. In River transportation (Locks systems) FORCES OF HYDROSTATIC PRESSURE ON PLANE SURFACES IMMERSED IN FLUIDS CONT.  Forces Of Hydrostatic Pressure On Plane Surfaces Immersed In Fluids Cont. Consider in Fig. (above) an open container, filled with a fluid and an inclined plane OM. On the inclined plane OM is an arbitrary plane figure AB with area A Our task is two folds: 1. to find the magnitude of the force on the plane surface due to water pressure. 2. to find the point (position) of action of this force Forces Of Hydrostatic Pressure On Plane Surfaces Immersed In Fluids Cont. Let us choose an arbitrary point m on the surface AB immersed in the fluid at a depth h, and at a distance z from the axis OZ At the point m, we choose an elemental area Da surrounding the point m, such that the pressure within it is the same. The hydrostatic force on the area dA is given by: Plane Surfaces Immersed In Fluids Cont. Forces Of Hydrostatic Pressure On Plane Surfaces Immersed In Fluids Cont. Finding (Centre of Pressure) To find the centre of pressure, we are going to use the theory of moments which states that the moment of the resultant force about a point (or axis) equals the sum of moments of all the forces about the same point (or axis) Let ZD be the centre of pressure and let us write the equation of moments about the axis Ox. The moment d M of the elemental force dF about Ox equals Finding (Centre of Pressure) Cont. The sum of moments of all the individual forces is given by Finding (Centre of Pressure) Cont. Finding (Centre of Pressure) Cont.  Finding (Centre of Pressure) Cont. The point of action of the resultant force F is given by: Summary Graphical Method For Finding Hydrostatic Force on Plane Surfaces Cont. Pressure Diagram Method Simple method for finding the force, FR and the point of action of the force, yR Very useful for rectangular planes or areas The properties of a pressure diagram include:  Every ordinate on the pressure diagram gives the hydrostatic pressure at the point.  The area under the pressure diagram gives the value of the hydrostatic force per unit width of the gate.  The force F passes through the centre of gravity of the pressure diagram Pressure Diagram Method Cont. Pressure Diagram Method Cont. Pressure Diagram Method Cont. Pressure Diagram Method Cont. Pressure Diagram Method Cont. Hydrostatic Pressure Forces On Curved Surfaces ES,GL 264: FLUID MECHANICS 102 Hydrostatic Pressure Forces On Curved Surfaces Hydrostatic Pressure Forces On Curved Surfaces Cont. Consider a curved surface ABC with length b. Let Px and Pz be the horizontal and vertical components of the force due to hydrostatic pressure acting on the curved surface. To find these components lets erect the plane DE. The plane DE will isolate that volume of liquid ABCED whose equilibrium we wish to investigate Hydrostatic Pressure Forces On Curved Surfaces Cont. The volume ABCED is acted upon by the ff:  the force Ph acting on the vertical side DE  the force RD -reaction of the base EC RD=[area (C1CED)] bγ  the reaction R from the curved surface. Rx, Rz  the horizontal and vertical components respectively.  force due to liquid’s own weight G G=[area (ABCED)] bγ Hydrostatic Pressure Forces On Curved Surfaces Cont. Horizontal/Vertical Component 1. The horizontal component Px of the force on a curved surface equals the force of hydrostatic pressure on the plane vertical figure DE, which is a projection of the curved surface on the vertical plane 2. The vertical component Pz equals the weight of the imaginary free body of the fluid ABCC1. This imaginary free body of the fluid we shall called "pressure body". The weight of the pressure body represented by [area ABCC'] bγ = Go Procedure For Determining The Horizontal Component 1. Place a vertical plane DE behind the curved surface 2. Project the curved surface onto the vertical plane to obtain a plane surface 3. Determine the horizontal component in a similar manner as in plane surfaces immersed in fluid. Procedure For Determining The Vertical Component The cylindrical surface ABC is the surface whose pressure body is to be found 1. First fix the extreme ends A and C of the curved surface 2. Draw vertical lines from these points to the water surface 3. finally note the contour of the pressure body A'ABCC' ie the body of fluid between the two vertical lines, the curved surface and the surface of the fluid  Procedure For Determining The Vertical Component The cross-section of the pressure body (positive or negative) is the area between the two verticals, the cylindrical surface ABC and the surface of the fluid (or their continuation). If the pressure body does not wet the cylindrical surface, then we have negative body pressure; however if the pressure body wets the surface, then the pressure body is positive  Buoyancy FLOATING BODIES: ARCHIMEDES PRINCIPLE: the force, which a fluid exerts on a body immersed in it equals the weight of the fluid displaced by the body or when a body is placed (submerged) in a fluid, it experiences an upward (upthrust) force which is equal to the weight of fluid the body displaces Buoyancy Cont.  Buoyancy Cont. The body AB with volume V completely submerged in a fluid. The resultant of all forces due to pressure acting on the surface element of the body is determined by the principle of forces on a curved surface. But Rx=0; Ry=0 Buoyancy Cont. Equation of floating bodies Stability Of Submerged Bodies  STABILITY OF SUBMERGED BODIES Stability Of Submerged Bodies A body is said to be in a stable equilibrium, if a slight displacement generates forces which oppose the change of position and tend to bring the body to its original position Criterion Of Stability For Submerged Bodies Cont. The criterion of stability for submerged bodies is the relative positions of D and C. For submerged body to be stable, I. the weight of the body G must be equal to the buoyancy force Fb and II. the centre of buoyancy D must always be above the centre of gravity C of the body. Submarines are submerged bodies, which use balancing tanks to make Fb equal to G and trimming tanks to bring the centre of buoyancy above the centre of gravity Some Basic Terms In Floating Bodies O – O – axis of floatation W-L: - water line – the line of intersection of the free surface of the fluid with the body. C – centre of gravity of the body D – centre of buoyancy of the body when it is upright D1 – centre of buoyancy of the body when body is rotate through a small angle θ M- Metacentre – is the point of Floating Bodies Fb>G intersection of the axis of floatation and the vertical through D1 Some Basic Terms In Floating Bodies Cont. MC – metacentric height-the distance between the metacentre and the centre of gravity. MD – metacentric radius: - distance between the metacentre and the centre of buoyancy when object is upright. h – height of floating body d – draft of floating body  FLOATING BODIES FB>G The figures shown above represent floating bodies. Fig a represents a body in equilibrium. The net force on the body is zero so it means the buoyancy force Fb equals in magnitude to the weight of the body. There is no moment on the body so it means the weight acting vertically downwards through the centre of gravity C must be in line with the buoyancy force acting vertically upwards through the centre of buoyancy D  FLOATING BODIES CONT. Fig (a*) shows the situation after the body has undergone a small angular displacement (angle of heel θ). It is assumed that the position of the centre of gravity C remains unchanged relative to the body. The centre of buoyancy D, however, does not remain fixed relative to the body During the movement, the volume immersed on the right side increases while that on the left side decreases; so the centre of buoyancy moves to the new position D 1. The line of action of the buoyancy force will intersect the axis of floatation at the point M  FLOATING BODIES On the other hand in Fig (b*), the point M is below the point C and the couple thus formed is an overturning couple and the original equilibrium would be unsafe. The distance MC is known as the metacentric height and for stability of the body, it must be positive (i.e. M above C). The magnitude of MC serves as a measure of stability of floating bodies.  FLOATING BODIES CONT. The distance MC is known as the metacentric height and for stability of the body, it must be positive (i.e. M above C). The greater the magnitude of MC, the greater is the stability of the body. The magnitude of MC serves as a measure of stability of floating bodies It is important that all floating bodies do not capsize in water. It is therefore essential that we are able to determine its stability before it is put in water Experimental Determination Of Metacentric Height  Determination Of Metacentric Height The experiment consists of moving a weight P across the deck through a certain distance x and observing the corresponding angle of heel or roll θ The shifting of the weight P through a distance x produces a moment Px which causes the vessel to tilt through an angle θ. This moment Px is balanced by the righting moment G x CM θ Determination Of Metacentric Height Cont. It must be noted that the vessel, before the weight was moved, was in an upright (vertical) position and G is the total weight of the vessel (including the weight P) The metacentric radius DM = I/V0  Metacentric Radius The metacentric radius DM = I/V0 Where I – second moment of area of the plane of floatation about centroidal axis; V0 – immersed volume  The Hydrometer The hydrometer is an instrument for measuring the specific gravity of liquids. It is based on the principle of buoyancy. The hydrometer consists of a bulb weighted at the bottom to make it float upright in liquid and a stem of smaller diameter and usually graduate. ES,GL 264: FLUID MECHANICS 129