physics notes for neet chapter 11.pdf
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60 516 Fluid Mechanics Chapter E3 11 Fluid Mechanics ID Fluid is the name given to a substance which begins to flow when external force is applied on it. Liquids and gases are fluids. Fluids do not have their own shape but take the shape of the containing vessel. The branch of physics which deals wi...
60 516 Fluid Mechanics Chapter E3 11 Fluid Mechanics ID Fluid is the name given to a substance which begins to flow when external force is applied on it. Liquids and gases are fluids. Fluids do not have their own shape but take the shape of the containing vessel. The branch of physics which deals with the study of fluids at rest is called hydrostatics and the branch which deals with the study of fluids in motion is called hydrodynamics. U Pressure. (6) Hydrostatic pressure depends on the depth of the point below the surface (h), nature of liquid ( ) and acceleration due to gravity (g) while it is independent of the amount of liquid, shape of the container or cross-sectional area considered. So if a given liquid is filled in vessels of different shapes to same height, the pressure at the base in each vessel's will be the same, though the volume or weight of the liquid in different vessels will be different. D YG The normal force exerted by liquid at rest on a given surface in contact with it is called thrust of liquid on that surface. The normal force (or thrust) exerted by liquid at rest per unit area of the surface in contact with it, is called pressure of liquid or hydrostatic pressure. If F be the normal force acting on a surface of area A in contact with liquid, then pressure exerted by liquid on this surface is P F / A (1) Units : N / m 2 or Pascal (S.I.) and Dyne/cm (C.G.S.) 2 (2) Dimension : [P] 2 [F] [MLT ] [ML1T 2 ] [ A] [L2 ] U (3) At a point pressure acts in all directions and a definite direction is not associated with it. So pressure is a tensor quantity. PA PB PC but WA(B) WB WC (C) (A) Fig.pressure 11.2 (7) In a liquid at same level, the will be same at all points, if not, due to pressure difference the liquid cannot be at rest. This is why the height of liquid is the same in vessels of different shapes containing different amounts of the same liquid at rest when they are in communication with each other. ST (4) Atmospheric pressure : The gaseous envelope surrounding the earth is called the earth's atmosphere and the pressure exerted by the atmosphere is called atmospheric pressure. Its value on the surface of the earth at sea level is nearly 1.013 10 5 N / m 2 or Pascal in S.I., other practical units of pressure are atmosphere, bar and torr (mm of Hg) 1atm 1.01 10 5 Pa 1.01bar 760 torr The atmospheric pressure is maximum at the surface of earth and goes on decreasing as we move up into the earth's atmosphere. (5) If P0 is the atmospheric pressure then for a point at depth h below the surface of a liquid of density , hydrostatic pressure P is given by P P0 h g (8) Gauge pressure : The pressure difference between hydrostatic pressure Fig. 11.3 P and atmospheric pressure P is called gauge pressure. 0 P P0 hg Density In lim V 0 a fluid, m dm V dV at a point, density is defined as: Fluid Mechanics (1) In case of homogenous isotropic substance, it has no directional properties, so is a scalar. (2) It has dimensions [ML3 ] and S.I. unit kg/m while C.G.S. unit 3 g/cc with 1g / cc 10 kg / m 3 3 (3) Density of substance means the ratio of mass of substance to the volume occupied by the substance while density of a body means the ratio of mass of a body to the volume of the body. So for a solid body. Density of body = Density of substance While for a hollow body, density of body is lesser than that of substance [As Vbody Vsub. ] and also to the walls of the container, provided the effect of gravity is neglected. Example : Hydraulic lift, hydraulic press and hydraulic brakes Working of hydraulic lift : It is used to lift the heavy loads. If a small force f is applied on piston of C then the pressure exerted on the liquid P f /a [a = Area of cross section of the piston in C] This pressure is transmitted equally to piston of cylinder D. Hence the upward force acting on piston of cylinder D. f Load C (6) If m1 mass of liquid of density 1 and m 2 mass of density 2 are mixed, then as [As V m / ] ID (7) If V1 volume of liquid of density 1 and V2 volume of liquid [As m / V ] D YG m 1V1 2 V2 and V V1 V2 (1 2 ) / 2 = Arithmetic Mean (8) With rise in temperature due to thermal expansion of a given body, volume will increase while mass will remain unchanged, so density will decrease, i.e., V0 (m / V ) V0 [As V V0 (1 ) ] 0 (m / V0 ) V V0 (1 ) If V1 V2 V 0 ~ – 0 (1 ) or (1 ) U (9) With increase in pressure due to decrease in volume, density will increase, i.e., ST (m / V ) V0 0 (m / V0 ) V Accidentally Archimedes discovered that when a body is immersed partly or wholly in a fluid, at rest, it is buoyed up with a force equal to the weight of the fluid displaced by the body. This principle is called Archimedes principle and is a necessary consequence of the laws of fluid statics. When a body is partly or wholly dipped in a fluid, the fluid exerts force on the body due to hydrostatic pressure. At any small portion of the surface of the body, the force exerted by the fluid is perpendicular to the surface and is equal to the pressure at that point multiplied by the area. The resultant of all these constant forces is called upthrust or buoyancy. To determine the magnitude and direction of this force consider a body immersed in a fluid of density as shown in figure. The forces on the vertical sides of the body will cancel each other. The top surface of the body will experience a downward force. U 2 1 2 Harmonic mean 1 2 of density 2 are mixed, then as: [As F1 AP1 A(h1g P0 ) F2 AP2 A(h2g P0 ) 1 p ~ p So 0 1 0 1 B B Pascal's Law It states that if gravity effect is neglected, the pressure at every point of liquid in equilibrium of rest is same. or The increase in pressure at one point of the enclosed liquid in equilibrium of rest is transmitted equally to all other points of the liquid h1 F1 m ] V p p i.e., V V0 1 B V [As P hg P0 ] While the lower face of the body will experience an upward force. But as by definition of bulk-modulus B V0 f A Fig. 11.4 A f a a Archimedes Principle mi m m1 m 2 V (m1 / 1 ) (m 2 / 2 ) (mi / i ) F PA As A a , therefore F f. So heavy load placed on the larger piston is easily lifted upwards by applying a small force. m m1 m 2 and V (m1 / 1 ) (m 2 / 2 ) If m1 m 2 F E3 Densityof body Densityof water D 60 (4) When immiscible liquids of different densities are poured in a container, the liquid of highest density will be at the bottom while that of lowest density at the top and interfaces will be plane. (5) Sometimes instead of density we use the term relative density or specific gravity which is defined as : RD 517 h2 F2 Fig. 11.5 As h2 h1 , F2 will be greater than F1 , so the body will experience a net upward force F F2 F1 Ag(h2 h1 ) If L is the vertical height of the body F AgL Vg [As V AL A(h2 h1 )] i.e., F = Weight of fluid displaced by the body. This force is called upthrust or buoyancy and acts vertically upwards (opposite to the weight of the body) through the centre of gravity of 518 Fluid Mechanics displaced fluid (called centre of buoyancy). Though we have derived this result for a body fully submerged in a fluid, it can be shown to hold good for partly submerged bodies or a body in more than one fluid also. (1) Upthrust is independent of all factors of the body such as its mass, size, density etc. except the volume of the body inside the fluid. (2) Upthrust depends upon the nature of displaced fluid. This is why upthrust on a fully submerged body is more in sea water than in fresh water because its density is more than fresh water. Weight will be more than upthrust so the body will sink If density of body is equal to that of liquid (3) Apparent weight of the body of density ( ) when immersed in Apparent weight = Actual weight – Upthrust W Fup Vg Vg V( )g Vg 1 water W1 W2 g (5) Relative density of a body density of body density of water D YG (R.D.)= Weight of body Weight of body = Water thrust Weight of equal volume of water Weight of body Loss of weight in water = W1 Weight of body in air = W1 W2 Weight in air – weight in water L Upthruston body in liquid W Upthruston body in water Loss of weight in liquid a Wair Wliquid b Wair Wwater Loss of weight in water ST U (6) If the loss of weight of a body in water is 'a ' while in liquid is 'b' V Vin Where Vin is the volume of body in the liquid (i) A body will float in liquid only and only if (ii) In case of floating as weight of body = upthrust U Then apparent (loss of weight) weight W1 W2 Vg Weight will be less than upthrust so the body will, move upwards and in equilibrium will float and partially immersed in the liquid Such that, W Vin g V g Vin g ID W2 = Weight of the body in E3 (4) If a body of volume V is immersed in a liquid of density then its weight reduces. V Weight will be equal to upthrust so the body will float fully submerged in neutral equilibrium with its top surface in it just at the top of liquid If density of body is lesser than that of liquid WAPP W 1 W1 = Weight of the body in air, 60 a liquid of density ( ). So WApp = Actual weight – upthrust = 0 (iii) In case of floating Vg Vin g So the equilibrium of floating bodies is unaffected by variations in g though both thrust and weight depend on g. (2) Rotatory Equilibrium : When a floating body is slightly tilted from equilibrium position, the centre of buoyancy B shifts. The vertical line passing through the new centre of buoyancy B and initial vertical line meet at a point M called meta-centre. If the meta-centre M is above the centre of gravity the couple due to forces at G (weight of body W) and at B (upthrust) tends to bring the body back to its original position. So for rotational equilibrium of floating body the meta-centre must always be higher than the centre of gravity of the body. Th M G B' G B B W Floatation (1) Translatory equilibrium : When a body of density and volume V is immersed in a liquid of density , the forces acting on the body are Weight of body W mg Vg, acting vertically downwards through centre of gravity of the body. Upthrust force = Vg acting vertically upwards through the centre of gravity of the displaced liquid i.e., centre of buoyancy. If density of body is greater than that of liquid (A) (B) Th G M W B B' However, if meta-centre goes(C) below CG, the couple due to forces at G and B tends to topple the floating body. Fig. 11.6 Fluid Mechanics That is why a wooden log cannot be made to float vertical in water or a boat is likely to capsize if the sitting passengers stand on it. In these situations CG becomes higher than MG and so the body will topple if slightly tilted. (3) Application of floatation The two streamlines cannot cross each other and the greater is the crowding of streamlines at a place, the greater is the velocity of liquid particles at that place. Path ABC is streamline as shown in the figure and v1 , v 2 and v 3 are the velocities of the liquid particles at A, B and C point respectively. (2) Laminar flow : If a liquid is flowing over a horizontal surface with a steady flow and moves in the form of layers of different velocities which do not mix with each other, then the flow of liquid is called laminar flow. (i) When a body floats then the weight of body = Upthrust Vg Ving Vin V i.e., Fraction of volume outside the liquid fout (ii) For floatation V Vin V out 1 V 60 In this flow, the velocity of liquid flow is always less than the critical velocity of the liquid. The laminar flow is generally used synonymously with streamlined flow. V Vin 1 V (3) Turbulent flow : When a liquid moves with a velocity greater than its critical velocity, the motion of the particles of liquid becomes disordered or irregular. Such a flow is called a turbulent flow. Vin fin V E3 Vout If two different bodies A and B are floating in the same liquid then A ( fin)A B ( fin)B (iii) If the same body is made to float in different liquids of densities and B respectively. V (Vin)A A (Vin)B B A (Vin)B B (Vin)A In a turbulent flow, the path and the velocity of the particles of the 11.8 liquid change continuously and Fig. haphazardly with time from point to point. In a turbulent flow, most of the external energy maintaining the flow is spent in producing eddies in the liquid and only a small fraction of energy is available for forward flow. For example, eddies are seen by the sides of the pillars of a river bridge. U Pillar ID A 519 Mg hAg D YG (iv) If a platform of mass M and cross-section A is floating in a liquid of density with its height h inside the liquid …(i) Now if a body of mass m is placed on it and the platform sinks by y then (M m)g (y h) Ag …(ii) Subtracting equation (i) from (ii), mg A y g , i.e., W y U Streamline, Laminar and Turbulent Flow ST (1) Stream line flow : Stream line flow of a liquid is that flow in which each element of the liquid passing through a point travels along the same path and with the same velocity as the preceding element passes through that point. Reynold's number is a pure number which determines the nature of flow of liquid through a pipe. It is defined as the ratio of the inertial force per unit area to the viscous force per unit area for a flowing fluid. NR Inertial force per unit area Viscous force per unit area If a liquid of density is flowing through a tube of radius r and cross section A then mass of liquid flowing through the tube per second dm volume flowing per second × density = A v dt Inertial force per unit area = dp / dt v(dm / dt) v Av = = A A A v2 Viscous force per unit area F / A A v1 The critical velocity is that velocity of liquid flow upto which its flow is streamlined and above which its flow becomes turbulent. …(iii) So we can determine the weight of a body by placing it on a floating platform and noting the depression of the platform in the liquid by it. Critical Velocity and Reynold's Number B C v2 Fig. 11.7 A streamline may be defined as the path, straight or curved, the tangent to which at any point gives the direction of the flow of liquid at that point. r So by the definition of Reynolds number v3 v NR v2 v r Inertial force per unit area v / r Viscous force per unit area If the value of Reynold's number 520 Fluid Mechanics (i) Lies between 0 to 2000, the flow of liquid is streamline or laminar. Energy of a Flowing Fluid (ii) Lies between 2000 to 3000, the flow of liquid is unstable and changing from streamline to turbulent flow. energy The equation of continuity is derived from the principle of conservation of mass. a2 v2 B a1 v1 A Fig. 11.9 flow passes through a tube AB of A non-viscous liquid in streamline varying cross section. Let the cross sectional area of the pipe at points A and B be a1 and a2 respectively. Let the liquid enter with normal velocity a1v1 1 a2v2 2 and a1v1 a2v2 [If the liquid is incompressible 2 1 ] a 1 v Potential energy of the liquid mgh Kinetic energy of the Pressure energy per unit mass of the P liquid Kinetic energy per unit mass of the Potential energy per unit volume of the liquid gh Kinetic energy per unit volume of the D YG C a U (1) The velocity of flow is independent of the liquid (assuming the liquid to be non-viscous) (2) The velocity of flow will increase if cross-section decreases and vice-versa. That is why : (a) In hilly region, where the river is narrow and shallow (i.e., small cross-section) the water current will be faster, while in plains where the river is wide and deep (i.e., large cross-section) the current will be slower, and so deep water will appear to be still. ST liquid 1 2 v 2 1 2 v 2 According to this theorem the total energy (pressure energy, potential energy and kinetic energy) per unit volume or mass of an incompressible and non-viscous fluid in steady flow through a pipe remains constant throughout the flow, provided there is no source or sink of the fluid along the length of the pipe. Fig. equation 11.10 This expression is called the of continuity for the steady flow of an incompressible and non-viscous liquid. v1 v2 Fig. 11.11 (b) When water falls from a tap, the velocity of falling water under the action of gravity will increase with distance from the tap (i.e, v 2 v1 ). P2 v2 A2 P1 A1 B h2 v1 h1 Mathematically for unit volume of liquid flowing through a pipe. Fig. 11.12 1 P gh v 2 constant 2 To prove it, consider a liquid flowing steadily through a tube of nonuniform area of cross-section as shown in fig. If P1 and P2 are the pressures at the two ends of the tube respectively, work done in pushing the volume V of incompressible fluid from point B to C through the tube will be W P1V P2 V (P1 P2 )V …(i) This work is used by the fluid in two ways. (a) In changing the potential energy of mass m (in the volume V ) from mgh to mgh , 1 So in accordance with continuity equation the cross section of the water stream will decrease (i.e., A2 A1 ), i.e., the falling stream of water becomes narrower. liquid Bernoulli's Theorem v A1 A2 1 mv 2 2 Potential energy per unit mass of the liquid gh Pressure energy per unit volume of the liquid P liquid U or av constant or Kinetic energy It is the energy possessed by a liquid by virtue of its motion or velocity. ID v 1 at A and leave with velocity v 2 at B. Let 1 and 2 be the densities of the liquid at point A and B respectively. Mass of the liquid entering per second at A = Mass of the liquid leaving per second at B Potential energy It is the energy possessed by liquid by virtue of its height or position above the surface of earth or any reference level taken as zero level. 60 Equation of Continuity Pressure Energy It is the energy possessed by a liquid by virtue of its pressure. It is the measure of work done in pushing the liquid against pressure without imparting any velocity to it. Pressure energy of the liquid PV E3 (iii) Above 3000, the flow of liquid is definitely turbulent. A flowing fluid in motion possesses the following three types of 2 i.e., U mg(h2 h1 ) …(ii) Fluid Mechanics i.e., K 1 1 mv 12 to mv 22 , 2 2 1 m(v 22 v12 ) 2 …(iii) Now as the fluid is non-viscous, by conservation of mechanical energy W U K i.e., (P1 P2 ) V mg (h2 h1 ) or P1 P2 g(h2 h1 ) 1 m(v 22 v12 ) 2 1 (v22 v12 ) 2 This is also based on Bernoulli's principle. The wings of the aeroplane are of the shape as shown in fig. Due to this specific shape of wings when the aeroplane runs, air passes at higher speed over it as compared to its lower surface. This difference of air speeds above and below the wings, in accordance with Bernoulli's principle, creates a pressure difference, due to which an upward force called 'dynamic lift' (= pressure difference × area of wing) acts on the plane. If this force becomes greater than the weight of the plane, the plane will rise up. 60 (b) In changing the kinetic energy from 521 (iii) Action of atomiser [As m / V ] P T C P1 gh1 or 1 P gh v 2 constant 2 The action of carburetor, paint-gun, scent-spray or insect-sprayer is based on Bernoulli's principle. In all these, by means of motion of a piston P in a cylinder C, high speed air is passed over a tube T dipped in liquid L to be sprayed. High speed air creates low pressure over the tube due to which liquid (paint, scent, insecticide or petrol) rises in it and is then blown off in very small droplets with expelled air. ID (i) Bernoulli's theorem for unit mass of liquid flowing through a pipe can also be written as: gh (iv) Blowing off roofs by wind storms 1 2 v constant 2 (ii) Dividing above equation by g we get P U Wind P v2 = constant h g 2g P0 v large so P < P0 P is called pressure head, h is called gravitational head and g D YG Here L Fig. 11.15 This equation is the so called Bernoulli's equation and represents conservation of mechanical energy in case of moving fluids. P E3 1 2 1 v1 P2 gh2 v 22 2 2 or v2 is called velocity head. From this equation Bernoulli's theorem can be 2g stated as. “In stream line flow of an ideal liquid, the sum of pressure head, gravitational head and velocity head of every cross section of the liquid is constant.” U Applications of Bernoulli's Theorem (i) Attraction between two closely parallel moving boats (or buses) P0 B ST 1 P
