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https://docs.google.com/document/d/1y6a4mUO7JPKWshljwV-gUOZji9ssee9J/edit?usp=sharing&ouid=109474854956598892099&rtpof=true&sd=true Bulk modulus, Modulus of rigidity. Pressure due to a fluid column; Pascal’s law and its applications. Viscosity, Stokes’ law, terminal velocity, streamline and turbulent flow, Reynolds number. Bernoulli’s principle and its applications. Surface energy and surface tension, angle of contact, application of surface tension - drops, bubbles and capillary rise. Surface energy and surface tension angle of contact application of surfactonsia drops, bubbles and capilars rise. INTERATOMIC AND INTERMOLECULAR FORCES The force between molecules of a substance is called intermolecular force. THIS CHAPTER COVERS : Inter-atomic and Inter-molecular U F r forces Hooke’s law Moduli of Elasticity Cohesion and r Adhesion Surface tension and surface energy The above graphs show the variation of potential energy and force with interatomic or intermolecular separation. For r = ∞, F = 0, U = 0 r > r , F is attractive as r decreases from ∞ to r , potential energy Capillary action Bernoulli's theorem Viscosity and 0 decreases. 0 terminal velocity r = r0, potential energy U = Minimum, F = 0. This is equilibrium position. r < r0, F is repulsive, therefore, potential energy increases. Elasticity : Property of a solid by which it tries to restore its original shape by developing a restoring force in it. Stress : Restoring force developed/Area. Strain : Change in dimension/original dimension. Stress Modulus of Elasticity = Strain Greater is modulus of elasticity greater is the stress developed i.e., greater is the restoring force. Such a body will be more elastic. That is why steel is called more elastic than rubber because its modulus of elasticity is more than that of rubber. TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 Types of Stress (1) F (2) A = 4πr2 Longitudinal Stress (Tensile) = F/A Volumetric Stress (Compressive) = P (pressure) A(area) F (3) Tangential Stress or shear stress = F A Types of Strain Δl Longitudinal strain = l Volumetric strain = − ΔV V Shear strain = φ = ΔL L Stress - Strain Curve : Stress Breaking B strength Elastic limit E Proportional P limit O HOOKE’S LAW Strain Within the proportional limit stress is directly proportional to strain. Stress ∝ strain Stress = Elastic constant Strain In region OE, material returns to original position after removal of stress. For deformation beyond E, material does not return to original size. This phenomenon is known as elastic hysteresis. At B, fracture of the solid occurs. TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 PRESSURE Pressure is defined as the force acting on a surface per unit area. It is a scalar quantity. PASCAL LAW P = F A SI unit Nm–2 It states that if effect of gravity is neglected, then the pressure at every point of a liquid in equilibrium is same. The increase in pressure at any point of the enclosed liquid in equilibrium is transmitted equally to all other points of the liquid and also to the walls of the container. Pressure difference between two points : The pressure difference between two points, which are at different horizontal level is given as, P2 – P1 = hρg P2 > P1 & P2 – P1 = hρg Following cases illustrate the common problems related to pressure difference : P = P0 + hρg P0(atmospheric pressure) TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 PB – PA = hρg PC – PB = 0 Stationary Fluid PB – PA = hρg PC – PB = Lρa A h a C L B Accelerated Fluid tanθ = a = 2h g L Hydraulic Lift : It is an arrangement to lift heavy objects by applying a small force. For equilibrium of the weight W, pressure at M should be equal to pressure at N, W = F A a hρg Area = a Area A Equilibrium of Different Liquids in a U tube PA = PB (as A & B are at same level) ⇒ P0 + h1ρ1g = P0 + h2ρ2g (where P0 is atmospheric pressure) h1ρ1g = h2ρ2g TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 When the U tube accelerates, difference of levels of liquid satisfies the relation, tanθ = a = h g L h a Buoyancy When a body is immersed wholly or partially in a fluid, it experiences a loss of weight, due to an upward force called upthrust or buoyant force. Archimedes Principle It states that when a solid body is immersed wholly or partially in a liquid, then there is some apparent loss in its weight. This loss of weight is equal to weight of liquid displaced by the body. Buoyant Force Consider a body (assumed cylinderical) of density σ and volume V immersed completely in a liquid of density ρ. As P2 – P1 = hρg ⇒ F2 – F1 = hρgA ⇒ Fupward = Vρg = loss of weight h P1 Area = A σ ρ P2 Following cases are possible depending on the relation between σ and ρ. Case - I : σ < ρ The body will float in the liquid with some part inside and remaining out side. V = volume of body Vi = volume of body inside liquid V0 = volume of body outside liquid Viρg = Vσg ⇒ Vi = σ V0 = ρ − σ V ρ V ρ TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 Case - II : σ = ρ Body floats, completely immersed in the liquid. V0 = 0 Body remains at rest wherever it is left Case - III : σ > ρ (Body will sink to the bottom) For figure-1, R = Normal reaction between body and bottom of container R = Vσg – Vρg Vρg R σ MODULI OF ELASTICITY Young’s modulus of elasticity Y = Tensile stress = Longitudinal strain Fl AΔl Vσg Figure-1 Bulk modulus of elasticity K = Normal or compressive stress = −V ΔP or, K = −V dP Compressibility = 1 K Volumetric strain ΔV dV Modulus of rigidity η or G = Shear stress = F = Shear strain Aφ FL AΔL TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 Thermal Stress : Rod Fixed between Rigid Support If Δθ = Rise in temperature Compressive strain = Δl = αΔθ l Compressive stress = Y × strain = YαΔθ ⇒ F = YαΔθ × A Heated Poisson’s Ratio Longitudinal strain Lateral strain = − Poisson’s ratio σ Theoretically – 1 ≤ σ ≤ 0.5 Practically 0 ≤ σ ≤ 0.5 When density of material is constant ⇒ σ = 0.5 (4) 9 = 3 + 1 Y η K (5) K = Y 3(1 − 2σ) (6) η = Y 2(1 + σ) (7) σ = 3K − 2η 2η + 6K Young’s modulus of a wire is numerically equal to stress required to double the length of wire. When a pressure dP is applied on a substance, its density changes from ρ to ρ′ so that ρ' = ρ⎛1+ ΔP ⎞ ⎜ ⎟ ⎝ K ⎠ TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 The energy density of water in a lake h meter deep is 1 (hρg )2 U = where ρ is density of water, K is Bulk modulus. 2 K In case of a rod of length L and radius r fixed at one end. Angle of shear φ is related to angle of twist θ by the relation Lφ = rθ. COHESION AND ADHESION The force of attraction between similar molecules is known as cohesive force. It is very strong in solids, weak in liquids and very weak in gases. The force between dissimilar molecules is known as adhesive force. Corresponding phenomenon is known as adhesion. SURFACE TENSION AND SURFACE ENERGY Property of a liquid due to which it behaves like a stretched membrane. A free liquid drop tries to acquire spherical shape (minimum surface area) due to surface tension. Surface tension is force/length. T = F l F = T × l F1 = T × a × 2 F2 = T × b × 2 Surface energy = T × surface area (N/m) b F2 a F1 F1 F1 (Two surfaces) Liquid drop of radius R ⇒ Surface Energy = T × 4πR2 Soap bubble of radius R ⇒ Surface Energy = 2 × T × 4πR2 Angle of Contact It is the angle between solid surface inside the liquid and the tangent drawn to the liquid surface at the point of contact. It depends on Relative cohesive and adhesive force of solid liquid pairs Temperature Application of Surface Tension Work done to blow a soap bubble of radius r = 2 × T × 4πr2 A drop of radius R breaks up into n identical drops work done = ΔS.E. = [n × 4πr2 – 4πR2]T …(1) R3 = nr3 …(2) ⇒ Work done = 4πR2T [n1/3 – 1] n identical drops coalesce to form a single drop Heat produced = 4πR2T [n1/3 – 1] = mc Δθ where, c = Specific heat, m = mass = 4 πR 3ρ , Δθ = Rise in temperature. 3 TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 A needle floats on the surface of a liquid due to surface tension. Surface tension decreases with rise in temperature. Surface tension decreases by adding sparingly soluble impurities like detergents. Surface tension increases by adding soluble impurities like NaCl, sugar. Excess pressure If Po = Atmospheric pressure Pi = Inside pressure then Pi – Po = Excess pressure Liquid drop (1) (2) Soap bubble Po (3) Air bubble P = P + 2T P - P = 4T P = P + 2T i o r i o r i o r Capillary tube, concave meniscus Pi = Po 2T R O R r PO (b) Fa > c Pi Capillary tube, Concave Meniscus Capillary tube, convex meniscus. 2T Pi = Po + R (b) Fa < c Combining of Bubbles If the soap bubble coalesce in vacuum, then Po = 0 ⇒ r2 = r 2 + r 2 Convex Meniscus 1 2 If two soap bubbles come in contact to form a double bubble then r = radius of interface, r1 > r2 1 = 1 − 1 r r2 r1 TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 The interface will be convex towards larger bubble and concave towards smaller bubble because P2 > P1 > P0. P0 CAPILLARY ACTION Rise or fall of liquid in a tube of fine diameter. Ascent formula h = 2T Rρg = 2T cos θ rρg where, θ = angle of contact (as shown in figure) Stream Line Flow or Steady Flow The flow of a fluid is said to be steady if all particle of the fluid passes through or cross-section with same velocity. Turbulent Flow Above a certain critical speed, fluid flow becomes unsteady. This irregular flow is called turbulence. Equation of Continuity It is based on conservation of mass. According to it, mass entering per second = mass leaving per second That is, ρ1 a1 v1 = ρ2 a2 v2 P1, a1 v1 P2, a2 v2 For incompressible liquid ρ1 = ρ2 ⇒ a1v1 = a2v2 ⇒ v ∝ 1 a TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 Energy of a Liquid Various energies per unit mass : Potential energy/mass = gh Kinetic energy/mass = 1 v 2 2 P Pressure energy/mass = ρ Energy Heads Various energy heads per unit mass : Gravitational head = h v 2 Velocity head = 2g P Pressure head = ρg BERNOULLI’S THEOREM It is based on conservation of energy. For an ideal, non-viscous and incompressible liquid, P1 + v 2 gh1 = P2 v 2 + 2 + ρ 2 ρ 2 Applications of Bernoulli’s To find rate of flow of liquid Q = av [area × velocity]. Value of Q in various cases is given by Case - (a) : Q = a a 1 2 Case - (b) : Venturimeter Q = a a 1 2 Hole in a tank problem Speed of efflux ve = (If a << A) TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 If a is comparable to A then v e = H h ve = Time taken by water level to fall from h1 to h2 Area = A Hole area = ‘a’ t = A a [ h1 − ] Time taken to completely empty the container by a hole at bottom t ∝ [Put h1 = H, h2 = 0] (d) ve = shown in figure in the situation h1 ρ1 ρ2 ve Range of liquid R = 2 h Rmax = H when h = H H 2 ρ R h H h R = 2 h( H - h) for both holes If A0 = area of cross-section of mouth of tap A0 A = area of cross-section of water jet at a depth h h A0v0 = Av = Q [rate of flow] A TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 v 2 v 2 By Bernoulli’s theorem 0 + gh = 1 [Θ pressure is atmospheric at both points] 2 2 ⇒ Q = AA0 Reynold’s Number N = ρvD = R η Inertial Force Viscous force Value of NR for various cases : NR < 2000, flow is streamline NR > 3000, flow is turbulent 2000 < NR < 3000, flow is unstable When NR = 2000, flow is critical ρvD = 2000 η ⇒ v = 2000 η D (Critical velocity) Viscosity & Viscous Force The property of the liquid by virtue of which, it opposes the relative motion between its adjacent layers is known as viscosity. Fluid friction is due to viscosity. Fluid in contact with the plate is moving with velocity v. Plate F v y Stationary plate Fluid at rest Viscous force is given in this case by, F = − ηA dv dy dv η = coefficient of viscosity & dy = velocity gradient Units of η : SI → 1 Pa-s = 10 poise = 1 decapoise C.G.S → 1 dyne/cm2-s = 1 poise Poiseuille’s Equation Volume flow rate across a tube with pressure difference between its ends is, 4 Q = dV = π Pr dt 8 ηl TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 Where, P = P1 – P2 = pressure difference Comparing with I = V R (V → P1 – P2 & I → Q) P1 P2 ⇒ Resistance to fluid flow R = 8ηl πr 4 Series combination of two tubes Two tubes of radius r1, length l1 and radius r2, length l2 are connected in series across a pressure difference of P. Length of a single tube that can replace the two tubes is found using, l l1 l2 4 4 4 1 2 STOKES LAW When a small spherical body of radius r is moving with velocity v through a perfectly homogeneous medium having coefficient of viscosity η, it experiences a retarding force given by F = 6πηrv. Important cases : A body of radius r released from rest in a fluid If σ = density of body ρ = density of liquid or fluid Terminal velocity is given by, 6πηrv Vρg σ v v = 2 T 9 r 2g η (σ − ρ) ρ Vσg Thus, velocity increases from 0 to vT .Variation of velocity is shown by the graph. V vT t A body is thrown downwards with speed greater than vT then its speed decreases, becomes equal to vT . t ❑ ❑ ❑ TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616