Elasticity and Fluid Mechanics is a branch of Physics in which we apply the principles learned under Mechanics to explain the elastic behavior of solids and various properties and behavior of fluids at rest (known as Fluid Statics) and fluids in motion (known as Fluid Dynamics). Elasticity and Fluid Mechanics is an important part of the NCERT syllabus.
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NCERT assigns 20 marks questions based on Elasticity and Fluid Mechanics plus Thermal Physics out of 70 marks in the Class XI CBSE exam paper (theory). Recent IIT-JEE Main and NEET papers assign not more than 10% on Elasticity and Fluid Mechanics.
The topics which come under Class 11 Elasticity and Fluid Mechanics as per the latest syllabus are as follows:"
Unit 1 : Properties of Bulk Matter
The property by virtue of which a body resists any change in its size, shape or both and tends to regain its configuration on withdrawal of the deforming forces is known as the elasticity of the body.
If different parts of a body do not change their positions with respect to each other under the action of a system of balanced forces or couples, however large they may be, the body is called a perfectly rigid body. Glass is a good approximation to that. A perfectly elastic body completely regains its original configuration once the deforming forces are withdrawn. A quartz fibre is almost perfectly elastic. A perfectly inelastic body or plastic body does not show any tendency to recover its original configuration even after removal of the deforming forces. Putty is a close example.
The average stress, σ, across any given cross section of a body is defined as the ratio of the resultant internal force acting on the section to the area of that section. Thus, if the resultant internal force is F and the area of the cross section is A then:
σ = F / A
Suppose we resolve F into a component vector F_{n} normal to the section and a component vector F_{t} tangent to the section. Normal stress is defined as:
σ_{n} = F_{n} / A
Tangential stress or shear stress is defined as:
σ_{t} = F_{t} / A
The dimensions of stress are [ML^{-1}T^{-2}], and its SI unit is newton per square metre (N/m^{2}).
Suppose two equal and opposite forces, F and - F, are applied to the ends of a solid rod of uniform cross section. If the forces act normally outward, the rod is said to be under tension. The stress at every perpendicular section of the rod is a normal tensile stress. On the other hand, if the forces act normally inward, the rod is under compression and the corresponding stress is a normal compressive stress. Both types of stress are known as longitudinal stress, because they act along the length of the rod.
A solid rectangular block, which attains static equilibrium under two pairs of forces applied tangentially to opposite faces, experiences shear stress. The same block, taken to a depth under the sea, experiences hydrostatic pressure of water acting normally inward on each of its six faces. The corresponding stress is called volume stress.
The strain of a body is the fractional change in its length, volume or shape relative to that in its original configuration. The ratio of two similar quantities, strain is a dimensionless number without units. As stress is the cause and strain is its effect, it is possible to relate each type of stress to a corresponding type of strain. Thus, longitudinal stress gives rise to longitudinal strain – either tensile strain or compressive strain, shear stress produces shear strain, and volume stress causes volume strain.
The fundamental law of elasticity is the Hooke's law which states that: "Provided the strain is small, the stress is proportional to the strain". The law implies that, within proportional limit, the ratio of stress to strain is a constant known as the elastic modulus of the material. The dimensions of elastic modulus are [ML^{-1}T^{-2}], and its unit is N/m^{2}.
In accordance with three types of stress and strain mentioned above, three types of elastic modulus can be defined. Young's modulus is the ratio of longitudinal stress to longitudinal strain. Shear modulus is the ratio of shear stress to shear strain. Bulk modulus is the ratio of volume stress to volume strain. While only a solid can have Young's modulus and shear modulus, bulk modulus is a characteristic of solids, liquids and gases.
Another elastic constant of importance is Poisson's ratio. A longitudinal strain produced in a rod is always accompanied by a lateral strain produced simultaneously. The negative of the ratio of lateral strain to longitudinal strain is known as Poisson's ratio for the material of the rod. The relations among elastic constants will be discussed in due course.
Although the relation between stress and strain follows Hooke's law for small strains, it becomes complex if the strain is large. Therefore, it is instructive to plot a stress versus strain graph for a typical ductile metal wire. There are certain terms of interest regarding this graph, such as proportional limit, elastic limit or yield point, breaking stress, necking, breaking point, crushing point etc. The lack of coincidence between the curves for increasing and decreasing stress is known as elastic hysteresis.
The elastic potential energy stored in a wire stretched within proportional limit is given by the formula:
U = YA(∆l)^{2} / 2l
where Y is the Young's modulus, A is the cross-sectional area, ∆l is the extension and l is the original length of the wire. Another useful formula for elastic potential energy stored per unit volume, or energy density, is:
Energy density = 1/2 × stress × strain
This last relationship is valid also if the body is under shear stress or volume stress.
Matter can be classified into three types: solids, liquids and gases. A solid can withstand shear stress; it has definite volume and shape. Liquids and gases cannot withstand static shear stress and begin to flow under it; hence they are collectively referred to as fluids. None of the fluids has any definite shape of its own and eventually takes the shape of the vessel in which it is kept. While a liquid occupies a definite volume almost unaffected even by very high pressure, a gas can be compressed easily.
The mechanics of fluids is governed by a number of physical principles, which are based on Newton's laws of motion and other force laws. Our present topic of discussion is fluids at rest, or fluid statics. Later we shall deal with fluids in motion, or fluid dynamics. Two important fluid properties, surface tension and viscosity, will be introduced in between for a comprehensive understanding of fluid mechanics as a whole.
The density, ρ, of a homogenous substance is its mass per unit volume. If m is the mass of a substance of volume V, its density is:
ρ = m / V
The dimensions of density are [ML^{-3}], and its SI unit is kilogram per cubic metre (kg/m^{3}). The relative density or specific gravity of a substance is the ratio of its density to the density of water at 4°C and 1 atm pressure.
The hydrostatic pressure, P, at any point inside a fluid at rest is defined as the magnitude of the normal force exerted by the fluid on unit area containing the point. If the pressure is the same at all points of a finite plane surface of area A, we may write:
P = F / A
where F is the normal force acting on the area. This force is called thrust. The dimensions of pressure are [ML^{-1}T^{-2}], and its SI unit is N/m^{2} or pascal (Pa). The standard atmospheric pressure is taken to be 1.013 × 10^{5} N/m^{2}. Although in most cases absolute pressure is quoted, in case of car tyres and human heart gauge pressure is quoted which is the pressure in excess of normal atmospheric pressure.
The absolute pressure at a depth h below the surface of a liquid open to the atmosphere is:
P = P_{a} + ρgh
where P_{a} is the atmospheric pressure, ρ is the density of the liquid and g is acceleration due to gravity. This last equation has an interesting implication. If water is poured up to the same height in a number of vessels of different shapes but equal bottom areas, the thrusts on the bottom of all vessels will be equal regardless of different quantities of water in them! This phenomenon is called hydrostatic paradox. Note further that, if the vessel is accelerated, the above formula will not hold. Two special cases of interest are vessel moving with constant linear acceleration and vessel rotating about a vertical axis with constant angular velocity.
A mercury barometer is used to measure atmospheric pressure. A manometer is used to measure gauge pressure at a certain point in a fluid.
Pascal's principle states that the pressure applied to an enclosed liquid is transmitted undiminished to every point of the liquid and to the walls of the container. The principle has widespread applications in the design of hydraulic press, hydraulic jack, remote control etc.
The upward force exerted by a fluid on a body partially or totally immersed in it is called the buoyant force. The effect is known as buoyancy. The magnitude of buoyant force was first determined by Greek scientist Archimedes (287-212 BC). Archimedes' principle states that:
The buoyant force on an immersed body has the same magnitude as the weight of the fluid displaced by the body.
The buoyant force acts vertically upward through the centre of buoyancy which coincides with the centre of gravity of the displaced fluid before its displacement.
Next we examine the conditions of immersion and flotation. A body, which is either completely immersed in a fluid or floats on its surface, experiences two vertical forces: force of gravity, F_{g} , and buoyant force, F_{B} . The densities of the body and the fluid are assumed ρ and ρ_{f} respectively. Suppose the body is taken below the surface of the fluid and released there. If ρ > ρ_{f} then F_{g} > F_{B}, and the body sinks to the bottom of the vessel. If ρ < ρ_{f} then Fg < F_{B} , and the body rises towards the surface of the fluid. If ρ = ρ_{f} then F_{g} = F_{B} , and the body remains at rest exactly where it is released without sinking or rising.
When a body floats on a fluid, it is only partially immersed in it. The fraction of the total volume of a floating body which remains immersed inside a fluid is equal to the ratio of the density of the body to the density of the fluid. A beautiful example of flotation is a cartesian diver, an ingenious toy invented by René Descartes.
Surface tension is an important fluid property which can explain events like a piece of camphor dancing on the surface of water, a water spider skating on a pond without wetting its legs, a needle floating on water, and so on. It is a molecular phenomenon which occurs at the surface of separation between two phases, such as a liquid in contact with air.
We begin with some important definitions. The force of attraction between the molecules of two different substances is known as the force of adhesion or adhesive force. The force of attraction between the molecules of the same substance is known as the force of cohesion or cohesive force. The molecular range of a substance is the maximum distance between its two molecules up to which the cohesive force is effective. If a plane is imagined within the liquid parallel to its free surface at a distance equalling the molecular range, the portion of the liquid lying between the free surface and the plane is known as the surface film.
All molecules lying in the surface film of a liquid experience a net inward force. Every time an external agent wants to increase the area of the free surface of a liquid, it must do so by raising molecules from the bulk of the liquid to the surface against the said inward force. The work done by the external agent is stored as potential energy in the free surface and known as free surface energy. Thus, the larger the surface area of a liquid, the more free surface energy it possesses. Since stable equilibrium is attained at minimum potential energy, the liquid surface behaves like a stretched elastic skin always trying to contract in area.
The force of surface tension, or simply surface tension, is defined as the tensile force acting across and perpendicular to a short, straight line on the surface of liquid divided by the length of that line. The dimensions of surface tension are [MT^{-2}], and its SI unit is newton per metre (N/m). It can be shown that, in general, the surface tension of a liquid is equal to its free surface energy per unit surface area.
When a liquid comes into contact with a solid wall, the liquid surface near the point of contact is usually curved. The angle between the tangent to the liquid surface at the point of contact and the solid surface measured from within the liquid is known as the angle of contact for that pair of liquid and solid. This angle of contact may be an acute angle (< 90°) as in the case of methylene iodide and glass, an obtuse angle (> 90°) as in the case of mercury and glass, or a right angle ( = 90°) as in the case of water and silver.
The curved liquid surface near a solid wall is known as a meniscus. The possible shapes of a meniscus can be explained on the basis of the intermolecular forces of adhesion and cohesion. A typical liquid molecule on the meniscus experiences adhesive forces exerted by the solid molecules, cohesive forces exerted by other molecules of the liquid, and the force of gravity which is comparatively small. Thus, the relative strengths of adhesive and cohesive forces determine whether the meniscus would be concave, convex or flat.
The excess pressure within a liquid drop is given by the formula:
P - P_{a} = 2γ / R
where P is the pressure inside the liquid drop of radius R, P_{a} is atmospheric pressure and γ is the surface tension of the liquid.
The excess pressure within a soap bubble is given by the formula:
P - P_{a} = 4γ / R
where the symbols have similar meanings as before. When we derive these two important formulas, we shall see why a multiplication factor of 2 is coming in the case of a soap bubble. More rigorous calculation will be required to determine the force between two plates separated by a liquid film such as water or mercury.
An important effect of surface tension is the rise or fall of a liquid in a capillary tube. An open tube of very small cross section is known as a capillary tube, and the said effect is called capillarity. If the angle of contact between the liquid and the material of the tube is less than 90°, as in the case of water and glass, the liquid rises in the tube. Conversely, if the angle is greater than 90° as with mercury and glass, the liquid level falls in the tube. The common formula for the rise or fall, as the case may be, is:
h = 2γ cosφ / rρg
In this expression, h is the elevation or depression, γ and ρ are the surface tension and density of the liquid respectively, φ is the angle of contact, r is tube radius, g is acceleration due to gravity. The inverse relation between h and r is sometimes referred to as Jurin's law. The rise of ink through the pores of a blotting paper or of oil through the wick of a lamp is an example of capillary action.
The factors affecting surface tension are contamination of the liquid surface, presence of dissolved substances, variation in temperature, and nature of medium in contact.
Fluid dynamics is the study of fluids in motion. Since the motion of a real fluid is complex, the treatment is simplified by making a few assumptions. We consider an ideal fluid which is non-viscous and incompressible. The flow of the ideal fluid is assumed to be steady and irrotational. Steady flow means the velocity of the fluid at a given point remains constant in time, and variation in velocity from point to point must be smooth.
The path followed by a fluid element under steady flow is called a streamline. The line can either be straight or curved such that the tangent drawn to it at any point gives the direction of flow of the fluid at that point. Steady flow is also called streamline flow or laminar flow. Example is water flowing through a tube at low speed.
On the other hand, if the fluid moves at high speed or the bounding surfaces cause abrupt changes in its velocity, the flow becomes irregular and complex. Such a flow is called a turbulent flow. Example is flow of water in a stream near boulders.
According to equation of continuity, the product of the cross-sectional area and the speed of an ideal fluid in steady flow remains constant at all points along a tube. The equation is based on the assumption that no fluid can flow across the wall of the tube, and there is no "source" or "sink" inside the tube which can create or destroy any fluid.
When a fluid flows steadily through a pipe of varying cross section and elevation, the pressure within the pipe changes along its length. The way the pressure depends on the fluid's speed and the pipe's elevation is governed by Bernoulli's principle. One way of writing Bernoulli's equation is:
P + ^{1}/_{2} ρv^{2} + ρgy = constant
Put in words, the sum total of the pressure, kinetic energy per unit volume and gravitational potential energy per unit volume of an ideal fluid in steady flow remains constant along a streamline.
There are interesting applications of Bernoulli's equation. One of them is to determine the speed of efflux, that is the speed with which a liquid emerges from a hole made at the wall of its container. Venturi tube is a horizontal constricted pipe used to measure the flow speed of an incompressible fluid. Pitot tube is another device to measure fluid speed. The ascent of an aeroplane is made possible by a net upward force — known as dynamic lift — on its wings. The shape and orientation of the wing are designed such that the speed of air is greater above the wing and hence, by Bernoulli's principle, air pressure is higher under the wing.
Unlike ideal fluids, a real fluid possesses a property called viscosity. When a real fluid flows through a pipe, the velocity of the fluid layer gradually increases from zero at the pipe wall to a maximum at the centre. The force of internal friction between the layers is known as viscous force. Newton's formula applicable to laminar flow of a viscous fluid is:
F_{t} = η A dv/dy
where F_{t} is the tangential force acting on fluid surface of area A, dv/dy is speed gradient, and η is coefficient of viscosity of the given fluid. The dimensions of viscosity coefficient are [ML^{-1}T^{-1}], and its unit is newton second per square metre (N-s/m^{2}).
Poiseuille's law on laminar viscous flow in a pipe is:
φ = πΔP R^{4} / 8ηl
In this expression, φ is volume flux (in m^{3}/s), ΔP is pressure difference between two ends of the pipe of length l and radis R, η is coefficient of viscosity of the fluid.
The minimum speed of a fluid in a pipe above which the fluid ceases to be steady is called the critical speed of the fluid. Reynolds number, N_{R} , suggests whether the flow of a viscous fluid in a pipe would be steady or turbulent. Experiments show that if N_{R} does not exceed 2000, the flow is steady; if N_{R} exceeds 3000, the flow is turbulent; if N_{R} falls between 2000 and 3000, the flow may be either steady or turbulent depending on the shape of the pipe entrance and the distance from it.
When a solid body moves through a viscous fluid, it experiences a resistive drag force exerted by the fluid on it. For a slow-moving spherical body, this drag force is given by Stoke's law as follows:
F_{D} = 6πηrv
On the right-hand side, η is the coefficient of viscosity of the fluid and r is the radius of the body moving with speed v relative to the fluid. Under the combined effects of this drag force, force of gravity and force of buoyancy, the sphere eventually falls with a constant terminal speed.
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