|
Elasticity
. ELASTICITY & PLASTICITY
1. When a body is subjected to external forces such that there is no acceleration then the effect of these external forces would be to deform the body. The extent of deformation depends on the elastic properties of the body.
2. Bodies which do not get deformed when subjected to external deforming forces are classified as ideal Rigid bodies which do not exist in reality.
3. a) The rigid bodies which we see in our daily life are not ideally rigid.
b) The behavior of solids, liquids and gases is different under the action of deforming forces.
4. When the deforming forces are withdrawn, the body may regain its original shape and dimension. This is because of the restoring forces (internal forces) acting within the body, which oppose changes in the shape or dimensions of the body.
5. The property of a matter by virtue of which it tends to regain its original size and shape after withdrawing the deforming forces is called elasticity. No material is perfectly elastic. Quartz can be taken as perfectly elastic body. (nearest example)
6. The property of a matter by virtue of which it cannot regain its original shape or size after the withdrawal of deforming forces is called plasticity. The nearest example to a perfectly plastic body is putty. Chewing gum, wax, wet clay, lead solder, etc., are plastic bodies.
7. Steel is more elastic than rubber. Water is more elastic than air. {in comparison }
8. When a body regains its original size and shape completely after withdrawing the deforming forces, it is said a perfectly elastic body.
9. If a body does not regain its original shape or size even after the removal of deforming forces, then it is said to be a plastic body (inelastic body).
10. Almost all the bodies exhibit elastic nature up to some extent and later they exhibit plastic nature (degree of elasticities differ).
11. Decreasing order of elasticity of certain substances : Tungsten, Nickel, Steel, Iron, Copper, Phosphor bronze, German silver, Brass, Aluminum.
12. With in the elastic limit, the restoring force setup within the body is exactly equal and opposite to the deforming force.
13. Elasticity is the molecular property of matter.
14. Within elastic limit every body behaves as an elastic body and beyond the elastic limit, the body undergoes some permanent change in shape or size and exhibits plastic behavior.
15. The restoring force arising within the body increases as the deforming force on the body increases and opposes always in direction.
. STRESS & STRAIN
Figure - A
Figure - B
1. Figure A represents an elastic material, subjected to external forces in all possible directions, such that the acceleration of centre of mass is zero.
2.  S represents a small elemental area interior to the material, which is subjected to forces F and -F which are shown in opposite directions, which are the internal forces arising due to the influence of the external deforming forces.
3. The component of F normal to S is represented by Fn called normal force, while the component of F tangential to the surface S is represented by Ft.
4. Fn is responsible for normal stress, which could lead to tensile stress, compressive stress, bulk stress.
5. Ft is responsible for the shearing stress.
6. Note that these stresses are the ratio of internal restoring forces per unit area. The magnitude of the restoring forces however shall be equal to that of external deforming forces so long as the material is elastic in nature.
7. Stress : It is the restoring force per unit area (measured as the applied force per unit area)
Stress = 
Units : dy cm-2(CGS), Nm-2or Pascal (SI)
Its dimensional formula is ML-1T-2
8. If the stress is normal to the surface, it is called normal stress. If the stress is tangential to the surface, it is called tangential stress.
9. Stress = 
Normal stress = 
Shearing stress = 
10. Longitudinal stress: When a force is applied on a body such that there is a change in the length of the body, the longitudinal force per unit area is called longitudinal stress. A loaded wire or rod develops longitudinal stress.
11. Longitudinal stress is called tensile stress when there is an increase in length and compressive stress when there is a decrease in length.
12. Bulk stress : If a body is subjected to the same force normally on all its faces such that there is a change in its volume, the normal force per unit area is called bulk stress or volume stress.
13. Bulk stress is equal to pressure or change in pressure also called hydraulic pressure.
14. Shearing stress or Tangential stress : When a tangential force is applied on a body such that there is a change in shape of the body only, then the tangential force per unit area is called tangential stress.
15. Longitudinal stress and Bulk stress are normal stresses which produce change in size. Shearing stress is a tangential stress which produces change in shape.
16. Stress is a tensor quantity as it has different values in different directions.
17. Pressure and stress both are expressed as force per unit area. But pressure is always normal to area while stress can be either normal or tangential. Pressure is always compressive while stress can be compressive or tensile.
18. Strain : The deformation or change produced per unit dimension of the body is called strain. As strain is a ratio, it has no units and no dimensions.
19. Longitudinal strain : It is the ratio of change in length per unit original length or it is the fractional change in length.
If there is an increase in length, then the strain is called tensile strain. If there is a decrease in length, then the strain is called compressive strain.
20. Volume strain : It is the change in volume per unit original volume or it is the fractional change in volume.
21. Shearing strain : It is the ratio of relative displacement between two parallel layers (surface) of the body to the perpendicular surface distance between those two layers.
It can be expressed as the angle through which the line originally normal to the fixed surface is turned.
22. Shearing strain = 2 x longitudinal strain
Bulk strain = 3 x longitudinal strain
23. A shearing strain q is equivalent to an extension strain  /2 and a perpendicular compression strain (/2)
24. Strain = 
Longitudinal strain = 
Bulk strain = 
Shearing strain =  = tan
Note :- 'tan q' can be approximated to 'q' in radians if the angle is to small
Lateral strain of a wire stretched = 
25. Longitudinal and bulk strains do not change the shape of the body. They cause change in magnitude of volume or length. Shearing strain causes change only in the shape of the body.
26. Two forces one elongative and the other compressive acting mutually perpendicular on a body produce shearing strain.
27. Stress is the property possessed by elastic bodies but strain is possessed by all bodies.
28. Strain is the cause and stress is the effect.
. HOOKE'S LAW
Note : Elastic force within Elastic Limit is a conservative force.
1. Hooke's law : Within the elastic limit, stress is directly proportional to strain i.e. 
E is a constant known as modulus of elasticity or coefficient of elasticity of the body.
2. i) The value of E depends upon the nature of material and the manner in which the body is deformed. There are three types of modulii corresponding to three types of strains
a) Young's modulus b) Bulk modulus c) Rigidity modulus
ii) Young's modulus, which measures the resistance of a solid to a change in its length
iii) Shear modulus, which measures the resistance to motion of the planes of a solid sliding past each other.
iv) Bulk modulus, which measures the resistance of solids or fluids to changes in their volume.
3. Units of E are dyne cm-2(CGS) and Nm-2or Pa (SI). E is called modulus of elasticity.
4. The maximum value of stress up to which a body retains its elastic property is known as elastic limit.
5. Carbon becomes plastic when heated. But for rubber elastic property increases with increase in temperature.
6. To increase the elastic property; carbon is added to iron. Similarly potassium is added to increase elasticity of gold.
7. Quartz and phosphor bronze are used for suspension wires of moving coil galvanometers. These two have high tensile strength and low rigidity modulus. Elastic after effect is least for those two.
8. When a spring is stretched, the resulting strain is shear strain.
9. Beams used in constructing a house have greater thickness compared to breadth (since depression at the centre is inversely proportional to breadth and inversely proportional to cube of thickness)
10. Elasticity may increase or decrease by the addition of impurity.
11. The area under stress-strain diagram gives the work done per unit volume.
12. If stress is plotted on Y-axis and strain on X-axis, within elastic limit slope of the graph gives E.
13. Hammering, rolling and addition of impurity increases elasticity. (type of impurity added is important)
14. Annealing and rise in temperature decreases elasticity.
15. For invar steel the elastic property does not change with temperature. (changes are very marginal)
. YOUNG'S MODULUS (Y)
1. The ratio of longitudinal stress to longitudinal strain within the elastic limits is called Young's modulus (Y).
2. If a wire of length 'l' is fixed at one end and loaded at the other end by a mass M, then longitudinal stress =  (r is radius of circular cross-sectional wire)
If e or Dl is elongation of the wire, then longitudinal or linear strain = e/l
Young's modulus 
a) Elongation 
b)  (same stretching force is applied to different wires of same material)
c)  (for same elongation in the above case)
d)  F or Mg (different stretching forces on the same wire)
e)  (same stretching force on wires of different material having same dimensions)
3. When a mass is suspended from a wire, its elongation is e. Then
a) If same mass is suspended from a wire of same length, material but half the radius of cross section, its extension is 4e.
b) If same mass is suspended from a wire of same material but half the length, its extension is e/2)
c) If that mass is a sphere, and now sphere of same material with double the radius is suspended its new extension is 8e.
d) If the suspended mass is completely immersed in a non viscous liquid, its new extension is  where d2and d1are the densities of liquid and the suspended mass.
e) If e1is the extension when the suspended mass is in air and e2is the extension when that mass is completely submerged in a liquid (water), then its relative density is  (relative density = density of substance / density of water)
4. Two rods or wires of the same material and same volume are subjected to same stretching force. Then ratio of their elongations is given by
 (r1, r2are radii of cross section)
 (l1, l2are their length)
5. Thermal stress : If the ends of a rod are rigidly fixed so as to prevent expansion or contraction and the temperature of rod is changed, tensile or compressive stress produced is called thermal stress developed within that rod.
Note:- If the body is able to expand or contract while heating or cooling respectively then thermal stresses do not develop.
Thermal stress does not depend on the original length of the rod.
6. Thermal stress = 
( is the rise or fall in temperature)
Force required to prevent the rod from expansion is F = YA  ()
Here thermal strain = ( )
 Here Y is Young's modulus; a coefficient of linear expansion
is change in temperature Note : For the same change in temperature, if same thermal stress is produced in two different rods, Y11= Y22
7. If a wire of length l is suspended, its elongation due to its own weight is  d is density of the material of wire Y is Young's modulus 8. The length of a wire is l1when the tension in it is T1and l2when the tension is T2then natural length of the wire is 
9. Young's modulus is numerically equal to stress when the increase / decrease in the length equals the original length.
10. If same stretching force is applied to different wires of same material, elongation will be more for wire having greater ratio of its length to cross section area.
11. Glass is more elastic than rubber. For a given stress, the strain produced in glass is much less than that of rubber.
12. i) When a solid is compressed or stretched, in both cases, potential energy of the molecules increases.
ii) Uelastic=  (x2is always positive whether x is positive or negative)
13. When a wire is vertically suspended, it elongates due to its own weight.
Note : Weight always acts at the centre of gravity of the wire.
14. A spring is made of steel and not of copper as elasticity of steel is more than that of copper.
15. Elasticity can be different for tensile and compressive stress.
Eg: for bone Ecompressive< Etensile
for concrete Etensile< Ecompressive
16. A beam of metal supported at the two ends is loaded at the centre. The depression at the centre is inversely proportional to its Young's modulus.
. POISSON'S RATIO :
1. When a force is applied on a wire to increase its length, its radius decreases. So two strains are produced by a single force.
''The ratio of lateral strain to longitudinal strain is called Poisson's ratio''.
2. Poisson's ratio = 
3. Poisson's ratio has no units and dimensions. Theoretically lies between -1 and 0.5. But in actual practice it lies between 0 and 0.5.
4. For most of the solids value lies between 0.2 and 0.4) for rubber is nearly 0.5.
5. If there is no change in the volume of a wire undergoing longitudinal extension, the Poisson's ratio must be 0.5.
6. Practically no substance has been found for which is negative.
. FORCE CONSTANT
1. The product of Young's modulus of a material and the interatomic distance is called interatomic force constant.
2. Inter atomic force constant K = Yr0
Y is Young's modulus
r0 is inter atomic distance.
3. Force constant of a spring which obeys Hooke's law is  (Y is Young's modulus, A is area of cross section, l is length).
4. Inverse of force constant is called compliance.
5. A smaller wire has greater force constant than a longer wire of same thickness and material.
6. A thick wire has greater force constant than a thin wire of same material and same length.
7. A spring is made of steel and not of copper as elasticity of steel is more than that of copper.
. BULK MODULUS (K)
1. Bulk Modulus (K) : The ratio of bulk stress to bulk strain within the elastic limit is called bulk modulus (K) or coefficient of volume elasticity.
2. If the volume V of a body diminishes by . When the pressure on it is increased uniformly by P then, volume stress = P
and volume strain = (negative sign suggests that with increase in pressure, volume decreases)
Bulk modulus K = 
Adiabatic bulk modulus =  P
Isothermal bulk modulus = P (P is pressure and = CP/ CV)
Compressibility (C) = 
3. If a material is enclosed in a very rigid container so that its volume cannot change and the rise of temperature is accompanied by an increase in pressure , then = K() where 'g' is coefficient of volume expansion. K is bulk modulus.
4. When a fluid is compressed,
( r is density); 
 = 1 - = = 
.RIRIGIDITY MODULUS (n or )
1. Rigidity modulus (n) : The ratio of tangential stress to shearing strain within the elastic limit is called rigidity modulus (n) or coefficient of tensile elasticity.
2. If F is the tangential force on a surface of area A the shearing stress = F/A.
If is the angle of shear then
Rigidity modulus n = 
If l is the lateral displacement of the parallel layers separated by l then 
'' is called the angle of shear
If '' is too small than 
. FACTORS ON WHICH Y, n, K, DEPEND
1. The value of modulus of elasticity is independent of the magnitude of the stress and strain.
2. The value of modulus of elasticity does not change when the dimensions of the body change (e.g: If the length a wire or its radius are changed, Young's modulus does not change)
3. For a given material there can be different modulii of elasticity depending on the type of stress applied and the strain produced.
4. A material is said to be more elastic if its modulus of elasticity is more.
5. Elasticity of a perfectly rigid body is infinite (Y or K or n = ) (as strain is zero).
6. Modulii of elasticity Y and n exist only for solids. Liquids and gases do not have these two modulii.
7. Bulk modulus K exists for all states of matter i.e, for solids, liquids and gases.
8. Bulk modulus of gases is very low and for solids it is very high. Gases are least elastic as they are most compressible.
Egas< Eliquid< Esolid
9. The reciprocal of bulk modulus is called compressibility.
10. Gases have two types of bulk modulii
i) isothermal bulk modulus
KIsothermal= P (Pressure of the gas)
ii) adiabatic bulk modulus KAdiabatic= g P where
=  (ratio of specific heats)
Adiabatic bulk modulus > Isothermal bulk modulus.
11. For liquids and gases n = 0.
12. If a graph is drawn for stress versus strain with in the elastic limit, it will be a straight line passing through the origin. Slope of this graph gives modulus of elasticity.
13. Young's modulus of a perfect elastic material is infinite and that of a perfect plastic material is zero.
14. For a liquid or gas rigidity modulus is zero.
15. For incompressible liquids bulk modulus is infinite and Poisson's ratio is 0.5.
16. Units of Y, n, K : dyne cm-2(CGS), Nm-2or Pa (SI)
dimensional formula = ML-1T-2
17. Relations among Y, n, K and s
a) Y = 3K (1 - 2s)
b) Y = 2n(1 + s)
c)  or  and 
. FACTORS AFFECTING ELASTICITY
1. Change of temperature : If the temperature increases, generally the elastic property of a body decreases. In the case of rubber, the modulus of elasticity (Y) increase with increase in temperature. In case of invar steel, elasticity remains practically unaffected by change of temperature. When cooled in liquid air, lead becomes elastic like steel. A carbon filament is highly elastic at ordinary temperatures but it becomes plastic when heated by the current flowing through it.
2. Addition of impurities : Addition of impurity may increase or decrease the elasticity. If the impurity is more elastic than the metal to which it is added, then elasticity will be enhanced. If the impurity is more plastic than the metal, then the elasticity will be reduced.
3. Annealing : Annealing means slow cooling after heating. Annealing reduces the elastic property of the body (due to the formation of larger crystal grain)
4. Hammering or rolling : Hammering or rolling of a body increases the elasticity. (due to breaking up of the crystal grain)
. STRAIN ENERGY & WORK DONE
1. When a body is deformed, the work done is stored in the form of P.E. in the body. This potential energy is called strain energy. When the applied force is withdrawn, the stress vanishes and the strain energy appears as heat.
2. Strain energy per unit volume
= (1/2) (stress) (strain)
Elastic strain energy
= (1/2) (stress) (strain) (volume)
Strain energy per unit volume
= (1/2) 
3) If a force F acts along the length of the wire and stretches it by x, then work done is equal to W = ½ Fx = ½ Mgx (if F = Mg)
4) If a wire or rod extends longitudinally by an amount while the stretching force increases in value from F1to F2then within elastic limit work done W = 1/2 (F1+ F2) e where e is the extension.
5. If one end of a wire or rod is clamped and to the other end of the wire hanging freely, a torque t is applied so that it experience a twist of q radian, then the work done on the wire is (1/2)t.
6. In case of bending of a beam of length l, breadth b and thickness d, by a load Mg at the middle, depression at its centre is given by  (Y is young's modulus).
7. When a rod of length l and radius r is twisted, elastic restoring couple per unit twist is given by C =  (n is rigidity modulus).
8. If a rod of length l and radius 'r' is fixed at one end and twisted through angle q, then angle of shear is given by f = 
9. The total work done is stretching a wire is given by the area under the load-extension diagram.
. BEHAVIOUR OF A WIRE UNDER INCREASING LOAD
1) Behaviour of a wire under increasing load : The load suspended from a metal wire is gradually increased, and a graph is plotted for stress versus strain. Then 'a' = Proportional limit.
a = Proportional limit
b = Elastic limit (or) yield point
c = A point indicating a permanent set at which stress is greater than elastic limit
d = Fracture point
a) Proportionality limit : When the strain is small, the stress is proportional to strain. The wire obeys' Hooke's law within this region. When the deforming force is removed, wire regains its natural length. Point 'a' is called proportionality limit.
b) Elastic limit or Yield point: When the strain is increased further slightly beyond 'a', wire disobey's Hooke's law i.e, stress is not proportional to strain. However the wire exhibits elastic property. When the deforming force is removed, wire regains its natural length. Point 'b' is called elastic limit or yield point. The points 'a' and 'b' are very close. (They may coincide in some cases).
c) Permanent set : If the wire is loaded beyond the elastic limit, it will not regain its original length even after the deforming force is removed. When the external load is completely withdrawn, length of the wire increases permanently by some amount 'OP' (strain) called permanent set.
d) Fracture point: Beyond the yield point, the cross section of the wire decreases more rapidly. Due to this, wire ultimately breaks. This position is denoted by 'd', which is called breaking point. The required stress is called breaking stress. This is equal to stress corresponding to point 'd' on the curve.
. Conclusion :
1) The wire exhibits elasticity from O to b and malleability or plasticity from b to d. If the distance between b and d is more, such metal is ductile. If the distance between b and d is small, such metal is brittle.
2. The substances which break as soon as the stress is increased beyond elastic limit are called brittle substances eg: glass, cast iron, high carbon steel.
3. The substances which have a large plastic range are called ductile substances. Eg: copper, lead, gold, silver, iron, aluminium. Ductile materials can be drawn into wires. Malleable materials can be hammered into thin sheets. Eg: gold, silver, lead.
. BREAKING STRESS
1. Breaking stress depends on the nature of material only. The product of breaking stress and area of cross section is called breaking force. Breaking force is independent of length of the wire
2. The capacity of a substance to withstand large stresses without permanent set is called resilience.
3. A substance that can be elastically stretched to large strains is known as elastomer.
4. Elastic Hysterisis (Behavior of rubber under stress). For rubber, stress is not proportional to strain. For rubber, strain lags behind the stress. On decreasing the load, the stress-strain curve is not retraced but follows a different path. This lag of strain behind stress is called elastic hysterisis. Area of the hysterisis loop gives the energy dissipated during its deformation.
5. Breaking stress = 
breaking force (F0)  Area (A); FBA
or 
. ELASTIC AFTER EFFECT
The delay in regaining the original state after the removal of deforming force on a body is called elastic after effect. Most of the elastic materials exhibit this even within elastic limit. Elastic hysterisis is the result of elastic after effect.
Glass exhibits elastic after effect. Quartz and phosphor bronze are exceptions. Elastic after effect is least for quartz.
. ELASTIC FATIGUE
The state of temporary loss of elastic nature of a body due to repeated stresses over a long time intervals is called elastic fatigue.
Eg: when a copper wire is bent once, it may not break. But it breaks when bent repeatedly at the same point.
Surface Tension
. Adhesive force. It is the force of attraction acting between molecules of two different materials. For example, the force acting between the molecules of water and glass.
. Cohesive force. It is the force of attraction acting between molecules of the same material. For example, the force acting between the molecules of water or mercury etc.
1. If cohesive force is larger than the adhesive force, the liquid will not stick to the vessel containing it. It is so in case of mercury and glass.
2. If cohesive force is less than the adhesive force, the liquid will stick to the vessel containing it. It is so in case of water and glass.
3. Cohesive or adhesive force varies inversely as the eighth power of distance between the molecules i.e. [Fc or Fa 1/r8]
. Molecular range. It is the maximum distance up to which a molecule can exert some measurable attraction on other molecules. The order of molecular range is 10-9m in solids and liquids.
. Sphere of influence. It is an imaginary sphere drawn with a molecule as centre and molecular range as radius. All the molecules in this sphere attract the molecule at the centre and vice-versa.
. Surface film. It is the top most layer of liquid at rest with thickness equal to molecular range.
. Surface tension.
1. It is the property of the liquid by virtue of which the free surface of the liquid at rest tends to have the minimum surface area and as such it behaves as if covered with a stretched membrane.
2. Quantitatively, surface tension of a liquid is measured as the force acting per unit length of a line imagined to be drawn tangentially any where on the free surface of the liquid at rest. It acts at right angles to this line on both the sides and along the tangent to the liquid surface i.e. S = F/l.
3. Surface tension of a liquid is also defined as the amount of work done in increasing the free surface of liquid at rest by unity at constant temperature i.e. S = W / A. or
W = S x A = surface tension x area of liquid surface formed.
4. Surface tension is a molecular phenomenon and it arises due to electromagnetic forces. The explanation of surface tension was first given by Laplace.
5. S.I. Units of surface tension is Nm-1or Jm-2and c.g.s. unit is dyne cm-1or erg cm-2.
6. Dimensional formula of surface tension = [M1L0T-2]
7. Surface tension is a scalar quantity as it has no specific direction for a given liquid.
8. Surface tension does not depend upon the area of the free surface of liquid at rest.
. Surface energy.
1. It is defined as the amount of work done against the force of surface tension in forming the liquid surface of a given area at a constant temperature i.e.
Surface energy = work done = S.T. x surface area of liquid
2. The S.I. unit of surface energy is joule and cgs unit is erg.
3. When small drops combine together to form a big drop, the surface area decreases, so surface energy decreases. Hence the energy is released. If this energy is taken by drop, the temperature of drop increases.
4. When a big drop is splitted into number of smaller drops, the surface area of drops increases. Hence surface energy increases. So energy is spent.
. Work done in blowing a liquid drop or soap bubble
1. Work done in forming a liquid drop of radius R, surface tension S is, W = 4p R2S.
2. work done in forming a soap bubble of radius R, surface tension S is,
W = 2 × 4pR2S = 8 pR2S
3. Work done in increasing the radius of a liquid drop from r1to r2is,
W = 4pS (r22- r12).
4. Work done in increasing the radius of a soap bubble from r1to r2is, W = 8 pS (r22- r12)
. Formation of a bigger drop by a number of smaller drops
1. When n number of smaller drops of liquid, each of radius r, surface tension S are combined to form a bigger drop of radius R, then
volume of bigger drop = volume of n smaller drop
i.e.,  pR3= n ×  pr3 or R = n1/3r
2. The surface area of bigger drop = 4pR2= 4pn2/3r2. It is less than the area of n smaller drops.
3. In this process energy is released, given by W = S × [4pr2n - 4pR2] = 4pSr2n2/3(n1/3- 1) = 4pSR2(n1/3- 1) = 4pSR3
4. The increase in temperature of bigger drop Dq = 
a) Excess of pressure inside a liquid drop,
p = 2 S/R
b) Excess of pressure inside a soap bubble,
p = 4 S/R
where S is a surface tension and R is the radius of the drop or bubble.
3. Pressure difference (p) across curved surfaces of radii of curvature R1and R2. 
a) If the curvatures are in mutually opposite direction as shown in Figure
b) If the curvatures are in the same direction as shown in Figure
c) For a cylindrical surface,  , because R 1 = R and R2 =  .
d) For a spherical surface,  , because R1= R 2 = R.
. Angle of contact.
1. The angle of contact between a liquid and a solid is defined as the angle enclosed between the tangents to the liquid surface and the solid surface inside the liquid, both the tangents being drawn at the point of contact of the liquid with the solid.
2. The angle of contact depends upon
a) the nature of solid and the liquid in contact
b) the given pair of the solid and the liquid
c) the impurities
3. The angle of contact does not depend upon the inclination of the solid in the liquid.
4. The value of angle of contact () lies between 0oand 180o. For pure water and glass, = 0o. For ordinary water and glass, = 8o. For silver and pure water = 90o. For alcohol and clean glass, = 0o.
5. The value of the angle of contact is less than 90ofor a liquid which wets the solid surface and is greater than 900if a liquid does not wet the solid surface.
6. Angle of contact is independent of the angle of inclination of the wall in contact with liquid.
7. The increase in temperature increases the angle of contact.
8. The angle of contact increases with the addition of soluble impurities in the liquid.
9. The addition of detergent in water decreases both the angle of contact as well as surface tension.
10. The materials used for water proofing increases the angle of contact as well as surface tension.
11. If a liquid wets the sides of containing vessel, then the value of angle of contact is acute i.e. less than 90o.
12. If a liquid does not wet the sides of containing vessel, then the value of angle of contact is obtuse i.e. greater than 90o.
. Capillary action or capillarity.
1. It is the phenomenon of rise or fall of liquid in a capillary tube.
2. The root cause of capillarity is the difference of pressure on the two sides of liquid meniscus in the capillary tube.
3. The height h through which a liquid will rise in a capillary tube of radius r which wets the sides of the tube will be given by
Where S is the surface tension of liquid, is the angle of contact, is the density of liquid and g is the acceleration due to gravity. R is the radius of curvature of liquid meniscus.
4. If < 90o, cos is positive, so h is positive i.e. liquid rises in a capillary tube.
5. If > 90o, cos is negative, so h is negative i.e. liquid falls in a capillary tube.
6. If a capillary tube is of insufficient length as compared to height to which liquid can rise in the capillary tube, then the liquid rises up to the full length of capillary tube but there is no overflowing of the liquid in the form of fountain. It is so because the liquid meniscus adjusts its radius of curvature so that hR = a constant i.e. h R =  .
7. The height of the liquid column in a capillary tube on the surface of moon is six times than that on the earth.
8. Rise of liquid in a capillary tube does not violate law of conservation of energy.
. Dependence of surface tension
a) On temperature. The surface tension of liquid decreases with rise of temperature. In low temperature region, the variation of surface tension of liquid with temperature is linear and is given by 
where, St, S0are the surface tension at toC and 0oC respectively and is the temperature coefficient of surface tension. Surface tension of a liquid at its critical temperature is zero.
Surface tension of molten cadmium increases with the increase in temperature.
b) On impurities.
1. A highly soluble substance like sodium chloride (common salt) when dissolved in water, increases the surface tension of water.
2. When a sparingly soluble substance like phenol, dissolved in water, reduces the surface tension of water.
3. When a detergent or soap is mixed with water, the surface tension of water decreases.
Soap or detergent helps in better cleaning of clothes because when it is added to water, reduces the surface tension of water.
c) On electrification. The surface tension of the liquid decreases due to electrification, because a force starts acting outwards normally to the surface of the liquid. It is due to this reason that the soap bubble expands when given positive or negative charge.
d) On contamination. The presence of dust particles or lubricating materials on the liquid surface decreases its surface tension.
Consider a cylindrical glass tube closed at one end. It is made heavy at closed end so that it floats in the vertical position as shown in Figure. If the depth of heavy end is h below liquid surface ; then
. Radius of the new bubble formed when two bubbles coalesce.
Consider two soap bubbles of radii r1and r2respectively. If V1and V2are the volumes of two soap bubbles, then
 and 
Let S be the surface tension of the soap solution. If P1and P2are the excess of pressure inside the soap bubbles, then
 and 
Let r be the radius of the new soap bubble formed when two soap bubbles coalesce under isothermal conditions. If V and P be the volume and excess of pressure inside this new soap bubble, then
 and 
As the new bubble is formed under isothermal conditions, so Boyle's law holds good i.e.
. Radius of interface when two soap bubbles of different radii are in contact.
Consider two soap bubbles of radii r1and r2in contact with each other as shown in Figure. Let r be the radius of the common boundary. If P1and P2are the excess pressures on the two sides of the interface then the resultant excess pressure is
P = P1- P2
or  or 
HYDROSTATICS
. DENSITY & PRESSURE
1. Density of a substance is ratio between mass and its volume.
2. a) Relative density = 
b) Relative density = 
= 
= 
3. If two liquids of equal masses are mixed then density of the mixture = 
4. If two liquids of equal volumes are mixed then density of the mixture 
a) If 'm1' and 'm2' are the masses of two different substances of volumes V1and V2having densities d1and d2respectively then ...
. DENSITY OF MIXTURE
m = m1+ m2 total mass
V = V1+ V2 total volume
Density of mixture = d = 
 = density of mix.
b) R.D. of body = 
= 
= 
Imp:
If ice were to float on a liquid having density as that of water than after the melting of ice, the level of (liquid + water) does neither rise nor fall.
If ice were to float on a liquid having density greater than the water, then after the melting of ice, the level of (liquid + water) rises.
If ice were to float on a liquid having density lesser than the water, then after the melting of ice, the level of (liquid + water) falls.
5. The total force extended by a liquid on any surface in contact with it is called thrust of the liquid.
6. The thrust exerted by a liquid at rest per unit area of the contact surface is called pressure
P = F/A
Units of P : dyne cm-2(CGS), Nm-2or pascal (Pa) is SI.
7. The pressure on a small volume element of a fluid due to the surrounding fluid should be the same in all directions.
8. Pressure in a liquid increases with depth. If a vessel contains a liquid, the pressure on the top surface = P0(atmospheric pressure), the pressure at depth h below the top surface (free surface ) is P = P0+ hrg. Here h rg is gauge pressure, P is absolute pressure. Pressure difference between two points in a liquid separated by a distance y in vertical direction is rg y.
(here r is density of the liquid)
9. Pressure in a liquid at two points will be the same if they lie at the same horizontal level.
10. Pressure at a point in a liquid depends on the vertical depth and density of liquid only but it is independent of shape of the vessel containing the liquid.
11. The pressure difference between static pressure and atmospheric pressure is known as gauge pressure P - P0= hrg
12. If a given liquid is filled in vessels of different shapes to same height, the pressure at the base on each vessel is same.
13. Pascal's law states that : A change in the pressure applied to a fluid is transmitted undiminished to every point of the fluid and to the walls of the container.
14. Barometer is used to measure atmospheric pressure while manometer is used to measure pressure difference or gauge pressure. .
ARCHIMEDE'S PRINCIPLE
Archimedes's principle states that the magnitude of the buoyant force always equals the weight of the fluid displaced by the object.
1. When a body is immersed partially or wholly in to a liquid, it experiences an upward thrust, which is equal to the weight of liquid displaced by the body.
Here the up thrust is called buoyant force. It acts through the centre of buoyancy which is centre of gravity of the displaced liquid.
2. Apparent weight = True weight - Up thrust
If a body of volume V and density d1is fully immersed in a liquid of density d2, then True weight W = Vd1g
Weight of displaced liquid = Vd2g
Apparent weight W1= Vd1g - Vd2g
W1=  ;  W1= W 
a) If d2< d1, W1> 0 and body will sink to the bottom.
b) If d2= d1, W1= 0 and the body will just float or remain hanging at what ever position it is left inside the liquid.
c) If d2> d1, W1< 0 i.e. up thrust will be greater than the true weight. The body will move partly out of the free surface of the liquid until the up thrust becomes equal to true weight W.
3. Laws of flotation :
a) weight of the floating body is equal to weight of liquid displaced.
b) Centre of gravity of the floating body and centre of gravity of the displaced liquid are in same vertical line.
4. If a body of volume V1and density d1floats in a liquid of density d2and V2is the volume of the body immersed in the liquid, then
V1d1g = V2d2g
V1d1= V2d2
If the liquid is water, relative density or specific gravity = 
5. Specific gravity of a liquid = 
6. When a body floats to  part of its volume in a liquid of density d1and to  part in a liquid of density d2, then  .
7. If W is weight of a body in air, W1is its weight in a liquid and W2is weight of that body in water (in both cases completely immersed) relative density of liquid = 
8. A beaker contains a liquid. An ice block is floating on the surface of that liquid. When ice melts completely, (d1, dware densities of liquid and water respectively)
a) level of the liquid rises if d1> dw
b) level of the liquid falls if d1< dw
c) level of the liquid does not change if d1= dw
9. A block of ice with a steel balls inside floats on water, the level falls when ice melts. If the ice block has an air bubble or cork, the level does not change.
10. If a body of outer volume V has a cavity of volume V0in it, 
11. A block is placed in a vessel in contact with the base. When liquid is poured into that vessel, so that liquid does not go beneath it, no buoyant force acts on it. The downward force on it is (P0+ hdg) A(on upper surface, where d is density of liquid)
12. A man is sitting in a boat which is floating in a pond. If the man drinks some water from the pond, level of water in pond remains unchanged.
13. A boat containing some pieces of material is floating in a pond. On unloading the pieces in the pond the level as water remains same if those pieces float and falls if those pieces sink.
14. A small iron needle sinks in water while a large iron ship floats as density of ship due to its larger volume is lesser than that of water.
15. A block of wood is floating on water at 0°C with a certain volume x outside the water level. If the temperature of water is slowly increased from 0°C to 20°C, x will increase from 0°C to 4°C and then will decrease with further rise.
16. A cube of side 'a' remains suspended at the interface of two liquids in a vessel such that their densities are d1and d2(d2> d1). If cube is immersed up to x in the lower liquid, weight of that cube is W such that (true weight)
W = a2[xd2+ (a - x) d1]g
Mass of that cube 'm' is given by
m = a2[xd2+ (a - x)d1]
Imp. :
a) A cylindrical vessel with a liquid moves linearly with acceleration a. Then shape of the surface of the liquid in the vessel will be as shown Here tan q = a/g
b) A manometer contains a liquid of density d. If it rotates at a constant angular velocity w about one limb as shown then level in the here (h2- h1) = 
c) In the previous case if the manometer is moving with acceleration 'a' along the horizontal, then (h1- h2) = (la/g)
Hydrodynamics (Viscosity)
. Viscos Force
1. Viscous force between the two layers of a liquid is given as F =  .
(Negative sign shows that the direction of viscous force F is opposite to that of v)
A is surface area of the layer.
 is velocity gradient
h is velocity gradient
2. Units of h {coefficient of viscosity or simply viscosity}
a) In cgs system it is ''poise'' named after the French Physician JEAN -MARIE POISEUILLE (1799 - 1869) who first investigated the flow of viscous fluids through tubes as an aid in understanding the circulation of blood.
1 poise = 1 dyne S cm-2
b) The S.I. unit of viscosity is NS m-2
c) 1 NS m-2= 10 poise
. POISEUILLE'S RELATION
1. Volume of liquid flowing per second through capillary tube is 
here P is pressure difference between the ends of capillary tube
r is bore radius of capillary tube
l is length of the capillary tube
h is coefficient of viscosity of the liquid
If h is the height of surface of the liquid above the axis of horizontal capillary tube then
P = hdg.
a) V  or n  when P, r, l are constant
V1h1= V2h2where V1and V2are time rate of volume flow of two different liquids through the same capillary tube at the same pressure difference h1, h2are coefficients of viscosity of those liquids
b) V P when the same liquid flows through the same capillary tube at different pressures 
c)  when P, l, h are constant 
d)  when r, r, h are constant
V1l1= V2l2
e) velocity of the liquid flowing at a distance x from the axis is
2. Poiseuille's equation is applicable for fluids if
a) the flow is steady and luminar
b) the liquid in contact with the walls of the capillary tube must be at rest
c) the pressure at any cross section of the capillary tube must be same
3. The velocity profile for a non viscous and viscous liquid through a pipe are as shown
4. If a viscous liquid flows in a tube, the velocity is greatest at the centre of the tube that is nearer to the central axis of the tubular flow and decreases to zero at the wall.
5. From Poiseuille's equation,  or V 
where R =  is called fluid resistance.
Imp.:
In all these expressions P is the pressure difference across the ends of the tube.
6. Fluids which flow through capillary pipes are assumed to have laminar flow.
7. The layers of the fluid are assumed to be thin walled cylinders with varying radii.
8. The flow velocity varies with the radius; its maximum value occurs on the axis and its minimum value, which we assume to be zero, at the walls.
9. The variation of the velocity with location across the pipe is not linear.
 ,
10. The speed at the centre of the pipe is  ( put r = 0)
V0is maximum.
11. The time rate of mass flow of the fluid is given by the relation
(DP is the pressure different across the tube and r is the density of the fluid)
12. Capillary tubes in series : When two capillaries are connected in series across constant pressure difference P, the fluid resistance R = R1+ R2
NOTE :- P1- P2= P
a) Capillary tube of length l1with radius of cross-section r1is joined in series to another tube of length l2of cross-sectional radius r2as shown above
b) A, B and C are three points in the flow where the velocity of a particular fluid element is  , and  respectively.
c) P1, P0and P2are the respective pressures at these three points.
d) (P1- P2) = (P1- P0) + (P0- P2)
e) The time rate of volume flow however will be same throughout the series combination.
Imp.:
Analogous to electric circuits in the series combination of resistors where the electric current remains constant.
Here P = P1- P2= (P1= P0) + (P0- P2) = Pressure difference across the combination in series
Where, (P1- P0) and (P0- P2) are pressure differences across individual capillaries
13. Capillary tubes in parallel : when two capillaries are connected in parallel across a constant pressure difference P, then fluid resistance R for the combination is given by
R = Fluid resistance of the parallel combination
R = 
a) PA- PB= P = same for the both the tubes joined in parallel.
b)  = Fluid resistance of tube of length 'l1' and radius of cross-section r1.
c)  = Fluid resistance of tube of length and radius of cross-section 'r2'
d)  ;
Here volume of fluid flowing per second in each tube is different but pressure difference P = (P1- P2) is same for the combination.
Total volume flowing per second V = V1+ V2
 ; 
. Important Information
1. The velocity of the layer touching the walls of the cylindrical tube is negligible, i.e. almost zero.
2. The velocity of the layers increases as we go towards the axis of the capillary tube.
. STOKE'S LAW & TERMINAL SPEED
Note :- Stoke has derived the equation for spherical objects falling from great heights through a viscous medium
1. When a body is allowed to fall in a viscous medium, its velocity increases at first and finally attains a constant value called terminal velocity (Vt)
2. Viscous force on a spherical body moving in a fluid is F = 6p h r v.
h is coefficient of viscosity
r is radius of the sphere
v is velocity of the sphere
3. As a spherical body falls down in a viscous medium its velocity gradually increases and as a result viscous force on it increases (F V) when the sphere attains terminal velocity (V = Vt), the effective weight of the sphere = Viscous force
6 p h r vt= mg1= 4/3 pr3(r - s)g
or  ; Here vt r2; Vt  (at constant temperature)
Here r is density of spherical body
and s is density of the viscous medium
Here the forces acting on the spherical body are
a) gravitational force on it  ;
b) force of buoyancy on it FB= 
c) viscous force of the fluid FV= 6p h aV
initially w > (FB+ FV) and so the body moves down with acceleration 'a' and
Ma = W - (FB+ FV)
When the sphere attains terminal velocity i.e. V = Vt, a = 0 W = FB+ FV
(or) (W - FB) = FV
If s > P, sphere moves vertically up and terminal velocity will be in the upward direction
Eg: air bubble in water
4. Variation of velocity of a body moving down in a viscous medium with time will be as shown the graph between instantaneous speed and time.
5. A liquid drop of radius 'r' is moving down with terminal velocity Vtin a viscous medium. If 'n' such drops merge and form a bigger drop of radius 'R', terminal velocity of that bigger drop is n2/3Vt.
6. If r1and r2are the radii of two small spheres falling through a viscous medium, with the same constant speed, ratio of the viscous forces on them is r1/r2. Ratio of their velocities is (r1/r2)2.
. STREAM LINE FLOW & TURBULENT FLOW
1. The pressure difference determines the direction of flow of a fluid just like potential difference determines the direction of flow of electric current.
2. When a liquid flows such that each particle of the liquid passing a particular point moves along the same path and has same velocity as the proceeding particle at the same point, its flow is called streamline flow.
3. Streamline is the path such that tangent to which at any point gives the direction of the flow of liquid at that point. Stream line may be straight or curved.
4. In stream line flow a) there will be no radial flow b) the pressure on any cross section is constant c) the velocity at any point remains same but may change from point to point d) no stream lines intersect
5. The stream lines are crowded where the velocity is more.
6. A bundle of streamlines having the same velocity of fluid over any cross section perpendicular to the direction flow is called tube of flow.
7. When a liquid flows such that the motion of any particle of the liquid at a point varies rapidly in magnitude and direction, its flow is called turbulent flow.
8. The minimum velocity at which the flow a liquid changes from laminar or streamlines flow to turbulent flow is called critical velocity.
9. Critical velocity of a liquid is directly proportional to its coefficient of viscosity, inversely proportional to its density and inversely proportional to the diameter of the tube. 
10. Reynold's number (R) gives the idea about fluid flow
If R < 2000, flow is laminar
If R > 3000, flow is turbulent
If 2000 < R < 3000, the flow may change from streamline flow to turbulent flow
Reynolds number R = 
r is density of fluid
h is coefficient of viscosity of the fluid
D is diameter of the pipe
V is critical velocity of fluid
R is dimensionless quantity
{Reynolds number }
11. For a liquid having large value of h, low value of r and flowing through a tube of narrow bore (smaller D) the value of critical velocity will be very large. As the liquid may not possess such large velocity, its flow should be streamlined.
12. For a liquid having low value of h large value of r and flowing through a tube of wide bore (larger D), the value of critical velocity is very low. A liquid can attain such velocity very easily and its flow would be turbulent.
13. Differences between streamline flow and turbulent flow
. STREAM LINE FLOW
1. Regular and orderly flow
2. Velocity of liquid is less than critical velocity
3. Velocity of a liquid at a point is constant with time
4. Viscosity is effect in this flow
. TURBULENT FLOW
1. Irregular and zig zag flow
2. Velocity of liquid is more than critical velocity
3. Velocity of a liquid at a point is not constant with time
4. Density is effective in this flow
. CHARACTERISTICS OF FLUID FLOW
1. The flow of a fluid is said to be steady, laminar when the velocity of it at any given point is constant with time.
2. The flow of fluid is said to be incompressible if there is no change in its density along the line of flow. If the density changes, the fluid is compressible
3. During the flow, a fluid element may have angular velocity. Then such flow is called rotational flow. If it has no net angular velocity then flow is irrotational.
4. Fluid may have viscosity or it may not have. For perfect fluid (idealized fluid) viscosity is zero.
5. For an ideal fluid
a) viscosity is zero b) compressibility is zero
c) its flow should be irrotational d) its flow should be steady.
. EQUATION OF CONTINUITY
1. For an incompressible and non viscous fluid flowing steadily, (irrotational), the product of its velocity and area of cross section at all points during its flow through a tube remains constant. The consequence of this is that velocity of the fluid is inversely proportional to the area of cross section.
2. For an ideal liquid flowing under streamline condition, mass of the liquid flowing per second is constant.
a v r = constant
a1v1r1= a2 v2r2
a1, a2are areas of cross section at points 1 and 2, v1, v2are velocities of the fluid at points 1 and 2, r1, r2are the densities of the fluid at points 1 and 2.
As ideal fluid is incompressible r1= r2then a1v1= a2v2 3. If r1and r2are radii of cross section of a pipe at two points and v1, v2are the velocities of a fluid at those points than (in compressible fluid) 
4. Volumetric flow rate = av; a is area of cross section; v is fluid velocity.
5. If the hose pipe outlet is partially closed, velocity of water increases (a  ). Here volume flow rate does not change.
6. A garden hose having internal radius R is connected to a lawn sprinkler with N holes each of radius 'r'. If V is speed of water in hose, its speed while leaving the sprinkler holes is 
7. An ideal fluid flows through a horizontal tube bifurcating at the end as shown. Then the volumetric flow rate is constant Av = A1v1+ A2v2
. ENERGIES OF A FLUID
1. Pressure energy per unit volume = P
Pressure energy per unit mass = 
2. Kinetic energy per unit volume = 
Kinetic energy per unit mass = 
3. Potential energy per unit volume = r gh
Potential energy per unit mass = gh
4. Pressure head = 
5. Velocity head = 
6. Gravitational head = h
. BERNOULLI'S THEOREM
1. The total energy per unit volume of an incompressible, non viscous fluid in laminar flow is constant at every point.
P +  = constant or  constant; Here  is pressure head  is velocity head; h is gravitational head
2. 
If the liquid flows horizontally,
h1= h2then P + 
3. When water flowing in a broader pipe enters into a narrow pipe, its pressure decreases (Greater is the velocity smaller in the pressure according to Bernoulli's equation)
4. Venturimeter is used for measuring the flow rare of a liquid through a pipe. Pitot tube is also used for this purpose. These two are based on Bernoulli's theorem.
5. A man standing too close to a fast moving train or bus gets a push towards the train or bus. The fast motion of air between the train and man reduces the pressure.
6. The principle behind the working of a gas burner is Bernoulli's principle.
7. Bernoulli's theorem is based on the law of conservation of energy.
8. The liquid at greater depth is always calm because at that depth its pressure there is high and consequently velocity is very low.
|
