Properties of Matter

Properties of Matter
ELASTICITY

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

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.
Stress =

Normal stress =

Shearing stress =

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.

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.

A shearing strain is equivalent to an extension strain /2 and a perpendicular compression strain (/2)
Strain =
Longitudinal strain =
Bulk strain =
Shearing strain = = tan

Note :- 'tan ' can be approximated to '' in radians if the angle is to small



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.
The value of E depends upon the nature of material and the manner in which the body is deformed.

There are three types of moduli corresponding to three types of strains
a) Young's modulus b) Bulk modulus c) Rigidity modulus



YOUNG'S MODULUS (Y)
The ratio of longitudinal stress to longitudinal strain within the elastic limits is called Young's modulus (Y).

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 is elongation of the wire, then longitudinal or linear strain = e/l
or

Note:- If the body is able to expand or contract while heating or cooling respectively then thermal stresses do not develop.


POISSON'S RATIO :
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".
i.e. =


FORCE CONSTANT
The product of Young's modulus of a material and the interatomic distance is called interatomic force constant.
Interatomic force constant K = Yr₀
Y is Young's modulus, r₀ is interatomic distance.


BULK MODULUS (K)
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.
Bulk modulus K =
Adiabatic bulk modulus = P
Isothermal bulk modulus = P (P is pressure and = C_P/ C_V)
Compressibility (C) =


RIRIGIDITY MODULUS (n )
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.
If F is the tangential force on a surface of area A the shearing stress = F/A.
If q is the angle of shear then Rigidity modulus n =

Relations among Y, n, K and s
a) Y = 3K (1 - 2s)
b) Y = 2n(1 + s)
c) or and



STRAIN ENERGY & WORK DONE
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.
Strain energy per unit volume = (1/2) (stress) (strain)
Elastic strain energy = (1/2) (stress) (strain) (volume)
Strain energy per unit volume
= (1/2)

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
Breaking stress =
breaking force (F₀) Area (A); F_B A
or


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.
Surface tension -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.
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.
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.
Surface energy- 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

Work done in blowing a liquid drop or soap bubble
Work done in forming a liquid drop of radius R, surface tension S is, W = 4p R²S.
work done in forming a soap bubble of radius R, surface tension S is,
W = 2 × 4pR² S = 8 pR²S
Work done in increasing the radius of a liquid drop from r₁to r₂is,
W = 4pS (r₂²- r₁²).
Work done in increasing the radius of a soap bubble from r₁to r₂is, W = 8 pS (r₂²- r₁²)

Angle of contact.
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.

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

The angle of contact does not depend upon the inclination of the solid in the liquid.

The value of angle of contact () lies between 0⁰and 180⁰. For pure water and glass, = 0⁰. For ordinary water and glass, = 80. For silver and pure water = 90⁰. For alcohol and clean glass, = 0⁰.

Capillary action or capillarity.
The root cause of capillarity is the difference of pressure on the two sides of liquid meniscus in the capillary tube. 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, q is the angle of contact, r is the density of liquid and g is the acceleration due to gravity. R is the radius of curvature of liquid meniscus.
(a) If < 90⁰, cos is positive, so h is positive i.e. liquid rises in a capillary tube.
(b) If > 90⁰, cos is negative, so h is negative i.e. liquid falls in a capillary tube.

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, S_t, S₀are the surface tension at t⁰C and 0⁰C respectively and a 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.
A highly soluble substance like sodium chloride (common salt) when dissolved in water, increases 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.

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



PRESSURE

The total force extended by a liquid on any surface in contact with it is called thrust of the liquid.
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.
The pressure on a small volume element of a fluid due to the surrounding fluid should be the same in all directions.
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)

The pressure difference between static pressure and atmospheric pressure is known as gauge pressure P - P₀= hrg

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.

ARCHIMEDE'S PRINCIPLE

Archimede's principle states that the magnitude of the buoyant force always equals the weight of the fluid displaced by the object.

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 upthrust is called buoyant force. It acts through the centre of buoyancy which is centre of gravity of the displaced liquid.

Apparent weight = True weight - Upthrust

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

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.


Hydrodynamics (Viscosity)
Viscous Force

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
Units of h {coefficient of viscosity or simply viscosity}The S.I. unit of viscosity is NS m^-2
1 NS m^-2= 10 poise


POISEUILLE'S RELATION

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.

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.
(DP is the pressure different across the tube and r is the density of the fluid)

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
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 =

STOKE'S LAW & TERMINAL SPEED

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 (V_t)

Viscous force on a spherical body moving in a fluid is F = 6 h r v.
h is coefficient of viscosity
r is radius of the sphere
v is velocity of the sphere
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 = V_t), the effective weight of the sphere = Viscous force

6 h r v_t= mg¹= 4/3 r³(r - s)g
or
Here r is density of spherical body
and s is density of the viscous medium

Note :- Stoke has derived the equation for spherical objects falling from great heights through a viscous medium



EQUATION OF CONTINUITY

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.
For an ideal liquid flowing under streamline condition, mass of the liquid flowing per second is constant.
a v r = constant
a₁v₁r₁= a₂ v₂r₂



ENERGIES OF A FLUID

Pressure energy per unit volume = P
Pressure energy per unit mass =
Kinetic energy per unit volume =
Kinetic energy per unit mass =
Potential energy per unit volume = r gh
Potential energy per unit mass = gh
Pressure head =
Velocity head =
Gravitational head = h



BERNOULLI'S THEOREM

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