PRINCIPLES OF PHYSICS - II

UNIT II

ELASTICITY

• the quality or state of being elastic

• a body can be deformed by the suitable application external forces on it

ELASTICITY

• A body is said to be perfectly elastic, if it regains its original shape of size, when the applied forces are removed

• Property of a body to regain its original state or condition on removal of the applied forces

• A body which does not tend to regain its original shape or size, even when the forces are removed – perfectly plastic body

STRESS

• Stress is the force acting on the unit area of a material.

• Stress can deform the body

• Stress is defined as “The restoring force per unit area of the material”. It is a tensor quantity. Denoted by Greek letter ς. Measured using Pascal or N/m2. Mathematically expressed as –

• ς=F/A

STRESS

where,

• F is the restoring force measured in Newton or N.

• A is the area of cross-section measured in m2.

• ς is the stress measured using N/m2 or Pa.

• Types of Stress:

- mainly categorized into two forms that are Normal Stress and Tangential or Shearing Stress

STRAIN

• it is defined as the amount of deformation experienced by the body in the direction of force applied, divided by initial dimensions of the body. The relation for deformation in terms of length of a solid is given below.

• ϵ=δLlL • where, • ϵ is the strain due to stress applied

δL is the change in length L is the original length of the material.

• The strain is a dimensionless quantity as it just defines the relative change in shape.

STRESS-STRAIN CURVE

Under different loads, the stress and corresponding strain values.

STRESS-STRAIN CURVE

• The stress-strain graph has different points or regions as follows:

• Proportional limit

• Elastic limit

• Yield point

• Ultimate stress point

• Fracture or breaking point

STRESS-STRAIN CURVE

Proportional Limit • It is the region in the stress-strain curve that obeys

Hooke’s Law. In this limit, the ratio of stress with strain gives us proportionality constant known as young’s modulus. The point OA in the graph is called the proportional limit.

Elastic Limit • It is the point in the graph up to which the material

returns to its original position when the load acting on it is completely removed. Beyond this limit, the material doesn’t return to its original position and a plastic deformation starts to appear in it.

STRESS-STRAIN CURVE

Yield Point • The yield point is defined as the point at which

the material starts to deform plastically. After the yield point is passed, permanent plastic deformation occurs. There are two yield points (i) upper yield point (ii) lower yield point.

Ultimate Stress Point • It is a point that represents the maximum stress

that a material can endure before failure. Beyond this point, failure occurs.

STRESS-STRAIN CURVE

Fracture or Breaking Point

• It is the point in the stress-strain curve at which the failure of the material takes place.

Hooke’s Law

• In the 19th-century, while studying springs and elasticity, English scientist Robert Hooke noticed that many materials exhibited a similar property when the stress-strain relationship was studied.

• There was a linear region where the force required to stretch the material was proportional to the extension of the material. This is known as Hooke’s Law.

STRESS-STRAIN CURVE

• Hooke’s Law states that the strain of the material is proportional to the applied stress within the elastic limit of that material.

• Mathematically, Hooke’s law is commonly expressed as: F = –k.x where, • F is the force • x is the extension length • k is the constant of proportionality known as spring

constant in N/m

Experimental verification of Hooke's law

• A spring is suspended from a rigid support • A weight hanger and a light pointer is attached at its

lower end • The pointer can slide over a scale graduated in

millimeters • The initial reading on the scale is noted • A slotted weight of m kg is added to the weight hanger

and the pointer position is noted. • The same procedure is repeated with every

additional m kg weight. • extension of the spring is proportional to the weight.

Experimental verification of Hooke's law

THREE MODULI OF ELASTICITY

• (i) Young's modulus of elasticity • Consider a wire of length l and cross sectional area A

stretched by a force F acting along its length. Let dl be the extension produced.

• Longitudinal stress = Force / Area = F/A • Longitudinal strain = change in length / original length =

dl/l • Young's modulus of the material of the wire is defined as

the ratio of longitudinal stress to longitudinal strain. It is denoted by q.

• Young's modulus = longitudinal stress / longitudinal strain • q =F/A / dl/l = Fl /Adl

THREE MODULI OF ELASTICITY

THREE MODULI OF ELASTICITY

• (ii) Bulk modulus of elasticity • Suppose euqal forces act perpendicular to the six faces of a cube of

volume V as shown in Fig.. Due • to the action of these forces, let the decrease in volume be dV. • Now, Bulk stress = Force/Area = F/ A • Bulk Strain = change in volume / original volume = -dV/V • (The negative sign indicates that volume decreases.) • Bulk modulus of the material of the object is defined as the ratio of

bulk stress to bulk strain. • It is denoted by k • Bulk modulus = Bulk stress / Bulk strain • K = -PV/dV

THREE MODULI OF ELASTICITY

• (iii) Rigidity modulus or shear modulus • Let us apply a force F ,tangential to the top surface of a

block whose bottom AB is fixed • Under the action of this tangential force, the body suffers a

slight change in shape, itsvolume remaining unchanged.The side AD of the block is sheared through an angle θ to the position AD'.

• If the area of the top surface is A, then shear stress = F/A. • Shear modulus or rigidity modulus of the material of the

object is defined as the ratio of shear stress to shear strain. It is denoted by n.

• Rigidity modulus = shear stress / shear strain • (i.e) n = F/Aθ

THREE MODULI OF ELASTICITY

• Table lists the values of the three moduli of elasticity for some commonly used materials.

PASCAL’S LAW

• A change in pressure at any point in an enclosed

fluid at rest is transmitted undiminished to all points in the fluid.

• Pressure exerted on a fluid in an enclosed container is transmitted equally and undiminished to all parts of the container and acts at right angle to the enclosing walls.

• Alternate definition: The pressure applied to any part of the enclosed liquid will be transmitted equally in all directions through the liquid.

PASCAL’S LAW

• This principle is stated mathematically as, P = g. h • P is the hydrostatic pressure (given in pascals in the SI system), or

the difference in pressure at two points within a fluid column, due to the weight of the fluid;

• ρ is the fluid density (in kilograms per cubic meter in the SI system); • g is acceleration due to gravity (normally using the sea level

acceleration due to Earth's gravity, in meters per second squared) • h is the height of fluid above the point of measurement, or the

difference in elevation between the two points within the fluid column (in meters).

• Pascal's law can be interpreted as saying that any change in pressure applied at any given point of the fluid is transmitted undiminished throughout the fluid.

APPLICATIONS OF ELASTICITY

• To design safe and stable manmade structures such as skyscrapers and overbridges to make life convenient.

• Cranes used to lift loads use ropes - designed - the stress due to the maximum load does not exceed the breaking stress.

• A collection of thinner wire strands when compacted together make the rope stronger than a solid rope of the same cross-section.

• Structures such as bridges and tall buildings that have to support static or dynamic loads are generally constructed using pillars and beams to support them.

• The beams used in buildings and bridges have to be carefully designed so that they do not bend excessively and break under the stress of the load on them.

• Beams and pillars are designed to remain stable and safe within the range of the maximum load they are designed to carry.

VISCOSITY

Viscosity is the measure of a substance's resistance to motion under an applied force. The formula for measuring viscosity is fairly simple:

• Viscosity = shear stress / shear rate • Shear stress is the force per unit area required to move one

layer of fluid in relation to another. • Shear rate is the measure of the change in speed at which

intermediate layers move with respect to one another.

VISCOSITY

• When two parallel layers of a liquid are moving with different velocities, they experience tangential forces which tend to retard the faster layer and accelerate the slower layer – forces of viscosity

• Let v and v+dv be the velocities of two layers.

• Velocity gradient is dv/dz

• A – surface area of the layer

VISCOSITY

• Viscous force is directly proportional to the surface area A and velocity gradient dv/dz

F A (dv/dz)

Or

F = A (dv/dz)

• - constant for a liquid – coefficient of viscosity

• If A = 1, dv/dz = 1; F =

VISCOSITY

• The coefficient of viscosity is defined as the tangential force per unit area required to maintain a unit velocity gradient.

• Unit of is Nsm-2- pascal second

• Dimensions of = [F] / [A] (dv/dz)

= MLT-2 / L2 (LT-1 / L)

= ML-1T-1

TYPES OF FLOW

• Streamline flow

• Laminar flow or streamline flow in pipes (or tubes) occurs

when a fluid flows in parallel layers, with no disruption between the layers

• At low velocities, the fluid tends to flow without lateral mixing, and adjacent layers slide past one another like playing cards.

• The motion of the particles of the fluid is very orderly with all particles moving in straight lines parallel to the pipe walls.

TYPES OF FLOW

• Turbulent flow

• chaotic property changes

• includes rapid variation of pressure and flow velocity in space and time

• the fluid no longer travels in layers and mixing across the tube is highly efficient.

• Reynolds number is a dimensionless quantity that is used to determine the type of flow pattern as laminar or turbulent while flowing through a pipe. Reynolds number is defined by the ratio of inertial forces to that of viscous forces.

• At low Reynolds numbers, flows tend to be dominated by laminar (sheet-like) flow, while at high Reynolds numbers flows tend to be turbulent.

Reynolds Number Formula

It is given by the following relation:

• Reynolds Number=Inertial Force / Viscous Force

• If the Reynolds number calculated is high (greater than 2000), then the flow through the pipe is said to be turbulent.

• If Reynolds number is low (less than 2000), the flow is said to be laminar

Reynolds Number

• The plume from this candle flame goes from laminar to turbulent. The Reynolds number can be used to predict where this transition will take place.

Reynolds Number

• The predictions of the onset of turbulence and the ability to calculate scaling effects can be used to help predict fluid behaviour on a larger scale, such as in local or global air or water movement and thereby the associated meteorological and climatological effects.

Reynolds Number

• Determines the nature of flow of a liquid through a tube

• Turbulence sets in at the same constant value of Reynold’s number in all cases of liquid flow

• Flow will be steady and streamline until this number is not exceeded

• After exceeding, flow becomes turbulent

Stoke’s Law

• mathematical equation

• expresses the settling velocities of the small spherical particles in a fluid medium.

• derived by considering the forces acting on a particular particle as it sinks through the liquid column under the influence of gravity.

• force that retards a sphere moving through a viscous fluid is directly proportional to the velocity and the radius of the sphere, and the viscosity of the fluid

Stoke’s Law

F=6πηrv

• Stokes’s law finds application in several areas such as:

Settling of sediment in freshwater

Measurement of the viscosity of fluids