Project Report
PROJECT REPORT
is
Presented By
Prachi Vishwakarma (0612ec101033)
BRANCH : EC
Department of Electronics & Communication Engineering
ADINA INSTITUTE OF SCIENCE AND TECHNOLOGY, SAGAR (M.P)
Session-2014
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1.Rolle’s Theorem
The Extreme Value Theorem states that a continuous function on a closed interval
[a, b] must have both a minimum and a maximum on the interval.
Both of these values, however, can occur at the endpoints. Rolle’s Theorem,
named after the French mathematician Michel Rolle, gives conditions that
guarantee the existence of an extreme value in the interior of a closed interval
Rolle’s Theorem states that if f satisfies the conditions of
the theorem, there must be at least one point between
a and b at which the derivative is 0.
If a real-valued function ƒ is continuous on a closed interval [a, b], differentiable
on the open interval (a, b), and ƒ(a) = ƒ(b), then there exists a c in the open interval
(a, b) such that
In calculus, Rolle's theorem essentially states that any real-valued differentiable
function that attains equal values at two distinct points must have a stationary point
somewhere between them; that is, a point where the first derivative (the slope of
the tangent line to the graph of the function) is zero.
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• If f = 0 everywhere it’s easy
• Assume that f > 0 somewhere (case f< 0 somewhere similar)
• Know that f must attain a maximum value at some point which must be a
critical point as it can’t be an endpoint (because of assumption that f > 0
somewhere).
• The derivative vanishes at this critical point where maximum is attained
• .
• Examples
• f(x) = sin x on [0, 2]
• exp(-x2) on [-1,1]
• Any even continuous function on [-a,a] that is differentiable on (-a,a)
Q. Verify Rolle’s Theorem for the function f(x)=|x| ,-1 to +1
The graph of the absolute value function.
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If differentiability fails at an interior point of the interval, the conclusion of Rolle's
theorem may not hold. Consider the absolute value function
Then f(−1) = f(1), but there is no c between −1 and 1 for which the derivative is
zero. This is because that function, although continuous, is not differentiable at
x = 0. Note that the derivative of f changes its sign at x = 0, but without attaining
the value 0. The theorem cannot be applied to this function, clearly, because it does
not satisfy the condition that the function must be differentiable for every x in the
open interval. However, when the differentiability requirement is dropped from
Rolle's theorem, f will still have a critical number in the open interval (a,b), but it
may not yield a horizontal tangent (as in the case of the absolute value represented
in the graph).
Natural Rubber
The words rubber come from the materials from the rubber tree name
“Havea Brasiliensis”
In beginning all product from rubber are made from natural rubber that
produced from mater ials from natural rubber tree called latex..
Process of making vulcalized rubber by raw rubber
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Difference between raw rubber and vulcanized rubber
The different between raw rubber and vulcanized rubber or elastomer is :
1. Raw rubber either natural rubber or synthetic rubber are materials that
has plastic properties and can be reshaping at high temperature and
not sutaible for applications.
2. Elastomer is the words that used for vulcanized rubber, vulcanisate or
crosslinking rubber
 In beginning all product from rubber are made from natural rubber that
produced from materials from natural rubber tree called latex.Synthetic
rubber are produced from reactions of low molecular weight materials called
monomer to produced long chain molecule called polymer.Elastic properties
are produced by mix raw rubber with specific additives during rubber
compounding
Natural Rubber
 The difficulties with natural rubber
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 Strength
 Availability
 Bacterial breakdown
 Creep
NATURAL RUBBER
Mastication is mechanical shearing process using two roll mill or internal mixer)
for reduced the molecular weight, reduced the viscosity and to soften the raw
rubber. after mastication the processing will be much easier and increased the
effectiveness of dispersions of compounding ingredients.The mastication is
compulsory for natural rubber due to high molecular weight in nature (around 10 5106 ).Each ingredient has a specific function either in processing, vulcanization or
end use of the products. the various ingredients may be classified according to their
specific functions in the following group
Vulcanized Rubber:
The crosslinking produced can have a monosulphide and polysulphide or both
depending on the vulcanization systems usedUnvulcanized rubber products are not
very strong and have the consistency of chewing gum.Charles Goodyear invented
the first recognizable method of vulcanization.Heating natural rubber with sulfur in
1841Both natural and synthetic rubber are vulcanized today90% of all
vulcanization occurs with sulfur of natural rubber, ethylene-propylene-diene
(EPDM), butyl rubbers, and nitrile rubber.Vulcanization is a processWhich
increases elasticity and reduces plasticity by the formation of a crosslinked
molecular network.Which occurs by heating the rubber and vulcanizing agents
under pressureSulphur vulcanization systems can be divided into 3 systems
depending on the relative amount of sulphur & accelerator used.The three systems
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can be differentiate through the types of crosslinking produced and the main chain
modification after vulcanization
FREE BODY DIAGRAM
Newton’s 1st Law:An object in motion stays in motion in a straight line, unless
acted upon by unbalanced force. A push or pull will cause object to speed up, slow
down, or change direction.
FREE BODY DIAGRAM:Free-body diagrams are used to show the relative
magnitude and direction of all forces acting on an object. Free-body diagrams are
used to show the relative magnitude and direction of all forces acting on an object.
where Fnorm= normal force
Ffrict= friction force
Fapp= applied force
Fgrav= gravity force
Examples of Free body Diagram
A book is at rest on a table top. Diagram the forces acting on the book.
In this diagram, there are normal and gravitational forces on the book
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An egg is free-falling from a nest in a tree. Neglect air resistance. Draw a freebody diagram showing the forces in
Gravity is the only force acting on the egg as it fallsvolved.
Draw the free body diagram of the given question
We know that two sphere are given sphere A and sphere B Here in this diagram
there are two sphere i.e sphere A and and sphere B . the force acting on sphere A is
wt which is the weight of the sphere or the gravitational force, another force is
Rb which is due to the contact it with the sphere B,Rp is force acting due to the
contact of spere at the point P,Rq is force acting due to the contact point Q,Rb is
the force acting when the two sphere are in contact
Theforce acting on sphere B is wt which is the weight of the sphere or the
gravitational force, ,Rs is force acting due to the contact of spere at the point S ,Ra
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is
the force acting when the two sphere a
The force acting on sphere A is wt which is the weight of the sphere or the
gravitational force, another force is Rb which is due to the contact it with the
sphere B,Rp is force acting due to the contact of spere at the point P,Rq is force
acting due to the contact point Q,Rb is the force acting when the two sphere are
in contact For sphere A the free body diagram is
in contact
The force acting on sphere A are as
Rp ,Rb ,Weight of sphere , Rq
The force acting on sphere B are as
Rs ,Ra ,Weight of sphere
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Theforce acting on sphere B is wt which is the weight of the sphere or the
gravitational force, ,Rs is force acting due to the contact of spere at the point S ,Ra
is the force acting when the two sphere are in contact
3. Equation of state and ideal gases
An “ideal” gas exhibits certain theoretical properties. Specifically, an ideal gas
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obeys all of the gas laws under all conditions. In reality, there are no gases that fit
this definition perfectly. We assume that gases are ideal to simplify our
calculations.
Ideal Gas Equation
Boyle’s law: V ∝ 1/P
(at constant n and T)
Charles’ law: V ∝T (at constant n and P)
Avogadro’s law: V ∝ n (at constant P and T)
V∝ 𝑛𝑇/𝑃
V=constant x nT/P=R nT/𝑃 R is the gas constant
PV = nRT n, V and R are constant
The conditions 0 0C and 1 atm are called standard temperature and pressure
(STP). Experiments show that at STP, 1 mole of an ideal gas occupies 22.414 L.
PV = nRT
R=PV/nT
PV = nRT
P = Pressure (in kPa)V = Volume (in L)
T = Temperature (in K) n = moles
R = 8.31 kPa• L
K • mol
R is constant. If we are given three of P, V, n, or T, we can solve for the unknown
value
Equation of thermodynamic State
dS=(𝜕𝑆/𝜕𝑇)V dt+(𝜕𝑠/𝜕𝑉)T dV
TdS=(𝜕𝑆/𝜕𝑇)V dt+(𝜕𝑆/𝜕𝑇)T dV
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T(𝜕𝑆/𝜕𝑇)V=CV, heat capacity at constant volume, and
(𝜕𝑆/𝜕𝑇)t=(𝜕𝑃/𝜕𝑇)V,Maxwell third equation
T dS=CV dT +T(𝜕𝑃/𝜕𝑇)V dV
Where S is a entrophy
T is a temperature
V is a volume
P is a pressure
This is known as first TDS equation
If S=S(T,P)
dS=(𝜕𝑆/𝜕𝑇)P dT+(𝜕𝑆/𝜕𝑃)T dP
T dS= T (𝜕𝑆/𝜕𝑇)P dT+(𝜕𝑆/𝜕𝑃)T dP
T(𝜕𝑆/𝜕𝑇)P=Cp dT is a TDS Equation