Amplitude Modulation

Let carrier wave be represented by the equation
v~^c=V~^ccos(ω~^ct) ----- (1)
Let Modulating Signal be represented by the equation
v~^m=V~^mcos(ω~^mt) ----- (2)

After Amplitude Modulation Amplitude of the carrier varies according to the instantaneous value of modulating signal. The total Amplitude of the Modulated carrier is given as

V~^A~^M = V~^c+V~^mcos(ω~^mt)

The Expression for AM Wave is

v~^A~^M = V~^A~^Mcos(ω~^ct)
=(V~^c+V~^mcos(ω~^mt))cos(ω~^ct)
=V~^ccos(ω~^ct)+V~^mcos(ω~^mt)cos(ω~^ct)
=V~^c(cos(ω~^ct)+m~^acos(ω~^mt)cos(ω~^ct)) , ( Since V~^m/V~^c = m~^a)



Now use the formula CosA.CosB = $ 1 #_ 2 $ [Cos(A+B)+Cos(A-B)]

v~^A~^M = V~^c(cos(ω~^ct)+ $ m~^a #_ 2 $ [cos(ω~^mt+ω~^ct)+cos(ω~^mt-ω~^ct)]

=V~^ccos(ω~^ct)+ $ m~^aV~^c #_ 2 $ cos2π(f~^m+f~^c)t+ $ m~^aV~^c #_ 2 $ cos2π(f~^m-f~^c)t

Types of Corrosion

Different types of corrosion are:

1. Differential metal corrosion or Galvanic Corrosion
2. Differential Aeration corrosion
3. Stress Corrosion

1. Differential metal corrosion:

When two metals of different electrode potential or two dissimilar metals are in contact, the metal with lower electrode potential or more reactive metal act as anode and undergoes corrosion. Potential difference between the metals causes the corrosion.

The factors determining the rate of corrosion

2. Differential aeration corrosion:

Difference in the concentration of oxygen on the surface of oxygen causes the differential aeration corrosion. The part of metal over which oxygen concentration is less becomes anode and gets corroded. The metal surface over which oxygen concentration is high becomes cathode, because cathodic reaction involves consumption of oxygen.

3. Stress Corrosion:

The corrosion taking place at stress points of metals formed due to welding, hammering, bending, etc. the atoms under stressed part get high energy and get higher oxidation potential than that of stress free part. hence stressed part becomes anode and stress free part becomes cathode and corrosion takes place.

Corrosion Science - Definition and Theory

Definition:

Corrosion is defined as destruction and consequent loss of metals through chemical or electrochemical attack by the environment.

Main Types of Corrosion:
1. Dry Corrosion
2. Wet Corrosion

Theory of Corrosion:

Corrosion takes place in a metal due to the formation of anodic and cathodic regions in the same metal surface, or when two metals are kept in contact with each other.

Oxidation and Reduction reactions takes place at Anode and Cathode Respectively. Since Oxidation involves liberation of electrons, metal undergoes corrosion at anodic region. The liberated electrons move towards cathodic region constituting corrosion current. The corrosion product is formed due to the diffusion of metal ions liberated at anodic and cathodic regions.

Corrosion Reactions:

At anode oxidation takes place liberating electrons. Metal atoms are converted into metal ions.

M --> M^n^++ne^-

Cathodic Reactions are of several types. The most common types are explained here. They are:
1. Hydrogen Evolution ( Hydrogen type Corrosion )
2. Oxygen Absorption ( Oxygen type Corrosion )

1. Hydrogen Evolution Type:

It takes place in the Absence of oxygen

(i) Acidic Medium:

2H^++2e^--->H~^2

(ii) Basic or Neutral Medium:
2H~^2O+2e^--->H~^2+2OH^-

2. Oxygen absorption type

This type of reaction takes place in the presence of oxygen.

(i) Acidic Medium:
O~^2+4e^-+4H^+-->2H~^2O

(ii) Basic or Neutral Medium:
H~^2O+O~^2+e^---> 4OH^-

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Apparent frequency calculation

Source, sound, observer are in same direction
f' = f $ ( $ v-v~^o #_ v-v~^s $ )

if the direction of observer or source changes, then sign changes.

If freq. Appears be double the freq. of source, for a stationary observer, then
v~^s = $v#_2
If for a stationary observer freq. Appears to be half the real freq. Then

v~^s= -v

Simple Harmonic Motion

Velocity of particle executing SHM:
v=ωACosωt = ω√A^2-y^2

Accelaration:
a=-ω^2ASinωt = -ω^2y

Kinetic Energy:

E~^k= $ 1 #_ 2 $ mω^2A^2Cos^2ωt

Potential Energy:

E~^p=$1#_2$mω^2A^2sin^2ωt

Total Energy:
E=2π^2mf^2A^2

Conjugate Acid-Base pair

Pair of acid and base that differ by a proton.

Examples:
1. OH^- and O^-^2 # 2. HCO~^3^- and CO~^3^2^-

How to find conjugate base pair:

To form an base, the substance should loose a H⁺ ion (proton). So, remove one H⁺ and subtract 1 from the charge.

To form an acid, the substance should recieve a proton. So add an H⁺ ion.

Faraday's laws of Electrolysis

First law:
Amount substance discharged at an electrode during Electrolysis is directly proportional to the quantity of electricity passed.
W α Q
W = ZIt

Z = $ Eq.mass # _ 96,500

Second law:
when same quantity of electricity is passed to solutions of different Electrolytes amount of substances discharged at the electrodes is directly proportional to their equivalet masses.

W~^1 #_ W~^2 $ = $ E~^1 #_ E~^2

Calculation of Number of Pi and Sigma bonds in a Molecule

PI bond is formed by axial overlapping.
SIGMA bond is formed by lateral overlapping
A SIGMA bond alone can exist in a molecule, but PI bond cannot.

How to calculate number of Pi and sigma bonds in molecule.

Every single bond in a molecule is formed by sigma bond.
Every double bond in a molecule is formed by a sigma bond and a Pi bond.
On the basis of this number of sigma and Pi bonds can be calculated.

Eg: CH~^3-CH=CH-CH~^3

Number of Sigma Bonds: 3
Number of π bonds : 1

Angle of Banking

For a Vehicle taking turn on a Curved path, If It's velocity exceeds a certain value it may loose it's balace and skid due to a force called Centrifugal Force.To Avoid this it should be provided by necessary centripetal force. It is done by slightly rising outer edge of the road. This is known as banking of roads. The angle made by the surface of road with horizontal direction is known as angle of banking.For a Vehicle moving with a velocity 'V' on a curved path of radious 'R', the angle of banking 'θ' and Velocity of vehicle are Related as:

θ=tan-1(	v2	)
	rg

Projectile Motion

Equation of motion of Projectile:
y = xtanθ - $ gx^2 # _2u^2cos^2θ

Note: Equation of the Projectile motion is Similar to the Equation of parabola, y=ax-bx² where a=tanθ and b = $g # _2u^2cos^2θ

Maximum Height Attained by a Projectile:
H = $2uSinθ # _g

Range or Displacement of Projectile:
R = $ u^2Sin2θ # _g

For given angles θ~^1 and θ~^2 , if θ~^1+θ~^2 = 90° then R~^1 = R~^2

Focal Length, Radious, Magnification, Power of Lenses

Convex Lense(Converging Lense) - Surfaces are Convex towards rarer medium.
Concave Lense(Diverging Lense) - Surfaces are Concave towards rarer medium

Focal Length of Lenses

1) f = $R~^1 R~^2 #_ (n-1)(R~^2 +R~^1 )

R - Radii of curvatures n - Refractive Index of Lense

2) f = $uv #_u+v

u - Object Distance. v - Image Dist.

Note: If given lense is equiconvex lense, then
R~^1 = R~^2 = R and f = $R#_2(n-1)
$ => n = 1 + $R #_2f