## Sunday, April 26, 2015

### Explanation Of S-Block (Group IA) Elements

S – Block
(Group IA Elements)
Group I A Elements:

Hydrogen (Atomic number - 1)
Lithium (Atomic number - 3)
Sodium (Atomic number - 11)
Potassium (Atomic number - 19)
Rubidium (Atomic number - 37)
Strontium (Atomic number - 55)
Francium(Atomic number - 87)
•  Elements of group IA and IIA are known as s - block elements.
•  Except Hydrogen all elements are metals.
•  Group IA consists of Hydrogen, Lithium, Sodium, Potassium, Rubidium, Cesium, Francium.
•  Group IIA consists of Beryllium, Magnesium, Calcium, Stratium, Barium, Radium.
•   IA elements (except Hydrogen) - Alkali metals.
•   IIA Elements  → Alkaline earth metals.
•  Have one electron in the outermost orbit.
•  By loosing the electron they form stable + 1 ion.
•  They are univalent and show + 1 oxidation state.

General periodic trends:
•  As we move from up to down the atomic radius increases.
• Order of radius size = H < Li < Na < Rb < Cs < Fr
• Ionization potential coming from top to bottom decreases.
• Density generally increases with an increase in atomic mass.
• Exception  → Potassium is expected to have more density than sodium but actually has less than sodium.
• Electronegativity from top to bottom decreases.
• Alkali metals along their individual periods have largest atomic radius.
• Mobility of ions increases down the groupRb⁺ > K⁺ > Na⁺ > Li⁺ > H⁺  is the order of mobility of ions.
• Hydration enthalpy decreases down the group.
∴ H⁺ > Li⁺ > Na⁺ > K⁺ > Rb⁺  in case of hydration enthalpy.
Heating effects of carbonates of IA elements.
(M = Na, K, Rb, Cs), (Exception Li₂CO₃ → Li₂O + CO₂)
All metal carbonates of IA except Li₂CO₃ have no effect on heating.
Heating effect of nitrates of IA elements.
(M = Na, K, Rb, Cs), (LiNO₃ → Li₂O + NO₂)
Flame Test:
Li Crimson Red
Na Yellow
K → Violet
Rb Red Violet
Cs Blue

## Friday, April 24, 2015

### Indefinite Integrals

Explanation Of Indefinite Integrals.......
•  A function  $\phi \left( x \right)$ is called a primitive (or) an anti-derivative of a function f(x) of
•
$\phi '\left( x \right)=f\left( x \right)$
$\Rightarrow \,\,\,\,\,\,\int_{{}}^{{}}{f\left( x \right)}=\phi \left( x \right)+C$
Here  f(x) = integrand
$\frac{d\left( \int_{{}}^{{}}{f\left( x \right)}\,dx \right)}{dx}=f\left( x \right)$

Algorithm to solve integrals of form:

$\int_{{}}^{{}}{{{\sin }^{m}}x}\,{{\cos }^{n}}x\,dx ,$  $\int_{{}}^{{}}{{{\sin }^{m}}x\,dx}$
$\int_{{}}^{{}}{{{\cos }^{n}}x}dx,$          where  $m,n\in N$
Step 1:    Find m,n
Step 2:
*   If m is odd i.e. power or index of sin x is odd, put cos x = t and reduced the integrand in terms of it.
*  If n is odd then put sin x = t and reduce the integrand.
*  If both are odd you can use any of above two methods.

• To Evaluate integrals of form
$\int_{{}}^{{}}{{{\sin }^{m}}x}\,{{\cos }^{n}}x\,dx$    where     $m,n\in Q$ and m + n is negative even integer then put
tan x = t.
• To evaluate integrals of form $\int_{{}}^{{}}{{{\sin }^{m}}x}\,{{\cos }^{n}}x\,dx$   where m,n are positive even integers then.
Let Z = cos x + i sin x
$\frac{1}{Z}=cosx-i\text{ }sinx$
$Z+\frac{1}{Z}=2\,\cos x$
$Z-\frac{1}{Z}=2i\,\operatorname{sinx}$
$cos\,nx=\frac{1}{2}\left( {{Z}^{n}}+\frac{1}{{{Z}^{n}}} \right)$
$\sin nx=\frac{1}{2i}\left( {{Z}^{n}}-\frac{1}{{{Z}^{n}}} \right)$
• Some important standard integrals :
$\int_{{}}^{{}}{\frac{1}{{{x}^{2}}-{{a}^{2}}}dx=}\frac{1}{2a}\log \left| \frac{x-a}{x+a} \right|+C$

$\int_{{}}^{{}}{\frac{1}{{{a}^{2}}-{{x}^{2}}}dx=}\frac{1}{2a}\log \left| \frac{a+x}{a-x} \right|+C$

$\int_{{}}^{{}}{\frac{1}{\sqrt{{{a}^{2}}+{{x}^{2}}}}}dx=\log \left| x+\sqrt{{{a}^{2}}+{{x}^{2}}} \right|+C$
$\int_{{}}^{{}}{\frac{1}{\sqrt{{{x}^{2}}-{{a}^{2}}}}dx=}\log \left| x+\sqrt{{{x}^{2}}-{{a}^{2}}} \right|+C$

Integrals of form

$\int_{{}}^{{}}{\frac{dx}{{{\left( a+b\cos x \right)}^{2}}}},$     $\int_{{}}^{{}}{\frac{dx}{{{\left( a+b\sin x \right)}^{2}}}}$    to evaluate this type of integrals

Define     $P=\frac{\sin x}{a+b\cos x}$   (or)      $\frac{\cos x}{a+b\sin x}$

Depending on integral then find   $\frac{dP}{dx}$  in terms of   $\frac{1}{a+b\cos x}$  (or) $\frac{1}{a+b\sin x}$

Now integrate both sides of expression to get required integral.

Integration by parts :

$\int_{{}}^{{}}{u\nu }dx=u\left( \int_{{}}^{{}}{\nu dx} \right)-\int_{{}}^{{}}{\left\{ \frac{du}{dx}.\int_{{}}^{{}}{\nu dx} \right\}}dx$

Very IMP Note:
Choose first function as the function which come first in word “ILATE”
(1)  I – Inverse trigonometric
L – logarithmic functions
A – algebraic functions
T for Trigonometric functions
E for exponential functions.

è   Integrals of the form
$\int_{{}}^{{}}{{{e}^{x}}}\left( f\left( x \right)+f'\left( x \right) \right)dx={{e}^{x}}f\left( x \right)+C$
Note:
$\int_{{}}^{{}}{{{e}^{kx}}}\left\{ k\,f\left( x \right)+f'\left( x \right) \right\}={{e}^{kx}}f\left( x \right)+C$
è    $\int_{{}}^{{}}{{{e}^{ax}}}\sin bx=\frac{{{e}^{ax}}}{{{a}^{2}}+{{b}^{2}}}\left( a\sin \,bx+b\cos \,bx \right)+C$
è    $\int_{{}}^{{}}{{{e}^{ax}}}\,\cos \,bx=\frac{{{e}^{ax}}}{{{a}^{2}}+{{b}^{2}}}\left( a\,cos\,bx+b\sin \,bx \right)+C$

Some Very Important Integrals :

$\int_{{}}^{{}}{\sqrt{{{a}^{2}}-{{x}^{2}}}}dx=\frac{x}{2}\sqrt{{{a}^{2}}-{{x}^{2}}}+\frac{1}{2}{{a}^{2}}{{\sin }^{-1}}\left( \frac{x}{a} \right)+C$
$\int_{{}}^{{}}{\sqrt{{{a}^{2}}+{{x}^{2}}}}dx=\frac{x}{2}\sqrt{{{a}^{2}}+{{x}^{2}}}+\frac{{{a}^{2}}}{2}\log \left| x+\sqrt{{{a}^{2}}+{{x}^{2}}} \right|+C$
$\int_{{}}^{{}}{\sqrt{{{x}^{2}}-{{a}^{2}}}}dx=\frac{x}{2}\sqrt{{{x}^{2}}-{{a}^{2}}}-\frac{{{a}^{2}}}{2}\log \left| x+\sqrt{{{x}^{2}}-{{a}^{2}}} \right|+C$

Reduction Formulae :
$\int_{{}}^{{}}{{{\sin }^{n}}x\,dx}={{I}_{n}}$
${{I}_{n}}=\frac{-{{\sin }^{n-1}}x\,\cos \,x}{n}+\frac{n-1}{n}{{I}_{n-2}}$

$\int_{{}}^{{}}{{{\cos }^{n}}x\,dx={{I}_{n}}}$
${{I}_{n}}=\frac{{{\cos }^{n-1}}x\,\sin \,x}{n}+\frac{n-1}{n}{{I}_{n-2}}$

$\int_{{}}^{{}}{{{\tan }^{n}}x\,dx={{I}_{n}}}$
${{I}_{n}}=\frac{{{\tan }^{n-1}}x}{n-1}-{{I}_{n-2}}$

$\int_{{}}^{{}}{\cos e{{c}^{n}}x\,dx}={{I}_{n}}$
${{I}_{n}}=\frac{-\cos e{{c}^{n-2}}\,\cot x}{n-1}+\frac{n-2}{n-1}{{I}_{n-2}}$

$\int_{{}}^{{}}{{{\sec }^{n}}x\,dx}={{I}_{n}}$
${{I}_{n}}=\frac{{{\sec }^{n-2}}x\,\tan \,x}{n-1}+\frac{n-2}{n-1}{{I}_{n-2}}$

$\int_{{}}^{{}}{{{\cot }^{n}}x\,dx}={{I}_{n}}$
${{I}_{n}}=\frac{-{{\cot }^{n-1}}x}{n-1}-{{I}_{n-2}}$

## Tuesday, April 21, 2015

### Free, Forced, Damped and Resonant Vibrations

Free Vibrations:
If a given body is once set into vibrations and then let free to vibrate with its own natural frequency, the vibrations are said to be free vibrations. The natural frequency of free vibrations depends on the nature and structure of the body and in ideal situation, the amplitude, frequency and the energy of the vibrating body remain constant.
Forced Vibrations:
The vibrations in which a body oscillates under the effect of an external periodic force, whose frequency is different from the natural frequency of oscillating body, are called forced vibrations. In forced vibrations, the oscillating body vibrates with the frequency of external force and amplitude of oscillations is generally small.

If an external driving force is represented by
$F (t) =F_{0} \cos {{\omega }_d}t$
The motion of particle is under combined action of
(i)             Restoring force (-kx)
(ii)            Damping force (-bv), and
(iii)           Driving force F(t)

Now,  $ma=-kx-bv+ F_{0} \cos{{\omega }_{d}}t$

Or       $\frac{{{d}^{2}}x}{d{{t}^{2}}}=-\frac{kx}{m}-\frac{b}{m}\frac{dx}{dt}+\frac{{{F}_{0}}\cos {{\omega }_{d}}t}{m}$

The solution of this equation gives   $x=x_{0}\sin({{\omega }_{d}}t+\phi )$   with amplitude
$x_{0}=\frac{{}^{{{F}_{0}}}/{}_{m}}{\sqrt{(\omega _{0}^{2}-\omega _{d}^{2})+{{(\frac{b\omega }{m})}^{2}}}}$
$Tan\theta =\frac{\omega _{0}^{2}-\omega _{d}^{2}}{{}^{b{{\omega }_{d}}}/{}_{m}}$

And
${\omega }_{0}=\sqrt{\frac{k}{m}}$
= natural frequency

Damped Vibrations:
When a body is set in free vibrations, generally there is a dissipation of energy due to dissipative causes like viscous drag of a fluid, frictional force, hysteresis, electromagnetic damping force, etc.., and as a result amplitude of vibration regularly decreases with time. Such vibrations of continuously falling amplitudes are called damped vibrations.
In these oscillations, the amplitude of oscillation decreases exponentially and hence, energy also decreases exponentially.
If the velocity of an oscillator is v, the damping force
Fd = - bv
Where, b=damping constant.

Resultant force on a damped oscillator is given by     F=FR+FD=-k x –b v

Or      $\small \frac{m{{d}^{2}}x}{d{{t}^{2}}}+\frac{bdx}{dt}+kx=0$

Displacement of a damped oscillator is given by
$\small X={{x}_{m}}{{e}^{{}^{-bt}/{}_{2m}}}\sin ({{\omega }^{'}}t+\phi )$

$\small {{\omega }^{'}}=\sqrt{\omega _{0}^{2}-{{({}^{b}/{}_{2m})}^{2}}}$
Where, $\small \omega '$ = angular frequency of the damped oscillator for a damped oscillator, if the damping is small then the mechanical energy decreases exponentially with time as
$E=\frac{1}{2}kx_{m}^{2}{{e}^{{}^{-bt}/{}_{m}}}$
Resonant Vibrations:

It is a special case of forced vibrations in which frequency of external force is exactly same as the natural frequency of oscillator. As a result the oscillating body begins to vibrate with large amplitude leading to the resonance phenomenon to occur. Resonant vibrations play a very important role in music and tuning of station/channel in a radio/TV.