Wave And Oscillation

                              Wave And Oscillation

Periodic motion:

Periodic motion is one kind of motion that repeats itself after a fixed intervals of time. Periodic motions are abundant in nature. Example of periodic motion: Earth motion in its orbit around the

sun, wave in a water. 

Oscillatory motion:

Oscillatory motion is one kind of periodic motion. Oscillatory motion is define as the to and fro motion of an object from its mean position. Its rare in nature. All periodic motions are not oscillatory motion but all oscillatory motions are periodic motion. Oscillatory motion must be to and fro motion but periodic motions may be to and fro or circular motion

Example of Oscillatory motion: Oscillation of a pendulum. 

Classification of Oscillatory Motion:

1. Simple Harmonic Motion.(SHM)
2. Forced Harmonic Motion.
3. Damped Harmonic Motion.

Oscillation:

Oscillation is a process of repeating verities of any measure about its equilibrium value in time.

Simple Harmonic Motion:
 
Definition:

If a particle repeatedly oscillate left and right with a constant amplitude and its force not only proportional but also opposite to the direction of the displacement about its equilibrium value in time is called  simple harmonic motion.

If   "y"  be the displacement of an object performing SHM due to force "F" acting on the object then

F=-ky eq-1
where "k" is proportional constant and "-" sign indicate the direction of  "F" is opposite to the displacement.
Again according to newtons law,

F=ma eq2

where "m" means mass and "a" means acceleration  of the object.
from eq-1 and eq-2 we get,

y=Asin(⍵t+Ჶ)

Ჶ means phase constant and sometime phase difference.

How to find phase difference:

  

we know that wave length is  2 𝜫

=>𝛌=2 𝜫

so Ჶ=2 𝜫Δx/𝛌

Energy of simple harmonic motion:

kinetic energy Ek=1/2mv^2

Potential energy U=1/2kx^2

Mechanical energy E=U+k

Example of simple harmonic motion:

• Torsional Pendulum 
• LC Circuits 

                                                                  Torsional pendulum:

A body hanged by a wire which twist first in a direction and then  reverse direction is called torsional pendulum .

𝝉= -kθ

𝝉 means torsional force 

and k is a proportional constant and 𝜃 means angel of a torsional pendulum.

 According to newtons law,                                                                  Torsional pendulum 

𝝉= Iα 

 so Iα =-kθ

which is similar to simple harmonic motion equation.

so we can say that,

𝜭 = 𝜭⦁ cos(⍵t+𝞿)

So, T=2 𝜫/⍵

and ⍵^2=k/I

T= 2 𝜫√(I/K)

                                                                 LC Circuits 

A circuits with inductance and conductor is call LC circuits.  L means inductance of inductor and C means capacitance of capacitor.

Again according to capacitor equation we know that ,

V=Q/C

Q means charge store in capacitor and C means capacitance of capacitor 

According to Kirchhoff's 2nd law sum of the potential drops across the various component of a close circuits loop is equal to zero. 

 so,

which is similar to simple harmonic motion equation.

so we can say that,

Q = Q⦁ cos(⍵t+𝞿)

And Time period,

T=2 𝜫/⍵

T= 2 𝜫√(LC)

                                                   composition of simple harmonic motion

Case ii