Lecture 1 [03 Feb, 2020] PHY 115

 PHY 115 

Wave and Oscillation 

Syllabus:
• Simple Harmonic Motion 
• Different types pf Pendulum
• Spring-mass System
• Composition of two SHM 
• Damped Harmonic Motion
• Wave Motion 
• Forced Harmonic Motion 
• Progressive and Stationary waves

 Books: 
• Waves and oscillation 
• Physics for engineering 
• Fundamental of Physics  

  

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    (1) Oscillation is the to and fro motion of the particle about their mean position.
    (2) Wave is the disturbance that transfers energy from one place to another.
    (3) When the frequency of oscillation is high and amplitude is low is called vibration.

Simple Harmonic Motion (SHM)

Hook's Law:  $F \propto - x$  

It is called SHM because the power of x is 1. It is a linear equation.

    Differential equation for SHM:

From Hook's Law,  $$\begin{align*} & F \propto - x \\ \implies & F = - kx ~~~~~~~~~~~~ \bigg| ~ [k = \textrm{spring constant}] \\ \implies & ma = - kx \\ \implies & m \frac{d^2 x}{d t^2} = - kx \\ \implies & m \frac{d^2 x}{d t^2} + kx = 0 \\ \implies & \frac{d^2 x}{d t^2} + \frac{k}{m }x = 0 ~~ \bigg|~[ \omega^2 = \frac{k}{m}] \\ \implies & \frac{d^2 x}{d t^2} + \omega^2 x = 0 \end{align*}$$  

    The solution of differential equation:

  $$\begin{align*} & \frac{d^2 x}{d t^2} + \omega^2 x = 0 \tag{1}\label{eq:1}\\ \textrm{ Multiply eq. (1) by } 2 \frac{dx}{dt}, \\ & 2 \frac{dx }{dt} \frac{d^2 x }{d t^2} + \omega^2 2 \frac{dx}{dt} x = 0 \end{align*}$$  

Integrating with respect to time,

  $$\begin{align*} & \left(\frac{dx}{dy}^2\right) + \omega^2 x^2 = C \tag{2}\label{eq:2} \\ \textrm{When,}~~~~~~~~~~~ & x = A, ~~ \frac{dx }{dt} = 0 \\ \textrm{ From eq (2),}~~ & 0^2 + \omega^2 A^2 = C \\ \implies & C = \omega^2 A^2 \end{align*}$$  

By substituting the value of C,

  $$\begin{align*} & \left({dx }{dt}\right)^2 + \omega^2 x^2 = \omega^2 A^2 \\ \implies & \left({dx }{dt}\right)^2 = \omega^2 ( A^2 - x^2) \\ \implies & \frac{dx }{dt} = \pm ~\omega \sqrt{ A^2 - x^2} \tag{3}\label{eq:3} \\ \implies & \frac{dx}{\sqrt{A^2 - x^2}} = \omega~dt \end{align*}$$  

By integrating,

  $$\begin{align*} & \sin^{- 1} \frac{x}{A} = \omega t + \delta_1 \\ \implies & \frac{x}{A} = \sin ( \omega t + \delta_1) \\ \implies & x = A \sin ( \omega t + \delta_1) \end{align*}$$  

From equation (3),

  $$- \frac{dx}{\sqrt{A^2 - x^2}} = \omega~dt$$  

By integrating,

  $$\begin{align*} & \cos^{- 1} \frac{x}{A} = \omega t + \delta_2 \\ \implies & \frac{x}{A } = \cos ( \omega t + \delta_2) \\ \therefore ~ & x = A \cos ( \omega t + \delta_2) \end{align*}$$