Lecture 2 [10 Feb, 2020] PHY 115

 PHY 115

Wave and Oscillation

The solution of the differential equation of SHM: 
We know that, the differential equation of simple harmonic motion is,
  $x = A \sin (\omega t + \phi \tag{1}\label{eq:1})$  

    ♦ The Velocity:  $$\begin{align*} v = & \frac{dx}{dt} = a \cos (\omega t + \phi) \tag{2}\label{eq:2}\\ \implies v = & A \omega \sqrt{1 - \sin^2(\omega t + \phi)} \\ = & \pm A \omega \sqrt{1 - \left(\frac{x}{A}\right)^2} \\ = & \pm A \omega \sqrt{\frac{A^2 - x^2}{A^2}} \\ \therefore v = & \pm \omega \sqrt{A^2 - x^2} \end{align*}$$  

  $$\begin{align*} & v_{min} = 0,~~~ \textrm{when }~ x = a\\ & v_{max} = \omega A , ~~~ \textrm{when} ~ x = 0 \end{align*}$$  

    ♦ The Acceleration:  $$\begin{align*} a = & \frac{dv}{dt} = - a \omega^2 \sin (\omega t + \phi) \\ = & - \omega^2 x ~~\bigg| ~[x = A\sin (\omega t + \phi)] \tag{4}\label{eq:4} \\ \therefore a \propto & - x \end{align*}$$  

    ♦ Total Energy: Total Energy = Kinetic energy + Potential energy

Total energy of a body executing SHM:
    • Kinetic Energy: 
  $$\begin{align*} v = & \omega \sqrt{A^2 - x^2} = A \omega \cos (\omega t + \phi) \\ K.E = & \frac{1}{2} m v^2 = \frac{1}{2} m \omega^2 (A^2 - x^2) \tag{5}\label{eq:5} \\ K.E = & \frac{1}{2}m v^2 = \frac{1}{2} m A^2 \omega^2 \cos^2 (\omega t + \phi) \tag{6}\label{eq:6} \end{align*}$$  

    • Potential Energy:
  $$\begin{align*} F = & ma = - m \omega^2 x \\ F = & - m \omega^2 a \sin (\omega t + \phi) \\ & \\ P.E = & - \int_{0}^{x} F~ dx \\ = & - \int_{0}^{x} - m \omega^2 x ~ dx \\ = & \int_{0}^{x} m \omega^2 x ~ dx \\ = & m \omega^2 \int_0^x x ~dx \\ & \frac{1}{2} m \omega^2 x^2 \tag{7}\label{eq:7} \end{align*}$$  

  $$\therefore P.E = \frac{1}{2} m \omega^2 A^2 \sin^2 ( \omega t + \phi)$$ $$\begin{align*} \textrm{ Total Energy}~ = & ~\textrm{(5) + (7)} \\ = & \frac{1}{2} m \omega^2 ( A^2 - x^2) + \frac{1}{2} m \omega^2 x^2 \\ = & \frac{1}{2} m \omega^2 A^2 \\ = & \frac{1}{2}k A^2 ~~~~~~~~~~~~~ \bigg| \omega^2 = \frac{k}{m} \\ = & \frac{1}{2} m \left(\frac{2\pi}{T}\right)^2 A^2 \\ = & \frac{1}{2} m (2\pi n )^2 A^2 \\ = & 2 \pi^2 n^2 A^2 m \end{align*}$$  

    Assignment: Find out the average value of K.E and P.E of harmonic oscillator.