Lecture 4 [02 March, 2020] PHY 115

    PHY 115

Wave and Oscillation

1. Horizontal oscillations of a spring:

Fig:

Restoring Force,  $F = - kx$  
From Newton's second law,  $$\begin{align*} F = & ma \\ \implies ma = & - kx \\ \therefore a = & - \frac{k}{m}x \end{align*}$$  

comparing with equation of SHM,  $$\begin{align*} a = & - \omega^2 x \\ \therefore \omega^2 \frac{k}{m} = & \frac{g}{dl} ~~~\bigg|~ T = 2\pi \sqrt{\frac{m}{k}} \end{align*}$$

    2. Vertical oscillation of a spring:

  $$\begin{align*} k~dl = mg \\ F = & k (dl - y) - mg \\ = & k (dl - y) - k~dl \\ = & - ky \\ \therefore F = - ky \end{align*}$$ $$T = 2\pi \sqrt{\frac{dl}{g}}$$  

    ♦ Parallel System:
  $$\begin{align*} & F_1 \ne F_2 ~~~~~~~~~~~~~~~~~ \bigg| F_1 = - k_1 y \\ & y_1 = y_2 = y ~~~~~~~~~~~\bigg|~ F_2 = - k_2 y \\ \\ \\ & ~~~~~~~~~~~~~ F = F_1 + F_2 \end{align*}$$  

    ♦ Series system:

$$\begin{align*} & F_1 = F_2 = F ~~~~~~~~~~~~~~~~ \bigg| F = - k_1 y_1\\ & y_1 \ne y_2 ~~~~~~~~~~~~~~~~~~~~~~~~~ \bigg| F = - k_2 y_2 \\ \\ \\ & ~~~~~~~~~~~~~~~~ y = y_1 + y_2 \end{align*}$$