Newton's Law

• The slope of the string at each end of the section is given by
  

  $$_1 = \frac{\partial y_1}{\partial x} = \tan\theta_1$$

  $$_2 = \frac{\partial y_2}{\partial x} = \tan\theta_2$$

  
  

• If the displacement of the string from equilibrium is small, then angles $\theta_1$ and $\theta_2$ are small, and
  $$\sin\theta_1 \approx \tan\theta_1$$   and$$\quad \sin\theta_2 \approx \tan\theta_2$$

• The net vertical force can therefore be written as
  $$F_n = T\sin\theta_2-T\sin\theta_1=T\left(\mbox{m}_2 - \mbox{m}_1\right) = T\Delta \mbox{m}.$$

• By Newton's second law, $F=ma$,
  

  $$T\Delta = (\mu\Delta x) \left(\frac{\partial^2y}{\partial t^2}\right)$$

  $$T\frac{\Delta \mbox{m}}{\Delta x} = \frac{\mu\partial^2y}{\partial t^2},$$

  
  where acceleration is given by $a = \partial^2y/\partial t^2$.