PHYS212 : Homework


Homework Assignment 3 (due 2/9)

More coupled oscillators

  1. (a) Use a spreadsheet and the recursion relation from lecture

    \begin{displaymath} y_b = \left( 2-\frac{\omega^2 m \ell}{T_o} \right) y_{b-1} - y_{b-2} \end{displaymath}

    to find the first four transverse normal modes of a beaded string with $n=10$, $m = 1.0\mathrm{e}{-2}$ kg, $L = 0.55$ m ($\ell = 0.05$ m), and $T_o = 2.0$ N. Here is a suggested format for your spreadsheet:
      A B
    1 $\omega$ [rad/s] = (guess)
    2 m [kg] = 0.01
    3 $\ell$ [m] = 0.05
    4 T [N] = 2.0
    5 x [m] A [m]
    6 0 0
    7 0.05 1
    8 0.1 =(2-$B$1^2*$B$2*$B$3/$B$4)*B7-B6
    9 0.15 (fill down)
    ...
    17 0.55 (very small)
    Adjust the value of $\omega$ to find values giving zero displacement at the right boundary (B17 = $y_{11} \approx 0$). Either use the ``Goal Seek'' tool of your spreadsheet (to adjust the value of cell B1 to make the value of cell B17 zero) or adjust $\omega$ by hand. If you take the latter approach, refine your frequency values to at least four digits. In practice, you won't get precisely zero. Instead, adjust $\omega$ to make the value of cell B17 very small. As you find each mode, record the frequency and save a graph of $A$ vs. $x$. (If you use an XY scatter plot with symbols connected by straight lines, you'll get a diagram of the system at its maximum displacement.) Setting B7 = $y_1$ = 1 gets the process started. It is only important to put a non-zero value here. The size of the overall amplitude pattern is directly proportional to this value. Hand a hard copy of your graphs.

    (b) Compare the normal mode frequencies you find with the theoretical prediction

    \begin{displaymath} \omega_M = 2 \sqrt{\frac{T_o}{mL}} \sin \left( \frac{M \pi}{2(n+1)} \right), \end{displaymath}

    where $M$ is the mode number.

    \scalebox{0.7}{ \includegraphics{nSprings.eps} }
  2. (a) Find an expression for the frequencies of the longitudinal normal modes of oscillation of a system of $n$ beads of mass $m$ connected by massless springs of strength $k$ by referring to lecture notes and drawing parallels between this system and the transverse modes of a beaded string covered in class. You may use the same ``enlightened guess'' of the amplitude condition that we used for the transverse modes,

    \begin{displaymath} A_b = A \sin \left( \frac{M b \pi}{n+1} \right) \end{displaymath}

    where $M$ is the mode number.

    (b) Confirm that your results with $n = 2$ agree with your results from Homework Assignment 2.


Copyright © 2003-2009, Lewis A. Riley Updated Tue Apr 14 02:04:45 2009