PHYS212 : Homework

## Homework Assignment 3 (due 2/9)

More coupled oscillators

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

to find the first four transverse normal modes of a beaded string with ,  kg,  m ( m), and  N. Here is a suggested format for your spreadsheet:
1 2 3 A B [rad/s] = (guess) m [kg] = 0.01 [m] = 0.05 T [N] = 2.0 x [m] A [m] 0 0 0.05 1 0.1 =(2-$B$1^2*$B$2*$B$3/$B$4)*B7-B6 0.15 (fill down) ... 0.55 (very small)
Adjust the value of to find values giving zero displacement at the right boundary (B17 = ). 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 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 to make the value of cell B17 very small. As you find each mode, record the frequency and save a graph of vs. . (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 = = 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

where is the mode number.

2. (a) Find an expression for the frequencies of the longitudinal normal modes of oscillation of a system of beads of mass connected by massless springs of strength 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,

where is the mode number.

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

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