References

1

T.F. Zheng et al., Phys. Teach. 32, 248-251 (1994).

2

Bernard J. Weigman and Helene F. Perry, Am. J. Phys. 61, 1022-1027 (1993).

3

Keith R. Symon, Mechanics 3rd ed., (Addison-Wesley, Reading, MA, 1971), pp. 191-199.

4

Ultrasonic Motion Detector, Universal Lab Interface, and MACMOTION 4.0 software from Vernier Software Co., 2920 S.W. 89th St., Portland, OR 97225. EXCEL 4.0 software from Microsoft Corp., One Microsoft Way, Redmond, WA 98052.

5

Clifford E. Swartz, Phys. Teach. 31, pp. 544-545 and Centerfold (1993)

6

George Arfken, Mathematical Methods for Physicists, 3rd ed., (Academic Press, Orlando, 1985), pp. 760-791, William H. Press, et al., Numerical Recipes in C, 2nd ed., (Cambridge, 1992), pp. 496-536.

7

Clifford E. Swartz, Used Math, (AAPT, College Park, MD, 1993), pp. 214-217.

8

R. Kenneth Walter, Phys. Teach. 27, 173-175 (1989), Rick Guglielmino, ibid. 27, 175-178 (1989), Michael T. Frank and Edward Kluk, ibid. 28, 308-311 (1990), C.L. Enloe, Comput. Phys. 3 (1), 75-76 (1989), Marvin L. De Jong, ibid. 5 (1), 12-15 (1991).

9

Charles W. Misner and Patrick J. Cooney, Spreadsheet Physics (Addison-Wesley, Reading, MA, 1991), pp. 23-27.

10

Alejandro L. Garcia, Numerical Methods for Physics (Prentice Hall, Englewood Cliffs, NJ, 1994), pp. 48-50, Ian R. Gatland, Am. J. Phys. 62, 259-265 (1994).

11

Ref. 9 , pp. 57-58.

  
Figure 1: Experimental arrangement. The two masses are identical gliders on an airtrack, connected by springs. The sonic ranger was arranged to measure the displacement of the left-hand mass from its equilibrium position ( in the figure).

  
Figure 2: Upper panel shows the motion of the gliders in the antisymmetric normal mode, where they move equal and opposite amounts. The middle panel shows the symmetric normal mode, where the gliders move identically. The lower panel shows that the sum of these two modes (each at equal maximum amplitude A) leads to one glider displaced by 2A and the other at its equilibrium position, with both at rest.

  
Figure 3: Measured motion of glider 1 when the two normal modes were individually excited.

  
Figure 4: Measured motion of glider 1 when the two normal modes were both excited. The upper panel shows the displacement vs time for the glider, and the lower panel shows the Fourier transform of this data. The peaks in the FFT match the normal mode frequencies.

  
Figure 5: Heavy line is the experimental data from Fig. 4; light line is the simple theoretical model of Eq. (4), offset to make comparison easier.

  
Figure 6: Heavy line is the experimental data from Fig. 4; light line shows the results of numerical integration of Newton's Laws (including air resistance), offset to make comparison easier.

  
Table i: Frequencies of the two normal modes, once by exciting the individual mode and measuring the period, and then by a FFT of the superposition of both normal modes. Errors are discussed in the text.

  
Table ii: Parameters used in the numerical integration of Newton's Law for the coupled harmonic oscillator.