4.6MllmThe figure shows two identical masses of mass m connected to a third mass of mass Mby two identical springs of spring constant k. Consider vibrations of the masses alongthe line joining their centres where x1, x2 and x3 are their respective displacementsfrom equilibrium. (a) Without any mathematical detail, use your physical intuitionto deduce the normal frequency for symmetric-stretch vibrations. (b) Show that theequations of motion of the three masses are:+ wjx – wfx2 = 0,dr2d'x2- ažn + 20žx2 – ožra = 0andd'x3- wx2 + wjxs = 0.where w = k/m and wž = k/M. (c) Show that the normal frequencies of the systemare K/m and K(2m + M)/Mn. (d) Determine the ratio of normal frequencies form/M = 16/12 and compare with the vibrational frequencies of the CO2 moleculegiven in the text.

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A pendulum of length L=6 m and mass M=1 kg has a spring of force constant k=30 N/m connected to it at a distance h=0.3m below its point of suspension. Find the frequency of vibration of the system for small values of the amplitude ( for small angles, sine=0 , cose =1)). Assume the vertical suspension of length is rigid, but ignore its mass. State your answer in Hz to the nearest 0.01 (Use g=9.8m/s²) L www k M Your Answer:
The quantities A and φ (called the amplitude and the phase) are undetermined by the differential equation. They are determined by initial conditions -- specifically, the initial position and the initial velocity -- usually at t = 0, but sometimes at another time. In the oscillating part of the experiment, I measured only the time of 30 periods. I measured no position or velocity. Consequently, A and φ (and also y0) are irrelevant in the problem. We only compare the period T or the frequency ω with the theoretical prediction. You have (hopefully) derived (or maybe looked up) the relation between ω and k and m. This final question relates ω and T. If ω = 8.2*102 rad/s, calculate T in seconds. (Remember, that a radian equals one.) T might be a fraction of a second.
The quantities A and p (called the amplitude and the phase) are undetermined by the differential equation. They are determined by initial conditions -- specifically, the initial position and the initial velocity -- usually at t = 0, but sometimes at another time. In the oscillating part of the experiment, I measured only the time of 30 periods. I measured no position or velocity. Consequently, A and p (and also yo) are irrelevant in the problem. We only compare the period T or the frequency w with the theoretical prediction. You have (hopefully) derived (or maybe looked up) the relation between w and k and m. This final question relates w and T. If w = 5.8*10° rad/s, calculate T in seconds. (Remember, that a radian equals one.) T might be a fraction of a second.
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one-dimensional crystal assembled from three identical atoms connected to one another and to the walls by identical springs as shown.find the normal modes of given system.plot relative displacements versus time for all three normal modes.
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