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4.6 M ll m m X2 X3 The figure shows two identical masses of mass m connected to a third mass of mass M by two identical springs of spring constant k. Consider vibrations of the masses along the line joining their centres where x1, x2 and x3 are their respective displacements from equilibrium. (a) Without any mathematical detail, use your physical intuition to deduce the normal frequency for symmetric-stretch vibrations. (b) Show that the equations of motion of the three masses are: d²x1 + wfx1 – wjx2 = 0, %3D dz2 dx2 - žx1 + 203x2 – wžx3 = 0 dz2 and d²x3 - wfx2 + w}x3 = 0, %3D dz2 where w = k/m and w, = k/M. (c) Show that the normal frequencies of the system %3D

4.6 M ll m m X2 X3 The figure shows two identical masses of mass m connected to a third mass of mass M by two identical springs of spring constant k. Consider vibrations of the masses along the line joining their centres where x1, x2 and x3 are their respective displacements from equilibrium. (a) Without any mathematical detail, use your physical intuition to deduce the normal frequency for symmetric-stretch vibrations. (b) Show that the equations of motion of the three masses are: d²x1 + wfx1 – wjx2 = 0, %3D dz2 dx2 - žx1 + 203x2 – wžx3 = 0 dz2 and d²x3 - wfx2 + w}x3 = 0, %3D dz2 where w = k/m and w, = k/M. (c) Show that the normal frequencies of the system %3D

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4.6 k lell M т m X1 X2 X3 The figure shows two identical masses of mass m connected to a third mass of mass M by two identical springs of spring constant k. Consider vibrations of the masses along the line joining their centres where x1, x2 and x3 are their respective displacements from equilibrium. (a) Without any mathematical detail, use your physical intuition to deduce the normal frequency for symmetric-stretch vibrations. (b) Show that the equations of motion of the three masses are: d²x1 + wjx1 – w}x2 = 0, dr2 d?x2 - wž1 + 203x2 – wžx3 dr2 and d?x3 wix2 + w}x3 = 0, dr2 where w; = k/m and w, = k/M. (c) Show that the normal frequencies of the system are k/m and VK(2m + M)/Mn. (d) Determine the ratio of normal frequencies for m/M = 16/12 and compare with the vibrational frequencies of the CO2 molecule given in the text.
Transcribed Image Text:4.6 k lell M т m X1 X2 X3 The figure shows two identical masses of mass m connected to a third mass of mass M by two identical springs of spring constant k. Consider vibrations of the masses along the line joining their centres where x1, x2 and x3 are their respective displacements from equilibrium. (a) Without any mathematical detail, use your physical intuition to deduce the normal frequency for symmetric-stretch vibrations. (b) Show that the equations of motion of the three masses are: d²x1 + wjx1 – w}x2 = 0, dr2 d?x2 - wž1 + 203x2 – wžx3 dr2 and d?x3 wix2 + w}x3 = 0, dr2 where w; = k/m and w, = k/M. (c) Show that the normal frequencies of the system are k/m and VK(2m + M)/Mn. (d) Determine the ratio of normal frequencies for m/M = 16/12 and compare with the vibrational frequencies of the CO2 molecule given in the text.
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