Question 15.11: Find the normal frequencies for a linear, asymmetric molecul...
Find the normal frequencies for a linear, asymmetric molecule with the shape in Fig. 15.18.
Conservation of the center of gravity and of angular momentum now read
m_{A}x_{1} +m_{B}x_{2} +m_{C}x_{3} = 0, x-center of gravity,
m_{A}y_{1} +m_{B}y_{2} +m_{C}y_{3} = 0, y-center of gravity,
m_{A}l_{1}y_{1} = m_{C}l_{2}y_{3}, angular momentum conservation.
For the potential energy of bending, we write
V = \frac{K_{2}}{2} (lδ)^{2}, (2l = l_{1} +l_{2});for that of rotation,
V = \frac{K_{2}}{2}(x_{1} − x_{2})^{2} + \frac{K^{′}_{1}}{2} (x_{2} −x_{3})^{2}.
The analogous calculation as for Exercise 15.9 after some effort yields
ω^{2}_{T} = \frac{K_{2}l^{2}}{l^{2}_{1} l^{2}_{2}} \left(\frac{l^{2}_{1}}{m_C} + \frac{l^{2}_{2}}{m_{A}} + \frac{4l^{2}}{m_{B}}\right)for the frequency of the transverse vibration, and also the equation quadratic in ω^{2}
ω^{4} − ω^{2} \left[K_{1} \left(\frac{1}{m_{A}}+ \frac{1}{m_{B}} \right) +K^{′}_{1} \left(\frac{1}{m_{B}} + \frac{1}{m_{C}}\right)\right] + \frac{μK_{1}K^{′}_{1}}{m_{A}m_{B}m_{C}}= 0for the frequencies ω_{L_{1}},ω_{L_{2}} of the two longitudinal vibrations.