A uniform rod BC has mass M and length 2a. The end B of the rod is connected to a fixed point A on a smooth horizontal table by an elastic string of strength α1, and the end C is connected to a second fixed point D on the table by a second elastic string of strength α2.
In equilibrium, the rod lies along the line AD with the strings having tension T0 and lengths b, c respectively. Show that the frequency of the longitudinal mode is ((α1 + α2)/M)1/2 and that the frequencies of the transverse modes satisfy the equation
b2c2µ2 - 2bc(2ab + 3bc + 2ac)µ + 6abc(2a + b + c) = 0.
where µ = Maω2/T0. [The calculation of Vapp is very tricky.]
Find the frequencies of the transverse modes for the particular case in which a = 3c and b = 5c.