(Lateral vibration of a bead on a string) Consider a mass m, such as a bead, restrained by...

(Lateral vibration of a bead on a string) Consider a mass m, such as a bead, restrained by strings (of negligible mass), in each of which there is a tension tq, as shown in Fig. a.

We seek the frequency of small lateral oscillations of m. A lateral displacement x (Fig. b) will cause the length of each string to increase from l0 to l(x) = a/(02 + x2. Suppose that the tension r is found, empirically, to increase with I, from its initial value tq, as shown in Fig. c.

Linearize the equation of motion by retaining only the leading term of that Taylor series, show that the equivalent spring stiffness is keq = 2tq/Io, and that the frequency of