A rod AB of mass M is attached as shown in figure to a spring of constant k. A small block of mass mis placed on the rod at its free end A. In equilibrium, rod is horizontal, i.e., spring is stretched.(a) If end A is moved down through a small distance d and released, determine the period of vibration, (b) Determine the largest allowable value of d if the block m is to remain at all times in contact with the rod.

A rod AB of mass M is attached as shown in figure to a spring of const

1.	A rod AB of mass M is attached as shown in figure to a spring of constant k. A small block of mass mis placed on the rod at its free end A. In equilibrium, rod is horizontal, i.e., spring is stretched.(a) If end A is moved down through a small distance d and released, determine the period of vibration, (b) Determine the largest allowable value of d if the block m is to remain at all times in contact with the rod.
Answer» Solution :Let `theta` be the angular displacement of the rod when block is pushed by a small displacement X. If `x_0` is the extension of the SPRING here , `x_0 = l theta and x=) = btheta` RESTORING FORCE in the spring = `kx_0` Restoring torque `tau= (Kx_0) b = Kb^2 theta` But `tau = I ALPHA ` where `I = ((Ml^2)/(3) + mL^2) (M/3 + m) l^2 alpha = - kb^2 theta` `alpha = - {(Kb^2)/(M/3 + m)l^2}theta = 2pi sqrt(alpha/theta)`