Question

5. Consider the system illustrated in the figure. The pulley with radius R and moment of inertia I around its fixed axis is mounted on a frictionless axle which is fixed to the table. Th has one end fixed to the table, and the other end is attached to a massless inextensible rope. Th and a mass m hangs at its other end. Initially, the system is at rest e spring with stiffness constant k e rope passes over the pulley (a) (5 Pts.) How much is the spring stretched from its relaxed (natural) length when the system is at rest in equilibrium? (b) (15 Pts.) If the hanging mass is slightly displaced downward and then released, the system will oscillate around the equilibrium configuration. Find the frequency of small oscillations. Assume that the rope does not slide over the pulley, and the pulley can rotate without friction

Answer 1

In equilibrium condition when system is at rest, T_1 = T_2 and mg - T_1 = 0 t_2 = k delta X so mg = T_1 = T_2 = k delta X = > delta X = mg/k (where delta X is the stretch in the spring) delta X = mg/K Let any instant Total stretch in the spring = (X + delta x), then k (X + delta X) - T_2) = 0 (1) (T_2 - T_1) R = -I alpha = -Ia/R T_2 - T_1 = -Ia/R^2 (2) = > T_2 - T_1 = -Ia/R^2 and T_1 - mg = -ma (3) By adding equations (1), (2) and (3) we get kX + k delta X - mg = -(m + I/R^2) a = > a = -[k/m + I/R^2] times [negative sign of a is due directional X and a] OR a = -w^2 X where w = squareroot k/m + I/R^2 So frequency = f =