Question

(30pt) Two blocks connected by a spring of spring constant k are free to slide friction- lessly along a horizontal surface, as shown in Fig. 1. The unstretched length of the spring is a ure 1: Two masses connected by a spring sliding horizontally along a frictionless surface. (a) (10pt) Identify a set of generalized coordinates and write the Lagrangian. (b) (10pt) Find the equations of motion. (c) (10pt) Find a complete solution to the equations of motion

Answer 1

Spring constant - k unstreched length of the spring - a rightarrow Two masses connected by a spring as shown in question part a: - If x - is the position of the right edge of block 'mu' u is the sketch of the spring. x + mu + a = position of the left edge of block 'M' Rightarrow legrangian (L) = K.E - P.E = 1/2 Mx^2 + 1/2 (x + u)^2 m = 1/2 ku^2 position = x + u + a velocity = x + u rightarrow In this a - constant (a = 0) Therefore L = 1/2 (M + m)x^2 + 1/2 mu^2 + mx^u^- 1/2 ku^2 part b: - equation of motion: - p_x = partial differential L/partial differential x p_x = (M + m) x^+ m u_1^ p_u = partial differential L/partial differential u^= m (u^+ x^) The legragian equation of motion d/dt (partial differential L/partial differential q_j) = partial differential L/partial differential q_j) [q_i = generated co-ordinate]