(Solved): (a) Two objects of equal mass \( m \) are attached to two opposing walls by two identical springs ...

(a) Two objects of equal mass \( m \) are attached to two opposing walls by two identical springs of spring constant \( 2 k \) and coupled by a third of spring constant \( k \) (as illustrated in the figure below). By explicit consideration of the forces on each object show that the equations of motion of the two objects are: \( m \ddot{x}=-3 k x+k y \) and \( m \ddot{y}=-3 k y+k x \). (b) (i) Explain how to excite the system to vibrate in each of the two normal modes. (ii) Describe the motion of the masses in the two normal modes. (iii) Explain why the lowfrequency mode is independent of the coupling spring.