All the springs in fig (a), (b) and (c) are identical, each one having force constant K. Mass m is attached to each system. If T_a , T_b and T_c are the periodic time of oscillations of the three systems in fig (a), (b) and (c) respectively, then

Step-by-step solution

For figure (a), the equivalent spring constant is K_a = K . The time period of oscillation is T_a = 2 m K . For figure (b), the two springs are connected in series. The equivalent spring constant is K_b = K K K + K = K 2 . The time period is T_b = 2 m K/2 = 2 2m K = 2 ( 2 m K ) = 2 T_a . For figure (c), the two springs are connected in parallel. The equivalent spring constant is K_c = K + K = 2K . The time period is T_c = 2 m 2K = 1 2 ( 2 m K ) = T_a 2 . From the expression for T_c , we can write T_a = 2 T_c . Subs