A body is suspended from the two light springs separately. The periods of vertical oscillations are T and 3T respectively. The same body is suspended from two springs connected first in parallel and then in series. The periods of oscillation in both cases, respectively, are

Step-by-step solution

The time period of a spring-mass system is given by T = 2 m k , which implies k = 4 ^2 m T^2 . For the first spring: k₁ = 4 ^2 m T^2 For the second spring: k₂ = 4 ^2 m (3T)^2 = 4 ^2 m 9T^2 When the springs are connected in parallel, the equivalent spring constant is k_p = k₁ + k₂ . 1 T_p^2 = k_p 4 ^2 m = k₁ + k₂ 4 ^2 m = 1 T₁^2 + 1 T₂^2 1 T_p^2 = 1 T^2 + 1 9T^2 = 10 9T^2 T_p = 3T 10 When the springs are connected in series, the equivalent spring constant is given by 1 k_s = 1 k₁ + 1 k₂ . T_s^2 = 4 ^2 m k_s = 4 ^2 m