A block of mass M is suspended from a rigid support by two springs of spring constants k and 2k . In Case 1, the two springs are connected in series. In Case 2, the same two springs are connected in parallel. If f₁ and f₂ are the natural frequencies of vertical oscillation of the block in Case 1 and Case 2 respectively, then the ratio f₁ f₂ is:

Step-by-step solution

The natural frequency of a spring-mass system is given by f = 1 2 k_ eq M . In Case 1, the springs are connected in series. The equivalent spring constant is: k_ eq1 = k 2k k + 2k = 2k 3 In Case 2, the springs are connected in parallel. The equivalent spring constant is: k_ eq2 = k + 2k = 3k The ratio of their natural frequencies is: f₁ f₂ = k_ eq1 k_ eq2 = 2k/3 3k = 2 9 = 2 3 Answer: 2 3