If the numbers of coils per length of spring is constant tie

If the numbers of coils per length of spring is constant, tie stiffness of the spring is inversely proportional to the length, what is the ratio of natural sequencies in case 1 & case 2.

Solution

stiffness of spring is inversely proportional to length of coil.

so. K x l =C (constant)

stiffness of spring in case 1 = K (say)

so, stiffness in case 2 = 2K and 2K for each spring so, equivalent stiffness = 2K +2K = 4K

natural frequency of case1 = sqrt (K/m) natural frequency of case 2 = sqrt (4K/m)

ratio of natural frquencies = sqrt(K x m / m x 4K) =1/2