Figures below show four different spring arrangements. If the mass in each arrangement is displaced from its equilibrium position and released, what is the resulting frequency of vibration in each case ? Neglect the mass of the each spring. (See Fig).

Explanation

Fig.(a) and (b) represents springs in parallel.
Let x be the extension produced in each spring. The total restoring force is mm F. Let `k_1 and k_2` be the spring constants
`F=-k_1x-k_2x`
`= -(k_1+k_2)x=-kx`
where `k = k_1+k_2`
The period of oscillation
`T=2pisqrt(m/k)`
`= 2pisqrt(m/(k_1+k_2))`
Fig (c) and (d) represent springs in series

Let `x_1 and x_2` be the extension in the the two springs. The restoring force acting in the two spring is the same. [Note Instead of `x_1` & `x_2,y_1 & y_2` could also be used]
So `F=-k_1x_1=-k_2x_2`
`x_1=-F/k_1,x_2=-F/k_2`
Total extension `x = x_1+x_2`
`=-F/k_1-F/k_2=-F[1/k_1+1/k_2]`
`= - F[(k_2+k_1)/(k_2k_1)]`
`F=-((k_1k_2)/(k_1+k_2))x=-kx`
where `k = (k_1k_2)/(k_1+k_2)`
Period of oscillation `=T= 2pisqrt(m/k)=2pisqrt((m(k_1+k_2))/(k_1k_2))`