four different springs arrangements. If the mass m in each arrangement is displaced from its equilibrium position and released, what is the resulting frequency of vibration ini each case? Neglect the mass of the spring. Figure (a) and (b) represent an arrangement of springs in parallel and (c) and (d) represent springs in series?

Explanation

In the arrangements shown in figure, (a) and (b) , the two springs are in parallel. The effective spring constant of each of these arrangements is , `K=k_(1)+k_(2)`
Frequency of oscillation,
`v=(1)/(2pi)sqrt((K)/(m))=(1)/(2pi)sqrt((k_(1)+k_(2))/(m))`
In the arrangements shown in figure , the two springs are in series. The effective spring constant K of each of these arrangements is
`(1)/(K)=(1)/(k_(1))+(1)/(k_(2))=(k_(2)+k_(1))/(k_(1)k_(2))or K=(k_(1)k_(2))/(k_(2)+k_(1))`
Frequency of oscillaltion,
`v=(1)/(2pi)sqrt((K)/(m))=(1)/(2pi)sqrt((k_(1)k_(2))/(m(k_(1)+k_(2))))`