Find the period of small vertical oscillations of a body with mass `m` in the system illustrated in figure. The stiffness values of the springs are `x_(1)` and `x_(2)`, their masses are negligible.

Explanation

During the vertical oscillation let us locate the bock at a vertical down distance `x` from its equilibrium position . At his moment if `x_(1)` and `x_(2)` are the additional or further elongation of the upper `&` lower springs relative to the equilibrium position, then the net unbalanced force on the block will be `k_(2) x_(2)` directed in upward direction. Hence
`-k_(2)x_(2)=m ddot(x) ............(1)`
we also have `x=x_(1)+x_(2) ......(2)`
As the springs are massless and initially the net force on the spring is also zero so for the spring
`k_(1)x_(1)=k_(2)x_(2).......(3)`
Solving the Eqns `(1), (2)` and `(3)` simultaneoulsy, we get
`-(k_(1)k_(2))/(k_(1)+k_(2))x=m ddot(x)`
Thus`ddot(x)=-((k_(1)k_(2)//k_(1)+k_(2)))/(m)x`
Hence the sought time period `T=2pisqrt(m(k_(1)+k_(2))/(k_(1)k_(2)))`