Method Based on D'Alembert's Principle:

In dynamic condition the free body diagram of mass might be drawn as follows :

Figure 3

The free body diagram of mass is illustrated in above Figure 3. The force equation may be written as follows:

                          mx'+ mg = k ( x + Δ)                    . . . (4)

Incorporating equation. (1) in Eq. (4), the below relation is attained.

                                           mx' + kx = 0

This equation is similar as we attained earlier.