\({k_1} = \frac{{Force}}{{Extension}} = \frac{{YA}}{L}\;\;\;\left( {\because Y = \frac{F}{A} \times \frac{L}{{\Delta L}}} \right)\)

Spring constant of the spring \({k_2} = K\)

Hence spring constant of the combination (series) \({k_{eq}} = \frac{{{k_1}{k_2}}}{{{k_1} + {k_2}}} \)

\(= \frac{{(YA/L)K}}{{(YA/L) + K}}\)

\(= \frac{{YAK}}{{YA + KL}}\)

Time period \(T = 2\pi \sqrt {\frac{m}{k}} = 2\pi {\left[ {\frac{{(YA + KL)m}}{{YAK}}} \right]^{1/2}}\)