A tray of mass \(12 \mathrm{~kg}\) is supported by two identical springs as shown in figure. When the tray is pressed down slightly and then released, it executes SHM with a time period of \(1.5 \mathrm{~s}\). The spring constant of each spring is

Mass of tray, \(m=12 \mathrm{~kg}\)
Time period, \(T=1.5 \mathrm{~s}\)
If \(k\) be the spring constant of each spring, then
\(k_{1}=k_{2}=k\)
Since, springs are connected in parallel, hence

\(k_{\text {net }}=k+k \)
\( \text {Time period, } T=2 \pi \sqrt{\frac{m}{k_{\text {net }}}}=2 \pi \sqrt{\frac{m}{k_{1}+k_{2}}} \)
\( \Rightarrow 1.5=2 \pi \sqrt{\frac{12}{k+k}} \Rightarrow 1.5=2 \pi \sqrt{\frac{12}{2 k}} \)
\( \Rightarrow 1.5=2 \pi \sqrt{\frac{6}{k}}\)
Squaring both side, we have
\(2.25 =4 \pi^{2} \cdot \frac{6}{k} \)
\( \Rightarrow k =\frac{4 \pi^{2} \times 6}{2.25} \)
\( =105.17 \mathrm{Nm}^{-1} \simeq 105 \mathrm{Nm}^{-1}\)