A mass M attached to a spring oscillates with a period of 2 s. If the mass is increased by 2kg, the period increases by one second. The initial mass M (in kg) is ______.

We know that for a spring block system $\mathrm{T}=2\mathrm{\pi }\sqrt{\left(\frac{\mathrm{M}}{\mathrm{k}}\right)}$

Here k = spring constant.

$\text{ In first case, }2=2\mathrm{\pi }\sqrt{\left(\frac{\mathrm{M}}{\mathrm{k}}\right)}$  ……….(i)

$\text{ In second case, }3=2\mathrm{\pi }\sqrt{\left(\frac{\mathrm{M}+2}{\mathrm{k}}\right)}$ ………(ii)

Squaring of Eqs. (i) and (ii) and then dividing (ii) by (i), we have

$\frac{9}{4}=\frac{\mathrm{M}+2}{\mathrm{M}}=1+\frac{2}{\mathrm{M}}$

Solving we get M = 1.6 kg.