Time period of oscillation is given by $T=2\pi \sqrt{\frac{m}{k}}$, where $k$ is the force constant of the spring. 

$\because$ Spring is same, therefore, spring constant will be same.

Putting the different values,

In the first case          $T=1=2\pi \sqrt{\frac{m}{k}}$

In the second case    $T^{\text{'}}=2=2\pi \sqrt{\frac{m+3}{k}}$

 Dividing the equations, we get

 $\frac{T}{T^{\text{'}}}=\frac{1}{2}=\sqrt{\frac{m}{m+3}}$

$$\begin{gathered} \Rightarrow \frac{m}{m+3}=\frac{1}{4} \\ \Rightarrow 4m=m+3 \\ \Rightarrow m=1 \end{gathered}$$

Thus, $m=1 \mathrm{kg}$