The time period ($T$) of a spring-mass system oscillating in Simple Harmonic Motion (SHM) is derived from the restoring force $F = -kx$ . The period is given by the formula: $T = 2\pi \sqrt{\frac{m}{k}}$ where $m$ is the mass and $k$ is the spring constant.

1. Initial Condition: With mass $M$, the period is: $T = 2\pi \sqrt{\frac{M}{k}}$

2. New Condition: When another mass $M$ is suspended along with the first, the total mass becomes $m' = M + M = 2M$. The new period $T'$ is: $T' = 2\pi \sqrt{\frac{2M}{k}}$

3. Comparison: $T' = \sqrt{2} \left( 2\pi \sqrt{\frac{M}{k}} \right) = \sqrt{2}T$

Therefore, the new period of oscillation is $\sqrt{2}T$.