A particle undergoes simple harmonic motion driven by the restoring...

The time period of a particle in simple harmonic motion attached to a spring of spring constant $k$ is given by $T = 2\pi \sqrt{\dfrac{m}{k}}$.

The spring constant of a spring is inversely proportional to its length, so $k \propto \dfrac{1}{l}$.

When the spring is cut into two equal parts, the length of each part becomes $l' = \dfrac{l}{2}$.

The new spring constant for one part is $k' = 2k$.

The new time period $T'$ is given by $T' = 2\pi \sqrt{\dfrac{m}{k'}} = 2\pi \sqrt{\dfrac{m}{2k}}$.

Substituting the initial time period, we get $T' = \dfrac{T}{\sqrt{2}}$.