b
(b) \(T = 2\pi \sqrt {\frac{m}{k}} \).

Also spring constant \((k) \propto \frac{1}{{{\rm{Length (}}L)}}\), when the spring is half in length, then \(k\) becomes twice.

\(\therefore T' = 2\pi \sqrt {\frac{m}{{2k}}} \)

\(\Rightarrow \frac{{T'}}{T} = \frac{1}{{\sqrt 2 }} \)

\(\Rightarrow T' = \frac{T}{{\sqrt 2 }}\)