Solution:

Cutting a spring of constant \(k\) into 3 equal parts increases each part's spring constant to \(3k\). In a parallel connection, the equivalent constant is \(k_{\text{eq}} = 3k + 3k + 3k = 9k\). The new time period is \(T' = 2\pi\sqrt{\frac{m}{9k}} = \frac{T}{3}\).