A block of mass \(m\) on a frictionless horizontal surface is attached to a uniform ideal spring of force constant \(k\) and oscillates with period \(T_0\). The spring is then cut into two identical halves, and the same block is attached to one of the halves. When displaced and released, the block oscillates with a new period of \(T' = \dfrac{T_0}{\sqrt{2}}\). Which of the following statements best explains why the period decreases to this value?