A block of mass \(m\) attached to an ideal spring of spring constant \(k\) undergoes weakly damped simple harmonic motion. Due to a resistive force, the amplitude of oscillation as a function of time \(t\) is given by \(A(t) = A_0 e^{-\frac{b}{2m} t}\), where \(A_0\) is the initial amplitude and \(b\) is a positive damping constant. Which of the following expressions represents the time \(t_{1/2}\) required for the total mechanical energy of the block-spring system to decrease to half of its initial value?