AIIMS 2019 Physics Damped Oscillations MCQ Question

In damped forced oscillations, the energy of the system decreases exponentially over time, described by the equation $E(t) = E_0 e^{-bt/m}$. To find the time $t$ when the energy becomes half of its initial value, we set $E(t) = \frac{1}{2} E_0$ and solve for $t$, leading to the expression $t = \frac{m}{b} \ln 2$. This matches option D, which correctly represents the relationship between mass $m$, damping constant $b$, and the natural logarithm of 2. The other options are incorrect as they do not accurately reflect the mathematical derivation or the exponential decay behavior of energy in damped oscillations.