Question #43817

A block of mass $m$ is attached to a spring of force constant $k$ and undergoes lightly damped oscillations on a horizontal frictionless surface. The damping force is given by $F_d = -b(t)v$, where $v$ is the velocity of the block. The damping coefficient $b(t)$ is linked to the activity of a radioactive sample such that $b(t) = b_0 e^{-\lambda t}$, where $\lambda$ is the decay constant of the sample. Assume that the damping is very light ($b_0 \ll \sqrt{mk}$) and the radioactive decay is very slow compared to the oscillation frequency ($\lambda \ll \sqrt{k/m}$). If the initial amplitude of oscillation at $t=0$ is $A_0$, what is the final amplitude of the block as $t \to \infty$?