Question #42628

A block of mass $m \text{ (kg)}$ is attached to a spring of constant $k \text{ (N/m)}$ and oscillates horizontally in a thermally insulated container filled with a viscous liquid. The total heat capacity of the block and the liquid is $C \text{ (J/K)}$. The damping force on the block is $F = -b v$, where $v \text{ (m/s)}$ is the velocity. The damping coefficient $b \text{ (kg/s)}$ depends on the temperature increase $\Delta T \text{ (K)}$ of the system as $b = b_0 (1 + \alpha \Delta T)$, where $b_0 \text{ (kg/s)}$ and $\alpha \text{ (K}^{-1}\text{)}$ are positive constants. Assume all dissipated mechanical energy is entirely converted into internal energy, raising the system's temperature uniformly. The initial mechanical energy is $E_0 \text{ (J)}$. Given that $\alpha E_0 = C$, find the time $t \text{ (s)}$ taken for the mechanical energy to reduce to $E_0 / 2$. Assume the oscillator is lightly damped such that the average rate of energy loss over a cycle is $\langle \frac{dE}{dt} \rangle = -\frac{b}{m} E$.