Question #43063

A mechanical oscillator consists of a block of mass $m = 3 \text{ kg}$ attached to a spring of stiffness $k = 4 \text{ N/m}$ and a damper with damping coefficient $b = 3 \text{ N s/m}$ on a smooth horizontal surface. It is driven by a horizontal force $F(t) = 25 \sin(2t) \text{ N}$ (where $t$ is in seconds) and reaches a steady state. The instantaneous power $P(t)$ delivered by the driving force can be uniquely decomposed as $P(t) = P_{\text{active}}(t) + P_{\text{reactive}}(t)$, where $P_{\text{active}}(t)$ has a non-zero time average and $P_{\text{reactive}}(t)$ is a purely sinusoidal function with a zero time average. At the first instant $t > 0$ when the magnitude of the reactive power is maximum, a lump of putty of mass $M = 5 \text{ kg}$ is dropped vertically onto the block and sticks to it. What is the loss in the total mechanical energy of the oscillator during this collision?