Question #42334

A particle of mass $m = 2 \text{ kg}$ is bound in a one-dimensional potential $U(x) = \frac{1}{2} m \omega_0^2 x^2$ with natural frequency $\omega_0 = 5 \text{ rad/s}$. It experiences a viscous drag force $F_d = -b v$, where $b = 6 \text{ kg/s}$ and $v$ is its velocity. A transverse wave $y(z,t) = A \cos(kz - \omega t)$ with amplitude $A = 0.5 \text{ m}$ and frequency $\omega = 4 \text{ rad/s}$ propagates on a nearby string of linear mass density $\mu = 0.2 \text{ kg/m}$. The wave couples to the particle, exerting a driving force $F(t) = C \langle u \rangle \cos(\omega t)$, where $\langle u \rangle$ is the time-averaged energy density of the wave and $C = 50$ is a dimensionless coupling constant. In the steady state, what is the magnitude of the net force acting on the particle at the exact instant the driving force reaches its positive maximum?