Question #42609

A spherical particle of mass $m = 2 \times 10^{-3} \text{ kg}$ and radius $r = 1 \text{ mm}$ is placed in a conservative potential well $U(x) = \alpha x^2$, where $\alpha = 0.025 \text{ J/m}^2$. The well is filled with a viscous liquid of viscosity $\eta = \frac{2}{\pi} \text{ Pa}\cdot\text{s}$. At $t=0$, the particle is displaced to $x = x_0$ and released from rest. The ensuing displacement of the particle is modeled as $x(t) = A e^{-\gamma t} \cos(\omega_d t + \phi)$, where $\omega_d$ is the damped angular frequency and $\phi \in (-\frac{\pi}{2}, \frac{\pi}{2}]$. Assuming Stokes' law is valid, what is the value of the phase constant $\phi$?