Question #43776

A particle of mass $m$ (in $\text{kg}$) moves in a horizontal $x$-$y$ plane under the influence of a conservative central attractive force $\vec{F}_c = -f(r) \hat{r}$ (in $\text{N}$) directed towards the origin $O$. The region is filled with a viscous medium that exerts a drag force on the particle given by $\vec{F}_d = -\frac{\beta}{r^2} \vec{v}$, where $\beta$ is a positive constant (in $\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-1}$), $r$ is the distance from the origin (in $\text{m}$), and $\vec{v}$ is the velocity vector of the particle (in $\text{m/s}$). At time $t=0$, the particle is at a distance $r_0$ (in $\text{m}$) from the origin and is projected with a velocity $v_0$ (in $\text{m/s}$) perpendicular to its position vector. The particle eventually loses all its transverse velocity. Assuming the particle does not reach the origin before its transverse velocity vanishes, what is the total angle $\Delta \theta$ (in $\text{rad}$) swept by the position vector of the particle from $t=0$ until its motion becomes purely radial?