Question #42344

A small sphere of volume $V$ and density $3\rho$ is attached to a fixed center by a massless inextensible string of length $R$ and moves inside a vertical circular tube of mean radius $R$. The tube contains an incompressible ideal fluid of density $\rho$ flowing steadily. The tube's cross-section varies such that the fluid velocity is $v_f(\theta) = v_0 \sqrt{\cos\theta}$ for $\theta \in [0, \pi/2]$, where $\theta$ is the angle measured from the lowest point. Due to the drag force from the fluid, the sphere moves at a constant speed $v_s = \sqrt{2gR}$. Given $v_0 = 2\sqrt{gR}$, let $T$ be the string tension at $\theta = \pi/3$, and $W_D$ be the magnitude of the work done by the drag force on the sphere from $\theta = 0$ to $\theta = \pi/3$. Find the ratio $\frac{T}{W_D/R}$.