Two spheres P and Q of equal radii have densities `rho_1` and `rho_2`, respectively. The spheres are connected by a massless string and placed in liquids `L_1` and `L_2` of densities `sigma_1` and `sigma_2` and viscosities `eta_1` and `eta_2`, respectively. They float in equilibrium with the sphere P in `L_1` and sphere Q in `L_2` and the string being taut(see figure). If sphere P alone in `L_2` has terminal velocity `vecV_p` and Q alone in `L_1` has terminal velocity `vecV_Q`, then

Options

A. `|(vecV_(P))/(vecV_(Q))|=eta_(1)/eta_(2)`
B. `|(vecV_(P))/(vecV_(Q))|=eta_(2)/eta_(1)`
C. `vecV_P. vecV_Q gt 0`
D. `vecV_P. vecV_Q lt 0`

Answer

A, D

Explanation

Force diagram of P and Q for equilibrium is shown in figure. It involves buoyant force, tension and weight of the spheres.

Equilibrium of P : `V sigma_(1) g=T+Vp_(1)g rArr T=V(sigma-p_(1))g ..........(i)`
Equilibrium of Q:
`T+V sigma_(2) g=Vp_(2)g rArr T=V(p_(2)-sigma_(2))g .........(ii)`
String is taut hence `T gt 0` as we can infer from equations (i) and (ii) that `sigma_(1) gt p_(1) and p_(2) gt sigma_(2)`. Using equations (i) and (ii) we can write the following
`siga_(1)-p_(1)=p_(2)-sigma_(2) ..........(iii)`