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. `|(v_(P))/(v_(Q))|=(eta_(1))/(eta_(2))`
B. `(vec v_(P))/(vec v_(Q))=(eta_(2))/(eta_(1))`
C. `vec v_(P).vec v_(Q)gt0`
D. `vec v_(P).vec v_(Q)lt0`

Answer

B, D

Explanation

As terminal velocity `v=(2r^(2)(rho-sigma)g)/(9eta)`
where `rho,sigma` be the density of spherical body of radius r and density of liquid respectively. `eta` is the coefficient of viscosity of liquid. Then
`v prop((rho-sigma))/(eta)`
As per question, if sphere P alone is in liquid `L_(2)` has terminal velocity `vec v_(P)` and Q alone in liquid `L_(1)` has terminal velocity `vec v_(Q)`, then
`(v_(P))/(v_(Q)) =((rho_(1)-sigma_(2))//eta_(1))/((rho_(2)-sigma_(1))//eta_(2)) =((rho_(1)-sigma_(2))//eta_(2))/((rho_(2)-sigma_(1))//eta_(1))` .....(i)
As the system is floating in equilibrium. Let `U_(P), U_(Q)` are the upthrust on P and Q respectively.
`U_(P)-W_(P) =T =W_(Q) -U_(Q)`
or `U_(P)+U_(Q)=W_(P)+W_(Q)`
or `sigma_(1) V_(g)+sigma_(2) V_(g) =rho_(1) V_(g)+rho_(2) V_(g)`
or `sigma_(1)+sigma_(2) =rho_(1)+rho_(2)` or `rho_(1)-sigma_(2) =sigma_(1) - rho_(2)` .....(ii)
From (i) and (ii) ,`(v_(p))/(v_(Q)) = (-eta_(2))/(eta_(1))` or `(|vec v_(P)|)/(|vec v_(Q)|) = (|eta_(2)|)/(eta_(1)|)`
`:.` option (b) is true.
As `vec v_(P)` and `vec v_(Q)` are of opposite sign
`:. vec v_(P).vec v_(Q) lt 0` . Thuse, option (d) is corrected.