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. (a) `(|vecV_P|)/(|vecV_Q|)=(eta_1)/(eta_2)`
B. (b) `(|vecV_P|)/(|vecV_Q|)=eta_2/eta_1`
C. (c) `vecV_P.vecV_Qgt0`
D. (d) `vecV.vecV_Qlt0`

Answer

A, D

Explanation

From the figure it is clear that
(a) `sigma_2gtsigma_1`
(b) `rho_2gtsigma_2` [ As the string is taut]
(c) `rho_1ltsigma_1` [As the string is taut]
`:.rho_1ltsigma_1ltsigma_2ltrho_2`
When P alone is in `L_2`
`V_P=(2pir^2(rho_1-rho_2)g)/(9eta_2)` is negative as `rho_1ltsigma_2`
Where r is radius of sphere.
When Q alone is in `L_1`
`V_Q=(2pir^2(rho_2-sigma_1)g)/(9eta_1)` is positive as `rho_2gtsigma_1`
Therefore `vecV_P.vecV_QltO` option (d) is correct

Also `V_P/V_Q=(rho_1-sigma_2)/(rho_2-sigma_1)xxeta_1/eta_2` ..(i)
For equilibrium of Q
`T+4/3pir^3sigma_2g=4/3pir^3rho_2g` ...(ii)
For equilibrium of P
`T+4/3pir^3rho_1g=4/3pir^3sigma_1g` ...(iii)
(iii)-(ii) gives
`rho_1-sigma_2=sigma_1-rho_2`
From (i) and (iv)
`V_P/V_Q=eta_1/eta_2` `:. (|V_P|)/(|V_Q|)=eta_1/eta_2`
`:.` A is also a correct option