Two sphere P and Q for 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 ` v _ P ` and Q along in ` L_1 ` has terminal velocity ` v_Q`, then :

Options

A. `|vec(V_(P))/vec(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. `vecV_(P).vecV_(Q)lt0`

Answer

A, D

Explanation

For floating
`(rho_(1) + rho_(2))V = (sigma_(1) + sigma_(2))V`
`rho_(1) + rho_(2) = sigma _(1) + sigma_(2)`
since strings in taut so
`rho_(1) lt sigma_(1), rho_(2) gt sigma_(2)`
`V_(P) = (2)/(9)(r^(2)(sigma_(2) - rho_(1))g)/(eta_(2))`
`V_(Q) = (2)/(9)((sigma_(1) - rho_(2))g)/(eta_(1))`
since `sigma_(2) - rho_(1) = -(sigma_(1) - rho_(2))`
`|(V_(P))/(V_(Q))| = (eta_(1))/(eta_(2))`
`vec(V_(P)).vec(V_(Q)) lt 0` because `V_(P)` and `V_(Q)` are oppsite