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Two spheres P and Q of equal radii have ...

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

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`

Text Solution

Verified by Experts

The correct Answer is:
A, D
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
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