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Question

Two spheres P and Q of equal radii have densities ρ1 and ρ2, respectively. The spheres are connected by a massless string and placed in liquids L1 and L2 of densities σ1 and σ2 and viscosities η1 and η2, respectively. They float in equilibrium with the sphere P in L1 and sphere Q in L2 and the string being taut. If sphere P along in L2 has terminal velocity VP and Q along in L1 has terminal velocity VQ, then

A
|VP||VQ|=η1η2
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B
|VP||VQ|=η2η1
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C
VP.VQ>0
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D
VP.VQ<0
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Solution

The correct option is D VP.VQ<0
For the body to float
(ρ1+ρ2)V=(σ1+σ2)V
ρ1+ρ2=σ1+σ2(1)
As the strings are taut
ρ1<σ1 and ρ2>σ2
Now using Stokes law
VP=29(σ2ρ1)r2gη2(2)
or
VQ=29(σ1ρ2)r2gη1(3)
From (1)
σ2ρ1=(σ1ρ2)
Dividing (1) and (2)
VPVQ=η1η2
The dot product VP.VQ<0 as VP and VQ are opposite.

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