Equal-radius spheres P and Q of densities ρ₁ and ρ₂ are connected by a taut massless string and placed in liquids L₁ and L₂ of densities σ₁, σ₂ and viscosities η₁, η₂. If P alone in L₂ has terminal velocity v(P) and Q alone in L₁ has terminal velocity v(Q), select the correct relation(s). Image
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Question

Equal-radius spheres P and Q of densities ρ₁ and ρ₂ are connected by a taut massless string and placed in liquids L₁ and L₂ of densities σ₁, σ₂ and viscosities η₁, η₂. If P alone in L₂ has terminal velocity v(P) and Q alone in L₁ has terminal velocity v(Q), select the correct relation(s). Image

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G1 Physics · Accepted Answer

Step 1: Use viscous-force relations: shear F = ηA(dv/dy), Stokes force 6πηrv, or Reynolds number Re = ρvD/η as appropriate. Step 2: At terminal speed set net weight/buoyancy equal to viscous drag and use the given geometry. Applying these relations to the given data gives A: |𝐕ₚ|/|𝐕q| = η₁/η₂; D: 𝐕ₚ·𝐕q < 0. Final answer: AD.