Two spheres P and Q of equal radii have densities ρ$$1$$ and ρ$$2$$ , respectively. The spheres are connected by a massless string and placed in liquids $$L1$$ and $$L2$$ of densities σ$$1$$ and σ$$2$$ and viscosities η$$1$$ and η$$2$$, respectively. They float in equilibrium with the sphere P in $$L1$$ and sphere Q in $$L2$$ and the string being taut (see$$ $$figure). If sphere P alone in $$L2$$ has terminal velocity V→P and Q alone in $$L1$$ has terminal velocity V→Q, then

Question

Two spheres  P and Q  of equal radii have densities ρ$$1$$  and ρ$$2$$ , respectively.  The spheres are connected by a massless string and placed in liquids $$L1$$  and  $$L2$$ of densities  σ$$1$$ and  σ$$2$$ and viscosities η$$1$$  and  η$$2$$, respectively.  They float in equilibrium with the sphere P  in $$L1$$  and sphere Q  in $$L2$$  and the string being taut (see$$ $$figure).  If sphere P  alone in $$L2$$  has terminal velocity V→P and  Q alone in  $$L1$$ has terminal velocity  V→Q, then

Infinity Learn · Accepted Answer

For equilibrium of P : σ$$1Vg$$ $$=$$ρ$$1Vg$$ $$+TFor$$ equilibrium of Q : σ$$2Vg$$ $$+T=$$ρ$$2Vg$$ Adding, σ$$1+$$σ$$2=$$ρ$$1+$$ρ$$2$$⇒σ$$2-$$ρ$$1=$$ρ$$2-$$σ$$1$$  Terminal velocity of P : $$VP=2r2g9$$η$$2$$σ$$2$$−ρ$$1$$→ Upward terminal velocity Terminal velocity of Q : $$VQ=2r2g9$$η$$1$$ρ$$2$$−σ$$1$$→ downword terminal velocity $$ratio=VPVQ=$$η$$1$$η$$2$$ And V→P  .V→Q   will be negative as they are opposite to each other

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