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Question

# Two non-mixing liquids of densities 1000 kg/m3 and 2000 kg/m3 having height of 1 m each are taken in a container, as shown in the figure. When a solid cylinder of length 0.5 m and density ρ is inserted in this container, it is observed that the cylinder floats with its axis being vertical and a length 0.2 m of it in the denser liquid. The density ρ of the solid cyclinder will be equal to

A
1500 kg/m3
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B
1300 kg/m3
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C
1600 kg/m3
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D
1400 kg/m3
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Solution

## The correct option is D 1400 kg/m3Given, Density of liquid in the upper half of container ρ1=1000 kg/m3 Density of liquid in the lower half of container ρ2=2000 kg/m3 From the figure given in the question, we can deduce that, at equilibrium, force of buoyancy due to both liquids should balance the weight of the cylinder (as shown in figure-2). Let Fb1 and Fb2 be the force of buoyancy due to liquids of density ρ1 and ρ2. ∴Fb1=m1×g=ρ1×V1×g =1000×g×[0.3×A]=300gA .......(1) Similarly, force of buoyancy due to liquid of density ρ2 Fb2=2000×g×[0.2×A]=400 gA .........(2) For equilibrium (floating) of cylinder (From figure-2) mg=Fb1+Fb2 By using (1) and (2), we get ⇒0.5Aρg=300Ag+400Ag ⇒0.5ρ=700 ⇒ρ=1400 kg/m3 Hence, option (d) is the correct answer.

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