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Question

Two non-mixing liquids of densities ρ and nρ (n>1) are put in a container. The height of each liquid is h. A solid cylinder of length L and density d is put in this container. The cylinder floats with its axis vertical and length pL (p<1) in the denser liquid. The density d is equal to:

A
[1+(n+1)p]ρ
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B
[2+(n+1)p]ρ
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C
[2+(n1)p]ρ
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D
[1+(n1)p]ρ
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Solution

The correct option is D [1+(n1)p]ρ

As we know,
Weight of the wooden block W=Vdg=ALdg
where A is the cross section area of the cylinder
Using Archimedes' principle,
Weight of wooden block = Upthrust due to liquid of density nρ + Upthrust due to liquid of density ρ
ALdg=(pL)A(nρ)g+(1p)LAρg
d=(1p)ρ+pnρ=[1+(n1)p]ρ

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