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Question

A container of large uniform cross-sectional area A resting on a horizontal surface, holds two immiscible, non-viscous and incompressible liquids of densities d and 2d, each of height H2 . The lower density liquid is open to the atmosphere having pressure P0.
(i) A homogeneous solid cylinder of length L(L<H2), cross-sectional area A5 is immersed such that if floats with its axis vertical at the liquid-liquid interface with length L4 in the denser liquid. Determine: (a) the density D of the solid, (b) the total pressure at the bottom of the container.

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Solution

a) Let ρ be the density cylinder

Volume of fluid having density 'd' displaced
Vd=A5×3L4=3AL20

Volume of '2d' displaced
v2d=A5×L4=AL20

Total Buoyant force qererted
FBuo=Vd×d×g+V2d×2d g

=3AL20dg+2AL20dg
=14ALdg

The cylinder remains in equilibrium
FBuo=Fweight
14ALdg=ρ×A5×Lg

14d=ρ5

ρ=54d

b) The pressure at the bottom surface:

P=Total weight of liquid+Solid Area of crossection of Container

P=A6(dgh/2+2dgh/2+5dgL4)A6

P=dg(6h5L)4

1145698_873742_ans_44ca6b5f00d3438bb26ac341f1cd04c6.png

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