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Question

A cubical metal block of edge 12 cm floats in mercury with one fifth of the height inside the mercury. Water in it. Find the height of the water column to be poured.
Specific gravity of mercury = 13.6.

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Solution

Given:
Length of the edge of the metal block, x = 12 cm
Specific gravity of mercury, ρHg= 13.6 gm/cc
It is given that 15th of the cubical block is inside mercury initially.
Let ρb be the density of the block in gm/cc.
(x)3×ρb×g=(x)2×x5×ρHg×g(12)3×ρb×g=(12)2×125×13.6ρb=13.65 gm/cc
Let y be the height of the water column after the water is poured.
∴ Vb = VHg + Vw = (12)3
Here,
VHg = Volume of the block inside mercury
Vw = Volume of the block inside water
(Vb×ρb×g)=(VHg×ρHg×g)+(Vw×ρw×g)(VHg+Vw)×13.65=VHg×13.6+Vw×1(12)3×13.65=(12-y)×(12)2×13.6+(y)×(12)2×112×13.65=(12-y)×13.6+(y)12.6y=13.612-125=(13.6)×(9.6)y=(9.6)×(13.6)(12.6)=10.4 cm

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