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A cubical block of each side equal to 10 cm is made of steel of density 7.8 gm/cm3. It floats on mercury surface in a vessel with its sides vertical. The density of mercury is 13.6 gm/cm3.

(a) Find the length of the block above mercury surface. 

(b) If water is poured on the surface of mercury, find the height of the water column when water just covers the top of the steel block.

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(a) Volume of steel block = (10)3 = 1000 cm3

Mass of steel block = 1000 × 7.8 = 7800 gm

Let l1 be the height of steel block above the surface of mercury

Height of block under mercury = 10 – l1

Weight of mercury displaced by block = (10 – l1) × 100 × 13.6 × g gm

Archimede’s principle shows that upward thrust equal to the weight of mercury displaced by block is equal to the weight of the block.

∴ (10 – l1) × 100 × 13.6 × g = 7800 × g

10 – l1 = 7800/100 x 13.6 = 5.74

∴ length of block above mercury surface = 10 – 5.74 = 4.26 am

(b) Let l2 be the height of water column above mercury surface so that water just covers the top of the steel block. The upward thrust due to mercury and water displaced is equal to the weight of the body. 

∴ weight of block = wt. of water displaced + wt. of mercury displaced

∴ 7800 g = l2 × 100 × 1 × g + (10 – l2) × 100 × 13.6 × g

7800 = 100 l2 + 13600 – 1360 l2

1260 l2 = 13600 – 7800 = 5800

∴  height of water column above mercury = l2 = 5800/1260 = 4.6 cm

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