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Fluid Mechanics

Centre of Buoyancy

A cylindrical...

Question

A cylindrical body of cross-sectional area A, height H and density $ρ_{s^{'}}$ is immersed to a depth h in a liquid of density $ρ$ , and tied to the bottom with a string. The tension in the string is

A

ρ g h A

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B

(ρ_{s} - ρ) g h A

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C

(ρ - ρ_{s}) g h A

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D

(ρ h - ρ_{s} H) g A

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Solution

The correct option is D $(ρ h - ρ_{s} H) g A$
Let T = Tension in string acting in the downward direction

Weight of body,
W = $M g (↓)$
= $ρ_{s} A H g$

Buoyant force,
$F_{B} = ρ g \forall (↑)$
= $ρ g A h$

For equilibrium,
Weight of cylindrical body + Tension in string
= Buoyant force
W + T = $F_{B}$
or T = $F_{B} - W$
= $ρ g A h - ρ_{s} A H g$
= $(ρ h - ρ_{s} H) g A$

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