First slide

Fluid statics

Question

A hollow cylinder of mass m made heavy at its bottom is floating vertically in water. It is tilted from its vertical position through an angle $θ$ and is left. The respecting force acting on it is

Easy

A

$mg cosθ$
B

$\frac{mg}{cosθ}$
C

$mg [\frac{1}{cosθ} - 1]$
D

$mg [\frac{1}{\cos θ} + 1]$

Solution

Let l be the length of the cylinder, when vertical, in water. Let A be the cross-sectional area of the cylinder. Equating weight of the cylinder with the upthrust, we get
Mg = $Alρg or m = Alρ$
When the cylinder is titled through an angle $θ, length$ of cyliner in water = $\frac{l}{\cos θ}$
Weight of water displaced = $\frac{l}{\cos θ} Aρg$
Restoring force = $\frac{lAρg}{cosθ} - lAρg$
= $lAρg [\frac{1}{cosθ} - 1] = mg [\frac{1}{cosθ} - 1]$

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