Water is filled up to a height $h$ in a breaker of radius $R$ . The density of water is $ρ$ , the surface tension of water is $T$ and the atmospheric pressure is $P_{0}$ . Consider a vertical section $A B C D$ of the water column through a diameter of the beaker. The force on water on one side of this section by water on the other side of this section has magnitude:

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$| P_{0} π R^{2} + R ρ g h^{2} + 2 R T |$

$| 2 P_{0} R h + R ρ g h^{2} - 2 R T |$

$| P_{0} π R^{2} + R ρ g h^{2} - 2 R T |$

$| 2 P_{0} R h + π R^{2} ρ g h - 2 R T |$

answer is B.

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Detailed Solution

Consider water on one side of the vertical section $A B C D$ as the system.

Note that this is the half-cylinder. Draw the forces on this half-cylinder by another half-cylinder.

At a depth below $x$ from the top surface consider a strip of width $d x$ .

Force on this strip is $(P_{0} + ρ g x) (2 R d x)$ .

Total force on one half-cylinder by the other half-cylinder is

$F = \int_{0}^{h} (P_{0} + ρ g x) 2 R d x - 2 R T = | 2 P_{0} R h + R ρ g h^{2} - 2 R T |$

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