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To what height should a cylindrical vessel be filled with a homogeneous liquid to make the force with which the liquid presses on the side of the vessel equal to the force exerted by the liquid on the bottom of the vessel

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equal to the radius

less than the radius

more than the radius

four times the radius

answer is A.

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Detailed Solution

Force acting per unit width of the vertical wall is

$\frac{F}{b} = \frac{1}{2} ρ {gh}^{2}$

$F = \frac{1}{2} ρ {gh}^{2} . b$

$F_{H} = \frac{1}{2} ρ {gh}^{2} (2 π r) \to (1) (b = 2 π r)$

As it is cylindrical wall volume $v = π r^{2} h$

Force at the base

$F_{B} = ρ gv$ Weight of liquid

$F_{B} = ρ {gr}^{2} h \to (2)$

$F_{H} = F_{B}$

$h = r$

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