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Physics

Fluid-Mechanics

Easy

Question

A cylinder of radius R, height h and density d has a hemispherical cut at its bottom. The top of the cylinder is kept at a depth h from the free surface of a liquid. Density of liquid is d/2 . At this position, hydrostatic force acting on the curved surface of the cylinder will be

$π R^{2} d g (h - \frac{R}{3})$
$π R^{2} d g \frac{h}{3}$
$π R^{2} d g \frac{R}{3}$
Zero

The correct answer is: $pi R squared d g open parentheses h minus R over 3 close parentheses$

As per free body diagram,
$F subscript 1 equals P subscript 1 A equals fraction numerator d g h over denominator 2 end fraction pi R squared straight F text is buoyant force such that end text F equals F subscript 2 minus F subscript 1 text Here end text F equals V d over 2 g equals open parentheses pi R squared h minus 2 over 3 pi R cubed close parentheses d over 2 g text Now end text F subscript 2 equals F plus F subscript 1 equals open parentheses pi R squared h minus 2 over 3 pi R cubed close parentheses d over 2 g plus fraction numerator d g h over denominator 2 end fraction pi R squared fraction numerator bold pi bold R to the power of bold 2 bold d bold g over denominator bold 2 end fraction begin bold style left square bracket h plus h minus fraction numerator 2 R over denominator 3 end fraction right square bracket end style bold equals fraction numerator bold pi bold R to the power of bold 2 bold d bold g over denominator bold 2 end fraction begin bold style left parenthesis 2 h minus fraction numerator 2 R over denominator 3 end fraction right parenthesis end style bold equals bold italic pi bold italic R to the power of bold 2 bold italic d bold italic g begin bold style left parenthesis h minus R over 3 right parenthesis end style$

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