Water is filled up to a height h in a beaker | vedclass

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(B) Consider a vertical rectangular strip of height $dx$ at a depth $x$ from the free surface of the water. The width of this strip is the diameter of

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$\left|2 P_0 Rh+R \rho gh^2-2 RT\right|$

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(B) Consider a vertical rectangular strip of height $dx$ at a depth $x$ from the free surface of the water. The width of this strip is the diameter of the beaker,which is $2R$.The pressure at depth $x$ is $P(x) = P_0 + \rho g x$.The force exerted by the pressure on this strip is $dF_p = P(x) \cdot (2R) dx = (P_0 + \rho g x) 2R dx$.Integrating this from $x = 0$ to $x = h$,the total force due to pressure is $F_p = \int_0^h (P_0 + \rho g x) 2R dx = 2R [P_0 x + \frac{1}{2} \rho g x^2]_0^h = 2 P_0 R h + R \rho g h^2$.Additionally,there is a force due to surface tension acting along the top edge of the section. The length of the section at the surface is $2R$,so the force due to surface tension is $F_T = T \cdot (2R) = 2RT$.Since the surface tension force acts in the opposite direction to the pressure force,the net magnitude of the force is $F = |F_p - F_T| = |2 P_0 R h + R \rho g h^2 - 2 RT|$.

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