Question

Water is filled up to a height $\text{h}$ in a beaker of radius $\text{R}$ as shown in the figure. The density of water is $\rho $ , the surface tension of water is $T$ and the atmospheric pressure is $p_{o}$ . Consider a vertical section $\text{ABCD}$ 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

• A. $\left|\right. 2 p_{0} R h + \pi R^{2} \rho g h - 2 R T \left|\right.$
• B. $\left|\right. 2 p_{0} R h + R \rho g h^{2} - 2 R T \left|\right.$
• C. $\left|\right. p_{0} \pi R^{2} + R \rho g h^{2} - 2 R T \left|\right.$
• D. $\left|\right. p_{0} \pi R^{2} + R \rho g h^{2} + 2 R T \left|\right.$

Answer 1

Answer: (B)

The force from right-hand side liquid on left-hand side liquid,
(i) Due to surface tension force
$\text{= 2RT}$ (towards the right)
(ii) Due to liquid pressure force
$=\displaystyle \int _{0}^{h}\left(p_{0} + \rho g x\right)2Rdx$
$=2Rp_{0}h+R\rho gh^{2}$
Net force is $\left|2 p_{o} R h + R \rho g h^{2} - 2 R T\right|$