Q51 / 66 · JEE Advanced 2007 (Paper 2) · Physics

Water is filled up to a height $h$ in a beaker of radius $R$ as shown in the figure. The density of water is $\rho$, the surface tension of water is $T$ and the atmospheric pressure is $p_0$. 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. Figure

  • A
    $\left|2 p_0 R h+\pi R^2 \rho g h-2 R T\right|$
  • Correct answer
    B
    $\left|2 p_0 R h+R \rho g h^2-2 R T\right|$
  • C
    $\left|p_0 \pi R^2+R \rho g h^2-2 R T\right|$
  • D
    $\left|p_0 \pi R^2+R \rho g h^2+2 R T\right|$

Solution

Force from right hand side liquid on left hand side liquid. (i) Due to surface tension force $=2 R T$ (towards right) (ii) Due to liquid pressure force $$ \begin{aligned} & =\int_{x=0}^{x=h}\left(p_0+\rho g h\right)(2 R \cdot x) d x \\ & =\left(2 p_0 R h+R \rho g h^2\right) \text { (towards left) } \end{aligned} $$ $\therefore$ Net force is $\left|2 p_0 R h+R \rho g h^2-2 R T\right|$ Correct option is (b).

Hence, the correct answer is option: 'B'.

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