JEE Advanced · Physics · 11. Properties of Fluids
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.
- A \(\left|2 p_0 R h+\pi R^2 \rho g h-2 R T\right|\)
- 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 \(\left|p_0 \pi R^2+R \rho g h^2+2 R T\right|\)
Answer & Solution
Correct Answer
(B) \(\left|2 p_0 R h+R \rho g h^2-2 R T\right|\)
Step-by-step Solution
Detailed explanation
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).
(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).
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