Question 448: Properties of Matter

Question

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. 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
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Accepted Answer

Pressure is acting on area $A B C D$, we have
Pressure Force $=\int_{0}^{h}\left(\mathrm{P}_{0}+ho g xight) 	imes d x 	imes 2 \mathrm{R}$
Force due to pressure (push)
$\begin{aligned}& =\left[\left(\mathrm{P}_{0}+ho g \frac{x^{2}}{2}ight) 2 \mathrm{R}ight]_{0}^{h} \\
& =\left(\mathrm{P}_{0} h+ho g h^{2} / 2ight) 2 \mathrm{R} \\& =2 \mathrm{P}_{0} h \mathrm{R}+ho g h^{2} \mathrm{R}\end{aligned}$
Surface tension force (push) $=T 	imes 2 R$
$	herefore \quad$ Net Force $=2 \mathrm{P}_{0} h \mathrm{R}+ho g h^{2} \mathrm{R}-2 \mathrm{TR}$
![](https://app-content.cdn.examgoal.net/fly/@width/image/1lguepxqv/b67a2f00-62e8-42e7-bd91-6edece896332/289b9270-e262-11ed-8ec0-d5a0bb2b1e44/file-1lguepxqw.png?format=png)

Answer

$\left|2 \mathrm{P}_{0} \mathrm{Rh}+\pi \mathrm{R}^{2} \rho g h-2 \mathrm{RT}\right|$

Answer

$\left|2 \mathrm{P}_{0} \mathrm{Rh}+\pi \mathrm{R \rho gh}^{2}-2 \mathrm{RT}\right|$

Answer

$\left|P_{0} \pi R^{2}+R \rho g h^{2}-2 R T\right|$

Answer

$\left|\mathrm{P}_{0} \mathrm{R}^{2}+\mathrm{R} \rho g \mathrm{~h}^{2}+2 \mathrm{RT}\right|$

Properties of Matter

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