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

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 $$=2 \mathrm{P}_{0} h \mathrm{R}+ho g h^{2} \mathrm{R}-2 \mathrm{TR}$$

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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