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QJEEAdv07Ph70cc: JEE Advanced 2007 Physics - Class 11 JEEAdv07Ph70cc Solution: Water is filled up to a height in a beaker of radius as shown…

Water is filled up to a height hh in a beaker of radius RR as shown in the figure. The density of water is ρ\rho , the surface tension of water is TT and the atmospheric pressure is P. Consider a vertical section ABCDA 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

IIT-JEE 2007 Paper 2 Offline Physics - Properties of Matter Question 4 English

Multiple Choice Options

  • A ∣2P0Rh+πR2ρgh−2RT∣\left|2 \mathrm{P}_{0} \mathrm{Rh}+\pi \mathrm{R}^{2} \rho g h-2 \mathrm{RT}\right|
  • B ∣2P0Rh+πRρgh2−2RT∣\left|2 \mathrm{P}_{0} \mathrm{Rh}+\pi \mathrm{R \rho gh}^{2}-2 \mathrm{RT}\right|
  • C ∣P0πR2+Rρgh2−2RT∣\left|P_{0} \pi R^{2}+R \rho g h^{2}-2 R T\right|
  • D ∣P0R2+Rρg h2+2RT∣\left|\mathrm{P}_{0} \mathrm{R}^{2}+\mathrm{R} \rho g \mathrm{~h}^{2}+2 \mathrm{RT}\right|

Step-by-Step Verified Solution

Pressure is acting on area ABCDA B C D , we have

Pressure Force =∫0h(P0+ρgx)×dx×2R=\int_{0}^{h}\left(\mathrm{P}_{0}+\rho g x\right) \times d x \times 2 \mathrm{R}

IIT-JEE 2007 Paper 2 Offline Physics - Properties of Matter Question 4 English Explanation

Force due to pressure (push)

amp;=[(P0+ρgx22)2R]0hamp;=(P0h+ρgh2/2)2Ramp;=2P0hR+ρgh2R\begin{aligned}& =\left[\left(\mathrm{P}_{0}+\rho g \frac{x^{2}}{2}\right) 2 \mathrm{R}\right]_{0}^{h} \\\\& =\left(\mathrm{P}_{0} h+\rho g h^{2} / 2\right) 2 \mathrm{R} \\\\& =2 \mathrm{P}_{0} h \mathrm{R}+\rho g h^{2} \mathrm{R}\end{aligned}

Surface tension force (push) =T×2R=T \times 2 R

∴\therefore \quad Net Force =2P0hR+ρgh2R−2TR=2 \mathrm{P}_{0} h \mathrm{R}+\rho g h^{2} \mathrm{R}-2 \mathrm{TR}

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