A pipe of diameter D = 2 cm is connected to an open tank of large cross-section area containing a non-viscous fluid which flows in a streamlined manner. The level of fluid in the tank from the base is maintained at H = 5 0 cm by some external means. A nozzle of diameter d = 1 cm fitted at the end of the pipe discharges water into the atmosphere.
Given that the density of the fluid is ρ = 2 . 5 g/cc , find the pressure ( to the nearest integer value in mm of Hg ) at point C which is at a height of h = 5 cm above the level of fluid in the tank.
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Couldn't we directly do
P(atmosphere) = P( at C ) - density x h x g ,
as velocity of the liquid through out the siphon remains the same
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The level of water is maintained in the tank which further is characterised to have a large cross-section area. Thus, the speed with which water level is theoretically observed to be coming down is almost negligible and not the same as that in the siphon tube.
Moreover, the velocity of efflux is different than that of the pipe because it has a different cross-sectional area.
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Does area of cross section even matter ?? Because according to Bernoulie's equation,ie. by conservation of energy, the velocity should be same through out the siphon, (as you said this velocity will be different from the rate at which the height column of the cylinder decreases) , shouldn't it ?
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@Aswin Ramesh – Yes, the area of cross-section does matter. Since equal mass of water should through all points at equal intervals of time, for two different cross-sections, we should have
Mass of water passing through a given cross section = density × volume ⟹ M = ρ ( A ⋅ s ) = ρ ( A ⋅ v ⋅ t )
thus
ρ A 1 v 1 t = ρ A 2 v 2 t ⟹ A 1 v 1 = A 2 v 2 = Φ
Thus, the volumetric flow rate Φ should be same and the velocities at different cross sections have to be different.
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@Tapas Mazumdar – Oh, i used to think that the velocity remains constant . Thanks for the help , but when i do solve the question using my method, I'm getting the correct answer. Is is a mere coincidence ?? I'm feeling is isn't.
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@Aswin Ramesh – Can you show me your method?
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@Tapas Mazumdar – My bad, i did a calculation mistake :P
Is that a typo in the 1st line? That H
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Thanks that was a big error. But the answer is still the same. Thanks for mentioning. :)
Hey I got 743 . And rhen tried 745. Do you think am correct?
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You should've got 744.7. I have already given all the values needed.
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Yeah ... Actually, I took g=10 I guess... and some other small errors :D.
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@Md Zuhair – Yes. Assuming you took g=10, I said it would have been 744.7. Nevermind the answer is correct. :)
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@Tapas Mazumdar – Yeah. Small errors come, Btw, You deleted Whatsapp?
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@Md Zuhair – Yes for JEE preparations
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@Tapas Mazumdar – I guess... you will get a nice IIT CLLGE
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Taking the mouth of the nozzle as the reference for zero gravitational potential and applying Bernoulli's theorem for the mouth of nozzle and the water in the tank, we get
⟹ P 0 + H ρ g + 0 = P 0 + 0 + 2 1 ρ v 2 v = 2 g H
Via equation of continuity we have
⟹ ⟹ A V = a v 4 π D 2 V = 4 π d 2 ⋅ 2 g H V = D 2 d 2 ⋅ 2 g H
Again taking mouth of nozzle as the reference for zero gravitational potential and applying Bernoulli's theorem for the mouth of nozzle and point C, we get
P C + ( H + h ) ρ g + 2 1 ρ V 2 ⟹ P C = = = = P 0 + 0 + 2 1 ρ v 2 P 0 − ( H + h ) ρ g + 2 1 ρ ( v 2 − V 2 ) P 0 − ( H + h ) ρ g + 2 1 ρ ( 1 − D 4 d 4 ) 2 g H P 0 − ρ g ( h + H ⋅ D 4 d 4 )
Putting required values, we get
P C = 9 9 3 3 4 . 3 7 5 Pa = 7 4 5 . 0 6 9 mm of Hg ≈ 7 4 5