Engineering
Physics
Relative Motion
Liquid Pressure
BernoulIis Equation and Equation of Continuity
Question

A fixed cylindrical tank having large cross-section area is filled with two liquids of densities ρ and 2ρ in equal volumes as shown in the figure. A small hole of area of cross-section a=6  cm2 is made at height h/2 from the bottom. Find the area of cross-section of stream of liquid in cm2 just before it hits the ground

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Solution
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Applying Bernaullis equation at cross-section 1 & 2
Patm + ρgh + 0 = P2 + 0 + 0 ⇒ P2 = Patm + ρgh    .....(1)
Again applying Bernaullis equation at section 2 & 3
P2+0+2ρg×h2=Patm+122ρv2                .....(2)
V=2gh
This is required velocity of efflux
Applying continuity equation between 3 & 4 cross-section.

aV=a1V1
This is required velocity of efflux
Applying Bernoullis equation between 3 & 4
Patm+12(2ρ)V2+2ρg×h2=Patm+12(2ρ)V12+0
ρV2+ρgh=ρV12V12=3gh
a1=aVV1=62gh3gh=2cm2