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Question: A cylindrical vessel contains a liquid of density r upto a height h. The liquidosed by a piston of m...

A cylindrical vessel contains a liquid of density r upto a height h. The liquidosed by a piston of mass m and area of cross-section A. There is a small hole at the bottom of the vessel. The speed v with which the liquid comes out of the hole is –

A

2gh\sqrt{2gh}

B

2(gh+mgρA)\sqrt{2\left( gh + \frac{mg}{\rho A} \right)}

C

2(gh+mgA)\sqrt{2\left( gh + \frac{mg}{A} \right)}

D

2gh+mgA\sqrt{2gh + \frac{mg}{A}}

Answer

2(gh+mgρA)\sqrt{2\left( gh + \frac{mg}{\rho A} \right)}

Explanation

Solution

Applying Bernoulli’s theorem at 1 and 2 : difference in pressure energy between 1 and 2 = difference in kinetic energy between 1 and 2

or rgh + mgA\frac{mg}{A} =12\frac{1}{2} rv2 or v = 2gh+2mgρA\sqrt{2gh + \frac{2mg}{\rho A}} = 2(gh+mgρA)\sqrt{2\left( gh + \frac{mg}{\rho A} \right)}