JEE Main 2024 · Physics · Fluid Mechanics, Elasticity and Surface Tension · medium

A large cylindrical tank of cross-sectional area 1 m^2 is filled with water to a depth of 1.8 m. A small hole of area 1 cm^2 is opened at its bottom.…

Numerical answer

A large cylindrical tank of cross-sectional area

1\ \mathrm{m^2}

is filled with water to a depth of

1.8\ \mathrm{m}

. A small hole of area

1\ \mathrm{cm^2}

is opened at its bottom. Treating the flow as ideal (Torricelli), the time taken (in s) for the tank to empty completely is (

g=10\ \mathrm{m\,s^{-2}}

)

Show answer and solution

Answer: 6000

Let the level be

h

. Outflow speed

v=\sqrt{2gh}

, so by continuity

A\left(-\dfrac{dh}{dt}\right)=a\sqrt{2gh}

, giving

\dfrac{dh}{\sqrt h}=-\dfrac{a}{A}\sqrt{2g}\,dt

. Integrating from

H

0

2\sqrt H=\dfrac aA\sqrt{2g}\,t

, so

t=\dfrac Aa\sqrt{\dfrac{2H}{g}}=10^{4}\sqrt{\dfrac{2\times1.8}{10}}=10^4\times0.6=6000\ \mathrm{s}

. Using the initial outflow speed throughout would give half this time.

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