A cylindrical vessel of cross-section A contains water to a height...

A cylindrical vessel of cross-section A contains water to a height

\mathrm{h}

. There is a hole in the bottom of radius '

a

'. The time in which it will be emptied is: A cylindrical vessel of cross-section A contains water to a height h . There is a hole in the bottom of radius ' a '.

Options

A $\frac{2 \mathrm{~A}}{\pi \mathrm{a}^2} \sqrt{\frac{\mathrm{h}}{\mathrm{g}}}$
B $\frac{\sqrt{2} \mathrm{~A}}{\pi \mathrm{a}^2} \sqrt{\frac{\mathrm{h}}{\mathrm{g}}}$
C $\frac{2 \sqrt{2} \mathrm{~A}}{\pi \mathrm{a}^2} \sqrt{\frac{\mathrm{h}}{\mathrm{g}}}$
D $\frac{\mathrm{A}}{\sqrt{2} \pi \mathrm{a}^2} \sqrt{\frac{\mathrm{h}}{\mathrm{g}}}$

Correct answer

B. $\frac{\sqrt{2} \mathrm{~A}}{\pi \mathrm{a}^2} \sqrt{\frac{\mathrm{h}}{\mathrm{g}}}$

Step-by-step solution

\text{ Let the rate of falling water level be }-\frac{d h}{d t} A cylindrical vessel of cross-section A contains water to a height h . There is a hole in the bottom of radius ' a '.

\\ Initially at

t=0 ; h=h

t=t ; h=0 Then,

A\left(-\frac{d h}{d t}\right)=\pi a^2 \cdot v

d t=-

\frac{A}{\pi a^2 \sqrt{2 g h}}

d h

[\because

velocity of efflux of liquid

v=\sqrt{2 g h}]

Integrating both sides \int_0^t d t=-

\frac{A}{\sqrt{2 g} \pi a^2} \int_

h^0 h^{-1 / 2} d h [t]_0^t=-

\frac{A}{\sqrt{2 g} \pi a^2} \cdot

\left[\frac{h^{1 / 2}}{1 / 2}\right]_

h^0 t=

\frac{\sqrt{2} A}{\pi a^2} \sqrt{\frac{h}{g}}

Practice Mechanical Properties of Fluids on Quantrex Academy →

More from Mechanical Properties of Fluids

Passage: A container of height 2 m, length 2 m and breadth 1 m is made of insulating… JEE Advanced 2026 A liquid of density 600 kg/m ^3 flowing steadily in a tube of varying cross-section. The… JEE Main 2026 A spherical liquid drop of radius R acquires the terminal velocity v_1 when falls through… JEE Main 2026 A cylindrical vessel of 40 cm radius is completely filled with water and its capacity is… JEE Main 2026 Eight mercury drops, each of radius r , coalesce to form a bigger drop. The surface… JEE Main 2026 A water spray gun is attached to a hose of cross sectional area 30 cm ^2 . The gun… JEE Main 2026 The surface tension of a soap solution is 3.5 × 10^ -2 N/m. The work required to increase… JEE Main 2026 The surface tension of a soap bubble is 0.03 N/m. The work done in increasing the… JEE Main 2026 Full Mechanical Properties of Fluids list All JEE Main PYQs