A large tank filled with water to a height h | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) According to Torricelli's law,the velocity of efflux is $v = \sqrt{2gh}$.The rate of change of height is given by $A \frac{dh}{dt} = -a \sqrt{2gh}

Vedclass · Accepted Answer

$\sqrt{2}-1$

Vedclass · Answer

(A) According to Torricelli's law,the velocity of efflux is $v = \sqrt{2gh}$.The rate of change of height is given by $A \frac{dh}{dt} = -a \sqrt{2gh}$,where $A$ is the area of the tank and $a$ is the area of the hole.This gives $dt = -\frac{A}{a\sqrt{2g}} h^{-1/2} dh$.The time taken to fall from height $h_1$ to $h_2$ is $t = \int_{h_2}^{h_1} \frac{A}{a\sqrt{2g}} h^{-1/2} dh = \frac{A}{a\sqrt{2g}} [2\sqrt{h}]_{h_2}^{h_1} = \frac{2A}{a\sqrt{2g}} (\sqrt{h_1} - \sqrt{h_2})$.Let $t_1$ be the time to fall from $h$ to $h/2$:$t_1 = \frac{2A}{a\sqrt{2g}} (\sqrt{h} - \sqrt{h/2}) = \frac{2A}{a\sqrt{2g}} \sqrt{h} (1 - 1/\sqrt{2})$.Let $t_2$ be the time to fall from $h/2$ to $0$:$t_2 = \frac{2A}{a\sqrt{2g}} (\sqrt{h/2} - \sqrt{0}) = \frac{2A}{a\sqrt{2g}} \sqrt{h} (1/\sqrt{2})$.The ratio is $\frac{t_1}{t_2} = \frac{1 - 1/\sqrt{2}}{1/\sqrt{2}} = \sqrt{2} - 1$.

$A$ large tank filled with water to a height $h$ is to be emptied through a small hole at the bottom. The ratio of the time taken for the level to fall from $h$ to $h/2$ and that taken for the level to fall from $h/2$ to $0$ is:

Similar Questions

Give the meaning of efflux.

The figure shows two holes in a wide tank containing a liquid column. The water streams coming out of these holes strike the ground at the same point. The height of the liquid column in the tank is $....\ cm$.

$A$ tank of height $5\, m$ is full of water. There is a hole of cross-sectional area $1\, cm^2$ in its bottom. The initial volume of water that will come out from this hole per second is

State Torricelli's law.

Vedclass Test Series

Exam Paper Generator

Online Exam Module