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

(C) The velocity of efflux is given by Torricelli's law as $v = \sqrt{2gh}$.Let $A$ be the area of the tank and $a$ be the area of the hole. The rate

Vedclass · Accepted Answer

$\sqrt{2}-1$

Vedclass · Answer

(C) The velocity of efflux is given by Torricelli's law as $v = \sqrt{2gh}$.Let $A$ be the area of the tank and $a$ be the area of the hole. The rate of change of height is given by $A \frac{dh}{dt} = -a \sqrt{2gh}$.Rearranging the terms,we get $dt = -\frac{A}{a \sqrt{2g}} h^{-1/2} dh$.Integrating from $h_1$ to $h_2$,the time taken is $t = \int_{h_2}^{h_1} \frac{A}{a \sqrt{2g}} h^{-1/2} dh = \frac{A}{a} \sqrt{\frac{2}{g}} (\sqrt{h_1} - \sqrt{h_2})$.For the first interval ($h$ to $h/2$): $t_1 = \frac{A}{a} \sqrt{\frac{2}{g}} (\sqrt{h} - \sqrt{h/2}) = \frac{A}{a} \sqrt{\frac{2h}{g}} (1 - \frac{1}{\sqrt{2}})$.For the second interval ($h/2$ to $0$): $t_2 = \frac{A}{a} \sqrt{\frac{2}{g}} (\sqrt{h/2} - 0) = \frac{A}{a} \sqrt{\frac{2h}{g}} (\frac{1}{\sqrt{2}})$.Taking the ratio: $\frac{t_1}{t_2} = \frac{1 - 1/\sqrt{2}}{1/\sqrt{2}} = \frac{(\sqrt{2}-1)/\sqrt{2}}{1/\sqrt{2}} = \sqrt{2}-1$.

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

Similar Questions

$A$ container has a hole of cross-sectional area $1 \, cm^2$ at its bottom. If water is poured into the container at a rate of $70 \, cm^3/s$, what is the maximum height (in $cm$) to which the water can be filled?

The top of a water tank is open to air and its water level is maintained. It is discharging $0.74\,m^3$ of water per minute through a circular opening of $2\,cm$ radius in its wall. The depth of the centre of the opening from the level of water in the tank is close to........ $m$

$A$ vessel is filled with water up to a height $h$. When a hole is made at the bottom,all the water drains out in time $t$. If the vessel is filled with water up to a height $4h$,how long will it take for the water to drain out completely when a hole is made at the bottom?

Vedclass Test Series

Exam Paper Generator

Online Exam Module