Question

A large cylindrical tank has a hole of area A at its bottom. Water is poured in the tank by a tube of equal cross-sectional area A ejecting water at the speed v.

• A. The water level in the tank will keep on rising
• B. No water can be stored in the tank
• C. The water level will rise to a height g and then stop
• D. The water level will oscillate

Answer 1

The correct option is C

The water level will rise to a height g and then stop

As established,
$\frac{dw}{dt}=\frac{d{w}_i}{dt}-\frac{d{w}^0}{dt}$
Now the area of cross-section of tube through which water is flowing into the tank = A with velocity = V
So$\frac{d{w}_i}{dt}$ AV {As in dt time interval, volume of water coming into the tank,
$d{w}_i=AVdt$ $\frac{d{w}_i}{dt}=AV$}
As clear from the Torricelli's theorem,
$Speedofefflux=$$\sqrt{2gh}$
Where h = level of water at that moment.
$\Rightarrow \frac{d{w}^0}{dt}A\sqrt{2gH}$
Let's assume that at any instant the level of water has reached a height H and volume of water does not change further,
$\frac{dw}{dt}=\frac{d{w}_i}{dt}-\frac{d{w}^0}{dt}=0$
$\frac{d{w}_i}{dt}=\frac{d{w}^0}{dt}$
$Av=A\sqrt{2gH}$
$H=\frac{v^2}{2g}$
$\Rightarrow$ This implies that water level rises till $\frac{v^2}{2g}$ and then stops fluctuating.
or input rate = output rate {after this instant is reached}.Hence (c)