Fluid mechanics water tank level

[utkarshakash]

Homework Statement

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 (v^2/2g) and then stop.
d)The water level will oscillate.

Homework Equations

The Attempt at a Solution

The velocity of water when it reaches the bottom of the tank is $\sqrt{2gh}$ if I assume h to be the height of cylindrical tank. But it seems difficult to answer the question on this basis.

 

[haruspex]

At first, water will only flow out of the hole slowly, so the level will rise.
What will determine the rate at which water flows out of the hole? (You may have been given an equation for this.)

 

[utkarshakash]

At first, water will only flow out of the hole slowly, so the level will rise.
What will determine the rate at which water flows out of the hole? (You may have been given an equation for this.)

Yes. The equation of continuity states that Av=constant. So it is the area of the hole that determines the rate of flow.

 

[haruspex]

The equation of continuity states that Av=constant. So it is the area of the hole that determines the rate of flow.

No, all that equation tells you is how to work out the volumetric rate from the linear rate. You also need an equation that helps you determine that linear rate. Does the name Bernoulli help?

 

[utkarshakash]

No, all that equation tells you is how to work out the volumetric rate from the linear rate. You also need an equation that helps you determine that linear rate. Does the name Bernoulli help?

Are you talking about Bernoulli's equation? If yes do you want me to write the equations for the water at the top and bottom level of the cylinder?

 

[haruspex]

Are you talking about Bernoulli's equation? If yes do you want me to write the equations for the water at the top and bottom level of the cylinder?

You need an equation that relates both the depth of the water in the tank and the size of the hole to the rate at which water leaves the tank.

 

[utkarshakash]

You need an equation that relates both the depth of the water in the tank and the size of the hole to the rate at which water leaves the tank.

I only know Bernoulli equation.

 

[haruspex]

I only know Bernoulli equation.

That's the one, but it might not be obvious to you how it simplifies for this situation.
Take a look at eqn (2) at http://www.engineeringtoolbox.com/bernouilli-equation-d_183.html. Even this you can simplify a bit because you have atmospheric pressure both above the water and outside the hole.
So this gives you the relationship between the velocity (distance/time) of the water coming out of the hole and the height of water above the hole. See if you can use that to get a differential equation relating the height of the water to its rate of change.