A cylindrical water tank is mounted on a platform with its central axis vertical

Question

A cylindrical water tank is mounted on a platform with its central axis vertical. The water level is distance d above the bottom of the tank, and the bottom is at height h above the ground. A small, circular hole of area A has formed in the bottom of the tank. Both the hole and the top of the tank are open to the air. Ignore air resistance and treat water as an ideal fluid. Let g denote the acceleration due to gravity.

When the depth of the water in the tank is d, at what rate is the tank losing water?

When the depth of the water in the tank is d, how fast is the water from the hole moving just as it reaches the ground?

Explanation / Answer

Torriceli's theorem syas that the speed of a liquid through a hole at a depth d below the surface of the liquid is same as that of a particle which has been in free fall through the height d under gravity.

Now, the speed v of a particle which has been in free fall through the height d under gravity is given by:

v = (2gd)

So, the liquid through the hole is flowing at a speed of (2gd) per unit time. So, in a unit time, length of water flown through area A of the hole is (2gd). So, in unit time, the volume of water flown through area A of the hole is

V = A(2gd)

So the tank is losing water at the rate of V = A(2gd) per unit time.

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The speed of water near the hole is v = (2gd). If we consider a small mass of water it's speed at hole will be v = (2gd) and

so, its kinetic energy will be KE = (1/2)m((2gd))2 = mgd

its potential energy at the hole which is at height h is PE = mgh

At the ground, let us suppose the speed of this small water mass be v. At the ground level the potential energy will be zero.

So, applying law of conservation of energy we have;

mgd + mgh = (1/2)mv2

or v2 = 2g(d+h)

or v = [2g(d+h)]

So when it reaches the ground it's moving at a speed of v = [2g(d+h)].

P.S. - If you want me to derive the Torriceli's theorem mentioned above, just tell me... I will do it.

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This concludes the answers. Check the answer and let me know if it's correct. If you need anymore clarification or correction I will be happy to oblige....

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