8.4a - Torricelli's Theorem

1. 1. Navigate to the simulation below and then click the “Water Tower” tab.
  2. Open the red gate on the water tower. Observe what happens. How does the water flow change over time?
  3. Refill the water tower. Place the pressure gauge in the bottom of the water tower. Once again, open the red gate. What happens to the pressure at the bottom of the tank as the water flows out? Why? Support your reasoning with a relevant equation.
  4. Refill the water tower. Place the “Speed” measurement device right where the water is initially leaving the water tower.
    1. Approximately how fast is the water leaving the tower when the tower is full?
      1. What happens to the speed of the water as the water tower drains? Why do you think this happens? Support your answer by discussing the pressure of the water within the tower.
      2. Measure the height of the water tower.
    2. Based on this height, how long is the water spending in the air?
      1. Does this time in the air change as the initial horizontal speed of the water changes? Briefly justify using your understanding of 2D kinematics.
  5. Let’s apply the Conservation of Energy to what we are observing. When does the water leaving the tower have the greatest kinetic energy?
    1. ____ When the tower is full ____ When the tower is nearly empty
    2. Justify your selection.

In 1643, Evangelista Torricelli crafted his theorem. This theorem, which can be derived using Bernoulli’s equation, predicted the velocity of the water coming out of a tall container based on the height of the water above the spout.

Torricelli, postulated that the velocity of a fluid flowing through a sharp edged hole at the bottom of a tank or vessel filled to a certain height h would be the same as any object of falling freely from that height.

The pressure in the water increases with depth. As the water leaves the tower, it converts this pressure energy into kinetic energy:

Can you solve for v in terms of g and H?

Use Bernoulli’s equation and set equal the energies at the red location equal to the energies at the green location.

The pressure in the water increases with depth. As the water leaves the tower, it converts this pressure energy into kinetic energy:

P1 + K1 + Ug1 = P2 + K2 + Ug2

P1 + 0 + 0 = P2 + K2 + 0

The pressure in the water increases with depth. As the water leaves the tower, it converts this pressure energy into kinetic energy:

P1 + K1 + Ug1 = P2 + K2 + Ug2

P1 + 0 + 0 = P2 + K2 + 0

What is the pressure equal to at P1?
What is the pressure equal to at P2?

P1 + K1 + Ug1 = P2 + K2 + Ug2

P1 + 0 + 0 = P2 + K2 + 0

where P1 = ρgH + P0, P2 = P0

ρgH + P0 = P0 + 0.5ρv2

ρgH = 0.5ρv2

v = √(2gH)

P1 + Ug1 + K1 = P2 + Ug2 + K2

Pressure energy inside converts to kinetic energy outside.

Pressure energy converts to kinetic energy, which powers the turbine.

The more pressure, the more kinetic energy your water will have when coming out of your tap!

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