A water tower supplies water through the plumbing in a...

A water tower supplies water through the plumbing in a house. A 2.54 -cm-diameter faucet in the house can fill a cylindrical container with a diameter of $44 \mathrm{cm}$ and a height of $52 \mathrm{cm}$ in 12 s. How high above the faucet is the top of the water in the tower? (Assume that the diameter of the tower is so large compared to that of the faucet that the water at the top of the tower does not move. $)$

Key Concepts

Volumetric Flow Rate

This concept refers to the volume of fluid that passes through a given cross-sectional area per unit time. In fluid mechanics, it is calculated as the product of the cross-sectional area and the fluid velocity. It’s essential for relating the time taken to fill a container to the speed of the water exiting the faucet.

Continuity Equation

The continuity equation states that for an incompressible fluid, the mass flow rate must remain constant from one cross-section of a pipe to another. This relationship links the small cross-sectional area of the faucet to the volumetric flow rate and is fundamental in computing the velocity of the fluid exiting the faucet.

Bernoulli’s Principle

Bernoulli’s Principle relates the pressure, velocity, and height in flowing fluids, asserting that an increase in the velocity of the fluid occurs simultaneously with a decrease in pressure or potential energy. This principle is applied to equate the kinetic energy of the water exiting the faucet to the gravitational potential energy corresponding to the height of the water in the tower, thereby determining the effective elevation of the water source.

Transcript

00:01 In this exercise, we have a faucet connected to a water tower.

00:05 The faucet has a diameter, df, of 2 .54 centimeters.

00:13 And the faucet fills a container, a cylindrical container, that has a diameter of 44 centimeters and a height of 52 centimeters.

00:28 I'm sorry, 52 centimeters in a time of 12 seconds.

00:38 Also, the diameter of the water tower that the faucet is connected to is much bigger than the diameter of the faucet.

00:46 So we should consider that the water up in the water tower is not moving.

00:54 And we have to find what is the height of the water tower relative to the faucet.

01:01 So the first thing we need to do here is to calculate the speed with which the water leaves the faucet.

01:09 So the speed of the water when leaving the water faucet is going to be one over the cross -sectional area of the faucet times delta v .t, and this is the volumetric flow rate of the water leaving the faucet.

01:28 So what we need to do here is to calculate the volumetric flow rate.

01:34 And that we can do by calculating how fast the container, the cylindrical container, is filled.

01:46 So we have the volume of the cylindrical container, and that's going to be pi times the radius of the cylindrical container, which is dc over two, squared times the height of the single container.

02:03 This is the volume.

02:05 And the volumetric flow rate is going to be the volume divided by the time it took for this volume to be filled, which is delta t.

02:17 Okay, so this is going to be pi dc squared, hc, divided by delta, by four delta t.

02:28 And now we can substitute the values that we have here.

02:31 So this is going to be pi times dc squared, which is 0 .44 meters squared, times the height, which is 0 .52 meters, divided by 4 times delta t.

02:52 Delta t is 12 seconds.

02:54 So here we're going to have a volumetric flow rate of 6 .59 times 10 to the minus 3...

Please give Ace some feedback