Figure 14-56 shows a siphon, which is a device for removing...

Figure $14-56$ shows a siphon, which is a device for removing liquid from a container. Tube $A B C$ must initially be filled, but once this has been done, liquid will flow through the tube until the liquid surface in the container is level with the tube opening at $A$. The liquid has density $1000 \mathrm{~kg} / \mathrm{m}^{3}$ and negligible viscosity. The distances shown are $h_{1}=25 \mathrm{~cm}$, $d=12 \mathrm{~cm},$ and $h_{2}=40 \mathrm{~cm} .$ (a) With what speed does the liquid emerge from the tube at $C ?$ (b) If the atmospheric pressure is $1.0 \times 10^{5} \mathrm{~Pa}$, what is the pressure in the liquid at the topmost point $B ?$ (c) Theoretically, what is the greatest possible height $h_{1}$ that a siphon can lift water?

Figure $14-56$ shows a siphon, which is a device for removing liquid from a container. Tube $A B C$ must initially be filled, but once this has been done, liquid will flow through the tube until the liquid surface in the container is level with the tube opening at $A$. The liquid has density $1000 \mathrm{~kg} / \mathrm{m}^{3}$ and negligible viscosity. The distances shown are $h_{1}=25 \mathrm{~cm}$, $d=12 \mathrm{~cm},$ and $h_{2}=40 \mathrm{~cm} .$ (a) With
what speed does the liquid emerge from the tube at $C ?$ (b) If the atmospheric pressure is $1.0 \times 10^{5} \mathrm{~Pa}$, what is the pressure in the liquid at the topmost point $B ?$ (c) Theoretically, what is the greatest possible height $h_{1}$ that a siphon can lift water?

Key Concepts

Siphon

A siphon is a device that uses fluid dynamics to move liquid from one container to a lower level without the need for a pump, relying on gravity and atmospheric pressure. In a siphon, a tube initially filled with liquid allows the liquid to be drawn up and then flow downhill, using the pressure difference between the start and end points.

Bernoulli's Principle

Bernoulli's principle is a fundamental concept in fluid dynamics that relates the pressure, velocity, and gravitational potential energy of a fluid. It states that along a streamline in steady flow, the sum of the pressure energy, kinetic energy per unit volume, and potential energy per unit volume remains constant, which is critical for analyzing the liquid's speed as it exits the siphon.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at rest due to the weight of the fluid above a given point. It increases with depth and is calculated as the product of the fluid density, gravitational acceleration, and the height of the fluid column, playing a key role in determining the pressure differences in the siphon system.

Atmospheric Pressure

Atmospheric pressure is the force per unit area exerted by the weight of the atmosphere. It serves as a reference pressure in many fluid dynamics problems, including siphoning, where the external atmospheric pressure helps to push the liquid through the system and defines limits such as the maximum height a siphon can theoretically lift liquid.

Maximum Siphon Height

The maximum siphon height refers to the theoretical limit to which a siphon can lift a liquid from a reservoir, governed primarily by atmospheric pressure and the properties of the liquid. This takes into account the balance between the upward force provided by atmospheric pressure and the gravitational force acting on the liquid column, establishing an upper boundary for the siphon’s operation.

Transcript

00:05 Okay, for this question, we have a few things given to us.

00:09 I kind of drew a little diagram over here on the right with the heights in meters rather than centimeters.

00:16 Then we have our density of the liquid, which is 1 ,000.

00:22 And then we'll have our atmospheric pressure, which is 1 times 10 to 5.

00:29 Okay, for the first part of this question, it wants us to find the velocity of the water exiting at sea.

00:35 So what we're going to do is we're going to use bernoulli's ecclation to find this.

00:41 And so we add all the pressures up together.

00:44 So let's split this diagram up.

00:47 So here we have the pressure of the atmosphere pressing down.

00:55 And we also have the pressure of the atmosphere pressing up at the bottom of the tube.

01:00 And then here in the water, we'll have the pressure of the water pressing down.

01:07 And then we also have the pressure of the water and the tube pressing down the same height and then we have this pressure coming down and then we have this whole pressure coming down the entire tube then we'll have the velocity exiting right there so let's let's add these things together so on the left side of the equation we'll split this um we'll split this up right here so that'll be the difference between the left side and the right side of the equation.

01:52 So for the left side of the equation, we'll have all of these pressures.

01:59 So let's write that out.

02:01 So we'll have the atmospheric pressure, pressure zero, going in the positive direction because the water is moving this way.

02:11 So that's the positive direction plus the pressure of the water in the container, just go in the positive direction minus the pressure of the water in the tube which is in the water so we'll that wc and then which is they're actually equal to each other plus oh sorry minus row g h1 of this bit here we split down the middle then we'll have plus row g h1 plus distance plus h2 which is all this here.

03:06 So the and that's one side of the equation.

03:09 So we set that equal to the pressure going that way plus one half mb squared.

03:31 So these pressures cancel out.

03:34 This one cancels out.

03:36 These cancel out.

03:39 And we're left with some things here.

03:42 And these can't these ones here cancel out because these pressures are the same going the opposite directions.

03:49 And same with that.

03:50 Atmospheric pressures.

03:52 So our final equation will be one half, sorry, not mass.

03:59 That's row.

04:00 We need row for there...