Torricelli's theorem has practical applications in everyday life. This physical law explains an important relationship between fluid exit velocity and the height of that fluid in a container. Here, you will learn about this relationship as well as see how to calculate exit velocity using Torricelli's theorem.
When you watch the weather report at night, you get lots of valuable information, like the high and low temperatures, the chance of rain, and the barometric pressure. This last one we have thanks to a man named Evangelista Torricelli, who was quite the scientist! Not only did he invent the mercury barometer, which measures atmospheric pressure, but he also was the first to correctly describe what causes wind, as well as designed and built some telescopes and microscopes.
Torricelli was interested in many different aspects of mathematics and physics. One of his greatest achievements is Torricelli's theorem. This theorem, or law, describes a relationship between fluid leaving a hole in a container and the height of the fluid in that container.
The relationship can be summed up this way: if you have a tank of fluid and there is a small hole in the bottom of the tank, the fluid will leave through that hole with the same velocity as it would experience if you dropped it from the same height to the level of the hole.
Whew! That doesn't really sound like a summary, does it? In its simplest form, what this means is that if you dropped the fluid from a certain height h, the fluid would have a certain velocity v as it fell from that height to the hole. What's cool is that this v is the same velocity at which the fluid leaves the hole in the container when the height of the fluid h is the same as if you dropped the fluid in the container.
Spouting Can Experiment
We can see how the velocity of the fluid leaving the hole relates to the height of the fluid in the container if we look at a simple experiment with a can of water. If you fill a can with water and you punch a small hole near the top, some water will come out of that hole at a certain velocity.
If you punch a hole a little further down, the water will come out of that hole with an even greater velocity because it is lower in the fluid, which is the same as saying that there is a greater height of fluid above that hole.
Punch a hole all the way down at the bottom, and it will spout out with an even greater velocity because now there is a much greater height of fluid above it.
It's important to note that we are making a few assumptions about the fluid in this container when we do this experiment. First is that the container itself is open to the atmosphere - there's no lid or anything else covering the top.
We are also assuming that this is an ideal fluid. This means that the fluid is incompressible, has laminar flow, and is non-viscous. These can't be neglected, because we can't apply the same rules to non-ideal fluids since their flow and viscosity may not be constant across the fluid itself.
Equation and Example
Since we're doing physics, you've probably already guessed that Torricelli's theorem can also be written as an equation. We write it as: v = √(2gh), where v is the velocity of the fluid, g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the fluid in the container. Since 2 is a constant, and g is always 9.8 m/s^2, if we want to determine the velocity of the fluid coming out of a hole in the container, all we need is the height of that fluid above the hole.
While the tank experiment looks cool, you may still be asking yourself that age-old question: 'when am I ever going to use this?' I mean, really, when would you ever need to calculate the velocity of a fluid coming out of a hole? Believe it or not, there are some very practical applications of Torricelli's theorem in everyday life. And I'll bet you've used it before without even knowing it! Every time you've opened a valve at the bottom of a tank to let some fluid out, like with a rain barrel or water cooler, you've used Torricelli's law!
But let's look at another way that we can use Torricelli's theorem. Say that you are in charge of a dam that has been built across a river. One day, that dam springs a leak, and water starts coming out of a small hole near the bottom. You probably want to figure out how fast the water is coming out so that you know how much time you have (or don't have) to fix that leak!
Torricelli's theorem to the rescue! All we need to do is plug in our values and solve for the velocity. We know that 2 is still 2 and that g is still 9.8 m/s^2. In this case, perhaps your hole is 50 m below the surface of the fluid, so this is the h of the equation. Remember, it's not the absolute height of the fluid, it's the height of the fluid above the hole that we use to calculate the exiting fluid's velocity.
Let's do the math! Once we plug in our values, we find that v = √(2*(9.8 m/s^2)*50 m). Once we solve, we find that v = 31.31 m/s. Sounds like you need to get that leak fixed pretty quickly!
At other times, you may want to find the height of the fluid above the hole. As long as you know the velocity of the fluid leaving the hole, Torricelli's theorem will serve you well again. Simply rearrange the equation so that the height is alone on one side. To do this, you will need to square both sides, then divide both sides by 2*(9.8 m/s^2). Now you have h = v^2/ (2 * (9.8 m/s^2)), and you can solve for the height.
Torricelli's theorem is useful because it describes a relationship between the velocity of a fluid leaving a hole in a container and the height of the fluid in that container. As we saw with the leaky dam, understanding this relationship can provide valuable information in real-life situations.
This theorem explains to us that the velocity of a fluid exiting a container through a small hole is the same velocity the fluid would have if you dropped it from a certain height. This height is the distance between the top of the fluid and the hole itself. Holes that are lower in the fluid will have a greater velocity than those closer to the top because the height of the fluid above the holes at the bottom is greater.
Torricelli's theorem also comes in equation form: v = √(2gh), where v is the velocity of the fluid, g is the acceleration due to gravity, and h is the height of the fluid above the hole. Using this equation, we can solve for either the height or the velocity of the fluid, as long as the other is known.
Once you have completed this lesson, you should be able to:
- State Torricelli's theorem and write it in equation form
- Use Torricelli's theorem to solve for the height or velocity of a fluid exiting a container