It’s amazing how much physics fun you can have with basic everyday stuff. Next time you’re watering the yard, stop and look at that simple spinning sprinkler. Watch the jets of water as they shoot out—why do they form those pretty arcing streams? A lot of times, a physicist’s job comes down to figuring out why the world is so beautiful.
Even more fun, once you know how something works, you can run little thought experiments by changing things up. That’s what Destin Sandlin and Steve Mould did on YouTube. Both of these guys have their own (excellent) science channels. But somehow they got to arguing about what would happen if you made a sprinkler that shot water inward, toward the center hub, instead of outward. If you took a snapshot from above, what shape would the water streams assume?
To settle the matter, Steve built just such an outside-in sprinkler, and the two got together on Destin’s Smarter Every Day channel to test it. First, though, to simplify the problem, they tried a different one that Steve had made that shot the water straight outward, away from the hub (i.e., without the angled nozzle these sprinklers usually have to produce rotational thrust).
But even on this one, their predictions were different. Destin said the water would keep moving in the direction of rotation it had when it left the tube—meaning it would go forward as well as out. Steve said it would curve back. What do you think?
Here’s what’s so interesting about this: These guys both have a solid understanding of physics, but they’re talking about two different things. Steve is focusing on the shape of the water stream—a snapshot view—and Destin isn’t quite hearing it. He’s thinking about the movement of individual particles of water. Those are very different questions!
You can watch the video and see what happens (and you should), but there is another option. What if we just model the water as a bunch of tiny balls instead of a continuous stream? Each ball will start with an initial exit velocity that depends on the rotation of the sprinkler and the speed of the water as it comes out of the tube. Then you can look at the motion of many of these “water balls” to see the aggregate pattern. That’s what I’m going to do. It’s true that you don’t really understand something until you can model it!
Squirters Pointed Outward
Let’s start with the case of the water shot radially outward from the spinning sprinkler. First, we need to recognize that once a water ball leaves the tube, there are no forces acting on it. (OK, there would be gravity, but it doesn’t affect what we’re interested in, so we’ll ignore that.) With zero net force, the ball will travel with constant velocity (same speed and direction).
I’m going to model this in Python, and I’ll use just two squirter tubes to make it easier. Then I just need to pick some values for the speed of the water, the length of the tubes, the rotation rate, and the number of water balls per second. Here’s the model. You can click on the pencil icon to see the code; hit Play to run it.
As you can see, the shape of the stream bends back from the spinning tubes. So Steve was right about that. You can go into the code and try different assumptions to see how things change. What happens if you turn the conceptual faucet handle to increase the water speed?
There are some other programing details in there that could be interesting, but I’ll gloss over that stuff for now. The important thing is that this does appear to model the spinning sprinkler using basic physics. That’s a win for me.
Squirters Pointed Inward
So back to the original question: What happens if we bend the tubes so they shoot water inward, toward the center of rotation? Steve built a separate device for this, but in our model we can get the same effect just by changing the sign on the water velocity from positive to negative. (In the code above, it’s parameter a in line 16.) Here’s what it looks like:
Is that awesome or what? If Steve wanted to patent and sell this sprinkler, I’d buy it. (Of course he’d have to think of a way to make it rotate on its own; they got pretty wet spinning it by hand.)
But thinking like Destin now, what is each water particle doing? It’s not so obvious. So in this next animation, I’m going to make one of the balls yellow and track it as it moves. Here’s what that looks like (and here is the code if you want to see it):
The balls move in a straight line! Specifically, the exit velocity of the ball is a combination of a velocity vector pointing inward and another one tangent to the circular path of the sprinkler outlets, pointing forward. The result is that the water particles move sort of diagonally inward and forward, which is pretty much what Destin said.
So there’s a real optical illusion going on here. Every water ball travels in a straight line, but our eyes see only the overall shape of the stream. And because each successive ball is released at a different point in space with a slightly different velocity vector, it looks like they’re circling around and curving back. In the end, both Steve and Destin were partly right.
Take It for a Spin
But wait! There is another way to model the motion of the water. So far we’ve been viewing this from above, as if with a stationary video camera, like they do in the video. That defines our “reference frame,” from which we describe how the water is moving. But just for fun, what if we looked at it in the rotating reference frame of the sprinkler?
This is what you’d get if you mounted a videocam above the sprinkler that rotated at the exact same angular velocity. You might remember Fred Astaire’s “Dancing on the Ceiling” scene in Royal Wedding, where they do something very much like this. (Watch this one if you want to see how the magic was made, but it’s more fun if you watch the scene with innocent eyes first!)
In this case, we need to build the model differently. In the previous model, the water balls moved with a constant velocity since they had zero net force acting on them and thus zero acceleration. We can write this as the following expression for force and acceleration.
However, that force-motion model only works in a non-accelerating reference frame (we call that an inertial reference frame). If we want to look at the water from the viewpoint of a rotating camera, that means that we now have an accelerating reference frame. That’s a problem.
There is a way to make the motion work in the accelerated frame. The key is to add fake forces. You actually know about these. Remember the time you were in the car and you pushed the gas pedal to speed up—that force pushing you back in your seat? Actually, there was no such force; your brain made it up to explain how you felt inside the accelerating reference frame of the car.
For a rotating reference frame, there are two fake forces we need to add. There is the centrifugal force that (supposedly) pushes outward from the center of rotation, and then there is the Coriolis force. This latter one takes a bit of explanation to fully understand. Instead, let me just show the fake force vectors on a ball of water in a rotating frame.
The key to the Coriolis force is that it is always perpendicular to the object’s velocity (in the rotating reference frame). Here is a model of the motion of a water ball in the rotating reference frame. It’s just an image this time, but if you want to see the code, it’s here.
See? That looks just like the streams of water we got before. But these are not streams—they’re the trails left by just two water balls! So in the stationary reference frame, the balls appear to move in a straight line. But from the sprinkler’s own rotating perspective, the balls appear to follow a curving path (with non-constant velocity).
Which way is better, a non-rotating reference frame with real forces or a rotating frame with fake forces? In the end they give the same result, but as a teacher I guess I prefer the first one, just because it doesn’t require any fakery.
Oh, don’t get me wrong. There are situations in which fake forces make things much easier—like tossing a ball in an accelerating elevator. But fake forces can be dangerous, since they aren’t due to fundamental interactions. That’s why most introductory physics courses avoid them.
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