Experiments with water, a scale, and a rubber duck • Minimum Viable Blog

I did some science experiments this morning, and I’m here to share my results. The experiments ultimately failed to produce any good data, but I think the process was interesting anyway.

Background

I was making pizza dough this week, which for me is a process involving a kitchen scale, a thermometer, and a three-day cold ferment. I always start with a bowl on a scale and add water, kosher salt, sugar, and yeast. I noticed that when I add the salt, its weight doesn’t register on the scale until it hits the bottom of the bowl (kosher salt doesn’t float). But when I add yeast, the weight registers while it floats on top of the water.

I’m sure this isn’t news to anyone with a basic knowledge of hydraulics, fluid mechanics, or even college physics. But for some reason it really piqued my curiosity. I started thinking about it and came up with a couple questions:

Is the yeast being measured accurately? Is its weight on top of the water the same as its weight would be if I dumped it on a bare scale?
How does the scale even know how much the yeast weighs? It’s separated from it by a couple inches of water, and we usually think of floating things as being weightless. The weight of the yeast registers long before it saturates or mixes with the water; it’s just floating there. Doing yeast stuff. Vibing. Farting.
How long does it take for the weight of the yeast to exert force on the scale? For me it seemed instant, but I know just enough science to be skeptical of anything that seems instant.

I then came up with a thought experiment, which I posed to my wife (who has a Physics degree) and two of my brothers (very smart software engineers).

Say you pick up a lake, Lake Michigan maybe, and place it on top of an enormous bathroom scale. The scale is infinitely precise and instantly registers any force on it. Then you put a single rubber duck on the surface of the lake, right in the middle. How long does it take for the weight of the duck to register on the scale?

This led to some spirited discussion on all sides.

One instinct you might have is to say it’s instant. After all, as soon as the duck is in the water, the combined mass of duck and water is immediately changed. But you have to remember that scales don’t measure mass. They measure contact force. The scale doesn’t know the duck is there and it doesn’t know anything about the combined mass of duck and water. All it knows is how much force is pressing down on it.

A basic principle of science is that nothing can travel faster than the speed of light. So if we take the depth of the lake and divide it by the speed of light, we know for certain that the force on the scale won’t change faster than that. But my instinct was that it would be even slower than that—a lot slower. I know that mechanical information (pushing and pulling and so forth) propagates through matter at the speed of sound. When you pick up a stick and wave it around, the far end of the stick lags behind the part you’re holding by a few tiny fractions of a second (the length of the stick divided by the speed of sound in wood). It looks instant but it definitely, 100% is not instant. Just really fast.

So I theorized that the time required for the weight of the duck on the lake to register on the scale would be at least the depth of the lake divided by the speed of sound in water. But could it be more than that? Did the duck’s weight need to propagate only to the bottom of the water? Or to all the bottom corners of the lake? Or all the way to the edges of the surface and then down? I needed to think about how the weight was propagating.

Another piece of the puzzle, I knew, was Archimedes’ Principle. The way I was taught this principle was:

An object will float when it has displaced as much water as its own mass.

In other words, for a 10 gram object to float, it needs to displace (push aside) 10 grams of water. If it’s too small to displace 10 grams of water, it will sink. If it’s big enough to displace 20 grams of water, it will float half in and half out of the water. That’s a simplified version of the principle, but good enough for us to work with.

I did some research on physics forums and learned that when an object displaces 10 grams of water, that water is pushed outward, which raises the water level. This increases the pressure on the bottom of the water by 10 grams, which increases the weight on the scale by 10 grams. You know how the pressure at the bottom of the ocean is greater than the pressure at the bottom of a swimming pool? That’s because the ocean is deeper. So when you make a body of water deeper, you increase the pressure (and therefore the weight) on the bottom.

To me it seems like magic that the weight of a floating object would increase the weight on the bottom of the water by the same amount. But there’s the beauty of physics for ya.

^{Okay, it’s not quite exactly the same amount. All objects are slightly buoyant in air (Archimedes’ Principle works for air too!), so a tiny tiny fraction of the weight of the object will be supported by air instead of water. But that amount will be too small to measure without extremely fine equipment.}

Hypothesis

I came up with two hypotheses:

If I weigh a rubber duck on a bare scale, then weigh it on top of a bowl of water (and subtract the weight of the water and the bowl), it will come out the same.
If I weigh a rubber duck on top of a tall vessel of water (like a vase), then pour the water into a short vessel (like a casserole pan) and weigh the rubber duck again, it will take longer for the weight to register in the short vessel than in the tall vessel.

I don’t stand by these hypotheses, these were just what I came up with on the spot. I needed to test them.

Design

I designed two experiments which I thought could prove or disprove my hypotheses.

Weigh a rubber duck on a scale. Fill a bowl with water and weigh it. Put the rubber duck on the water and write down the weight. Then subtract the weight of the bowl and water, and see if it’s the same.
Using a stopwatch camera, video myself putting the duck in a tall vessel of water, then video myself putting the duck in a short vessel of water. See how long it takes for the duck’s weight to register on the scale in each case.

Experiment #1

I grabbed one of my kid’s rubber duckies and weighed it. Then I weighed a flower vase, filled it with water, and weighed it again. Then I put the duck on top and weighed it again.

My first hypothesis was correct: the duck’s weight on the water was the same as its weight on a bare scale. (It actually registered 1 gram heavier on the water—a gram is about three hundredths of an ounce—but I’m pretty sure that’s because my scale isn’t quite accurate to the gram).

For funsies, I tried a couple other quick experiments here.

First, I touched the duck to the water and held it up a millimeter or so so that the the water would “stick” to the bottom of the duck. This reduced the weight on the scale by about a gram. That’s the power of cohesion, which is a whole other story, but very cool!

Second, I pushed the duck down in the water, deeper than it would normally float. I made sure not to get my fingers in the water, just the duck. I was able to make the weight on the scale 57 grams heavier by doing this.

Experiment #2

I downloaded the Video Stopwatch SC app on my iPhone. I filmed myself dropping the duck on a bare scale three times so I could figure out how long it takes for the scale to process a change in force. Then I put the vase full of water on the scale and filmed myself dropping the duck in the water three times, then poured the water into a short casserole pan, made sure I was dealing with the same water weight in both cases, and filmed myself dropping the duck in the water three more times.

Then I made a spreadsheet in Google Docs. I recorded the weight on the scale in all my experiments, then scrolled through each video a frame at a time and marked down the timestamps for when I let go of the duck, when the scale started to change, and when it stopped changing.

I took the average of each set of three attempts and compared them.

Unfortunately, the raw data didn’t indicate any significant change between dropping the duck on a bare scale, dropping it in a tall vase of water, and dropping it in a short pan of water. I calculated that my measurements were only accurate to one tenth of a second, so with this considered, the difference in the results was zero. This means I have to discard them, since I already know the change in weight can’t propagate faster than the speed of light.

At this point, my friends, I finally looked up the speed of sound in water…and discovered that my entire experiment had been a fool’s errand. Sound travels over 4800 feet per second in room temperature water. That means it can travel from the top of my vase to the bottom in just over one ten-thousandth of a second! My measurement accuracy was a thousand times too low. In order to measure a tenth-of-a-second difference with my stopwatch app, I would need a vase almost 500 feet tall - roughly the height of the Great Pyramid in Egypt, or half the height of the Eiffel tower.

If the weight propagates faster than the speed of sound for some reason, I’d need a much, much bigger vase.

So no, we will not be finding out today how quickly the weight of a rubber duck propagates to the bottom of a body of water. Sorry about that.

Data

Here’s my spreadsheet: https://docs.google.com/spreadsheets/d/1rCkqxmNEtIv32ExEiI3KSqHMolF4IfBAS4MPeNEVUlU/edit?usp=sharing

And here are all my photo and video files: https://drive.google.com/drive/folders/1hM_ljBW0YXJ0SSRRnKy5nZHlb8itBcfu?usp=sharing