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Let's Go For a Swim. Don't forget to Accelerate!

You take a deep breath, hold it, jump into the water, then slowly, your body comes up to the surface. This time, you took special care not to move a muscle once you entered the water. Four seconds later, you break the surface of the water and come up for air. If you take a deep breath before jumping off a cliff, and hold it, you don’t float in the air. Why is water so different?

Person jumping into water, swimming
Image by FREEP!K

It’s different, you say, because it’s heavier. Careful, I say. I think you mean water is more dense. Same thing, you retort. If you’re talking about the same volume of air vs. water, then yes, that much water is heavier than the same amount of air, but this is still referring to density, not weight. This is invoking inertia, not weight. On Earth, water feels heavier than air of the same volume, because water has more matter packed into that space. Matter packed into space is density. It feels heavier because it is being pushed upwards and it has more inertia, as discussed extensively in my last article, here.

Ok, we all know water is more dense than air, you say, so what’s your point? How can I float in water? Why do I feel lighter in the water? Let’s take this one step at a time. Do this simple experiment: mostly fill a clear water bottle with water, leaving some air space at the top and close the cap tightly. Now turn it on its side, and throw it up in the air without rotating it. Notice what happens to the air bubble. Try it and come back. I’ll wait. Before you launched it, the air bubble was at the top. While falling, the air bubble turns into a sphere, somewhere in the middle, not at the top of the bottle any more. Cool, huh?

spherical water, water in space
In space, water forms a sphere, and air doesn't rise, Credit: NASA

Einstein’s General Theory of Relativity explains that while falling, there is no support force on the fluids. Therefore, the only forces left, take over. Here this means the molecular forces of attraction of the water molecules cause surface tension where the water meets the air bubble, forming a nice sphere. When you catch it, you are providing a support force, pushing up on the system. The bubble moves up because the water is more dense. There is an awesome video by Smarter Every Day here, that helps us grasp this same concept using a plumb bob and a balloon. I highly recommend that you watch it before you read on. But, just in case you have no access, I’ll summarize the main points here: When you take a system and accelerate forward, the fluid inside that system moves backward due to its inertia. Objects more dense than the fluid move to the very back (plumb bob). Objects less dense than the fluid (i.e. the helium balloon) move toward the front.

Centripetal force machine at rest, inertia
Centripetal Force Machine at rest

This is also the principle that a centrifuge uses to separate materials based on their densities. Inspired by the Smarter Every Day video linked above, I invented this centripetal force machine to demonstrate the same principle in a classroom setting. This first link shows how the machine works, but in this second link, you can clearly see the effects of inertia and centripetal force on the system. The bobber, as expected, floats toward the center, whereas the fishing weight angles away from the center. Density separation is achieved with the centripetal force and the inertia of the water, bobber, and fishing weight.

Hopefully, you are now thinking, hey, it takes a force, it takes acceleration before

density separation of fluids and solids due to acceleration

objects will separate out into more dense (opposite direction of the force) and less dense (same direction as the force). Using the bottle example from above, when you catch the bottle and hold it in place, an upward force is constantly applied to the system (bottle/water/air). Inside the bottle, the water moves downward and the air moves upward. Why do they separate? How do they know which way to go? Inertia, pressure. As the bottle pushes up on the fluids (water and air), pressure builds up at the bottom due to inertia. Since water is more dense than air, it exerts more pressure (more force per area), which pushes on the air and the air has nowhere else to go but up because it is being pushed on with greater pressure at the bottom of its bubble and with less pressure at the top of its bubble.

There’s greater pressure the deeper you go in a fluid. Why? No, really think about it, why? Stop here, I’ll wait. Ok, so you may have figured it out, and this depends on a few things. 1. How much force is pushing on the fluid and in what direction. If the force is pushing from the left, then you measure how deep the fluid is starting from the right hand surface to the left side of the container. If the force is pushing up from the bottom of a container, then depth is measured in the normal way (top to bottom). 2. What type of fluid is it? Is it compressible? Air is a gas, which is compressible; the harder you push on it, the less space it occupies. Water, not so much. Push on it, it pushes back and doesn’t compress much at all.

Since water doesn’t compress much, you can imagine this: with a constant force pushing on a container, compare the pressure in the container at a depth of 1m to the pressure at 2m depth. You guessed it, the one two meters deep has double the pressure. If you do the same with air, the one that is twice as deep will be what? More than double the pressure, or less? It’s compressible, so more depth equals more compression, so that leads to more pressure. This means more than double. But be careful, just because air’s density increases more with depth, doesn’t mean its density is anywhere near water’s density. Water is more dense to begin with and even at sea level, where there’s an entire atmosphere’s worth of inertia resisting the sea’s upward push, air is still 838 times less dense (than salt water, and 816 times less dense than freshwater).

When you measure the air pressure at a certain elevation (or altitude), you are measuring the resistance that the column of air above you has to being pushed upward by

air pressure at altitude

the ground (or by the air below it). Imagine the column of air above you going straight up to space, until there are no more molecules to resist the force. It’s the inertia of all those molecules you are measuring. Getting technical, you’d have to go up 10,000km (6,200mi) to get above all the atmosphere, but in reality, you can get above most of it in the first 100km (62mi).

To measure water pressure, just add the pressure of the water above you, but if you want the total pressure under water, don’t forget to add the inertia of the column of the atmosphere above the water before you go under! For example, at sea level, it takes 100km of air to give you 1 atmosphere of pressure, but going under water 10.33 meters will add another whole atmosphere of pressure to that, making the total pressure at that depth 2 atmospheres. Put another way, because water is so much more dense, water takes only 10.33 meters to give the same pressure as air’s 100,000 meters!

water pressure at different depths, atmospheres

Ok, now that we’ve dealt with all that, can you imagine what causes you to feel lighter when you jump into the water? You can visualize buoyancy. When you are in a fluid and there’s a force on the fluid, it puts a force on your body. Jump into water and the water pushes with more pressure on the deeper parts of your body and less pressure on the shallow parts. Add the forces together and you come up with a combined (net) force pushing up on your body. If the net force is greater than the force of your inertia (weight), then you rise in the water. If the net force is less than your inertia (weight), you sink. If you want to know the value scientifically, Archimedes solved it for you. The upward buoyant force on your body is equal to the inertia (weight) of the fluid that you displace. So, if you swim in freshwater, you push aside a certain volume of water. Put it on a scale and measure its inertia (weight), and that is equal to the upward force on your body as you swim. Do the same thing in saltwater. Measure its inertia and you’ll figure out why you float better in the ocean. Saltwater is more dense!

Ok, now, let’s go back to the reason this is all happening in the first place. We learned earlier that in order for objects to separate into different densities when in a fluid there must be a force on the fluid, causing it to accelerate. What force causes all of those pressure differences in the first place? What force is ultimately causing you to accelerate when you jump into the water? What force is causing you to float on the surface? If you remember the name of this website, then you know the answer is not gravity. The force you are looking for is beneath your feet. Under the water. The core of the Earth supports the mantle which supports the crust, which supports the water, which supports you. All of these support forces (pushing up) are the cause of your acceleration (remember, you are accelerating when you take into account curved spacetime). At the submicroscopic level, they are due to electromagnetic repulsion (electrons repel electrons).

If everything on Earth’s surface was not accelerating upwards, if it were actually “at rest,” then everything would be floating. Just like in a falling bottle, dense and light objects would occupy whatever space they were in, not caring which direction was “up”. This is yet another reason that Einstein’s Equivalence Principle is so important to understand. Indeed, the world as we know it acts exactly like a spaceship which is constantly accelerating upward beneath our feet at a rate of 9.8m/s/s.

Welcome to spaceship Earth. In case of an emergency, and in the event of a water landing, your body can be used as a flotation device. Simply breathe in, hold your breath, and physics will take care of the rest for you. Have a pleasant swim!

Blue Marble, Earth from space, NASA, Apollo 17
NASA: Earth as seen from Apollo 17, 1972


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