The buoyant force does not get smaller as you sink

Imagine you’re hanging out with your friends on a Friday night, when your good friend Jacques texts you and asks you to join him on a trip to the bottom of the ocean. Jacques has a brand new submarine that he’s been itching to try out, and he wants you to come with him to check out some hydrothermal vents at the bottom of the Marianas Trench that he’s been talking about for weeks.

You don your itchiest wetsuit and climb aboard his submarine, which has suspiciously thick iron walls given how tiny and cramped it is inside. Jacques reminds you that the super-thick walls are necessary in order to survive the descent to the bottom of the Trench, since the external pressure down there is nearly 1000 times what you’re used to at sea level. The stiff walls hold the pressure inside constant even while the pressure outside increases. The walls themselves actually get compressed due to the gradually-increasing difference in pressure across them. The reason that they don’t collapse onto you and Jacques is that they compress (like springs), which counter balances the force arising from the pressure difference. So as the submarine dives deeper, the internal pressure (thankfully) stays the same… but the walls actually get thinner!

Recall that, at sea-level, you experience a pressure of about 101 kPa due to the weight of the Earth’s entire atmosphere sitting on top of you. Because air is not very dense this pressure only barely varies with elevation. For example, at the top of the Empire State Building the pressure is roughly 95 kPa. It’s only at colossal distances that the change in pressure becomes noticeable. At the top of Mount Everest the pressure is closer to 33 kPa. The decrease in pressure occurs simply because as you go higher there is less air above you pushing down on you.

Under water is a different situation. Because water is very dense, pressure rapidly increases with depth in the ocean. For every ten meters deeper you dive, the pressure of the surrounding water increases by an amount that’s equal to the total ambient pressure that you feel when you’re at the surface (101 kPa). So if you dive down to 10 m, the total pressure your body feels is now 202 kPa. If you dive even deeper to 20 m, you’ll feel 303 kPa! It’s important to remember that this pressure arises purely due to the combined weight of all the water that’s sitting above you. If you were diving on a planet with less gravity than Earth, the pressure you feel at 10 meters would be less than 202 kPa.

Jacque’s submarine is filled with air, which is much less dense than water. As you’d expect, the sub would float on the water’s surface for the same reason that boats and bubbles float. The force that allows the submarine to stay afloat is known as the buoyant force. In order to successfully descend, the submarine has to use a propeller that pushes against the buoyant force and drives the sub deeper into the ocean.

A strange property of the buoyant force is that it stays the same regardless of how deep you go; it is independent of the surrounding pressure. This means that, if you were watching Jacques’ submarine dive at a constant speed, it would appear that the propeller always spins at the same speed and that the engines consistently draw the same amount of fuel. Because water is incompressible, its density, stickiness, and other properties stay pretty much the same as you go deeper… and so the buoyant force stays the same as well.

Back on land, you decide to write down some equations to describe Jacque’s submarine. You start by making a free-body diagram describing the forces that push and pull on the submarine as it sinks. The first one is obviously gravity, which exerts a force F${}_g$‍ proportional to the mass of the sub (M${}_{sub}$‍)

where g is the acceleration due to gravity, 9.8 m/s/s. But you know that gravity isn’t the only force that acts on the submarine. There must be a buoyant force that acts to counteract it. The submarine is filled with air (like a balloon), so you expect it would float on top of the water were it not for the action of its propeller: even though the steel walls of the submarine make it heavy, this added weight is offset by a huge amount of trapped air. This effect is similar to a balloon, which doesn’t float until it’s filled with air. The submarine thus requires a propeller that acts together with gravity to make the submarine sink, despite the buoyant force acting on it.

Another way of thinking about this is to realize that the submarine floats because it is less dense than the surrounding water. Even though the steel walls are definitely denser than the surrounding water, the enclosed air is much less dense, and so the overall density of the submarine is less than water and so it floats. So unlike a rock (denser than water) that immediately sinks when it’s dropped into the ocean, the submarine has to actively use energy (in the form of a spinning propeller) to push itself against the buoyant force and in the direction of gravity. The equation that gives you the size of the buoyant force is called Archimede’s principle, and it states:

This says that the buoyant force F${}_b$‍ acting on the sub is determined by the density of water (d${}_{water}$‍, which is roughly 1 g/cm${}^{\text{3}}$‍), the volume of the submarine (V${}_{sub}$‍), and gravity (g). You might be surprised to see that “g” appears in the equation for buoyant force, which always points in the opposite direction to the weight of an object due to gravity. But there’s a good reason for this: the buoyant force on an object equals the total weight of water that it pushes out of the way. For bigger subs (larger V${}_{sub}$‍), more water needs to get pushed out of the way, and so the buoyant force is larger.

We can combine this equation with the weight of the sub to get the net force acting on the submarine.

F${}_{propeller}$‍ is the force that the propeller exerts during the dive. If F${}_{total}$‍ is positive, the submarine floats upwards, and if F${}_{total}$‍ is negative, the submarine sinks. Jacques can determine which one of these occurs by altering F${}_{propeller}$‍. The free-body diagram corresponding to this equation looks something like:

In the equation for F${}_{total}$‍ there is no term that is proportional to depth, which shows that the buoyant force does not get smaller as you sink. This is because water is incompressible. For this reason, a stiff object like a steel submersible (which has a roughly constant volume as it dives, since the compression of the walls is a tiny effect) displaces the same amount of water regardless of whether it’s just below the surface or right next to a sperm whale at the bottom of the Marianas trench. Because the volume is the same at any depth, and the density of water is the same at any depth, the total mass of displaced water (mass = volume x density) is the same at any depth—making the buoyant force constant.

The amount of dissolved gases in your bloodstream is related to the pressure that you’re sitting at. That means that when you scuba dive with a pressurized air tank, the amount of dissolved nitrogen in your bloodstream actually increases as you get deeper. If, at the end of a dive, you attempt to re-surface very quickly, this dissolved nitrogen will suddenly exit your bloodstream in the form of tiny bubbles that can disrupt or damage your blood vessels. This condition is known as decompression sickness or the bends, and it constitutes a life-threatening risk in deep-sea divers. The primary means by which divers can avoid developing the bends is by carefully controlling their rates of re-ascent during a very long dive. The reason that this condition is less of a risk for submarine-divers is because submarines have thick walls that hold out the external pressure, keeping the internal pressure pretty much constant.