
Positive, negative, and neutral buoyancy Buoyancy is easiest to understand thinking about a submarine. It has diving planes (fins mounted on the side) and ballast tanks that it can fill with water or air to make it rise or fall as it needs to. If its tanks are completely filled with air, it's said to be positively buoyant: the tanks weigh less than an equal volume of water and make the sub float on the surface. If the tanks are partly filled with air, it's possible to make the submarine float at some middle depth of the water without either rising up or sinking down. That's called neutral buoyancy. The other option is to fill the tanks completely with water. In that case, the submarine is negatively buoyant, which means it sinks to the seabed. Find out more about how submarines rise and fall. Photo: Submarines can rise to the surface or sink to any chosen depth by controlling their buoyancy. They do so by letting precise amounts of water or air into their ballast tanks. Photo courtesy of US Navy. Buoyancy on the surface Now most boats don't operate in quite the same way as submarines. They don't sink, but they don't exactly float either. A boat partly floats and partly sinks according to its own weight and how much weight it carries; the greater the total of these two weights, the lower it sits in the water. There's only so much weight a boat can carry without sinking into the water so much that it... does actually sink completely! For ships to sail safely, we need to know how much weight we can put in or on them without getting anywhere near this point. So how can we figure that out? Archimedes' Principle The person who first worked out the answer was Greek mathematician Archimedes, some time in the third century BCE. According to the popular legend, he'd been given the job of finding out whether a crown made for a king was either solid gold or a cheap fake partly made from a mixture of gold and silver. One version of the story says that he was taking a bath and noticed how the water level rose as he immersed his body. He realized that if he dropped a gold crown into a bath, it would push out or "displace" its own volume of water over the side, effectively giving him an easy way to measure the volume of a very complex object. By weighing the crown, he could then easily work out its density (its mass divided by its volume) and compare it with that of gold. If the density was lower than that of gold, the crown was clearly a fake. Other versions of the story tell it a slightly different way—and many people think the whole tale is probably made up anyway! Later, he came up with the famous law of physics now known as Archimedes' Principle: when something is resting in or on water, it feels an upward (buoyant) force equal to the weight of the water that it pushes aside (or displaces). If an object is completely submerged, this buoyant force, pushing upwards, effectively reduces its weight: it seems to weigh less when it's underwater than it does if it were on dry land. That's why something like a rubber diving brick (one of those bricks you train with in a swimming pool) feels lighter when you pick it up from underwater than when you bring it to the surface and lift it through the air: underwater, you're getting a helping hand from the buoyant force. All this explains why the weight of a ship (and its contents) is usually called its displacement: if the ocean were a bowl of water filled right to the brim, a ship's displacement is the weight of water that would spill over the edge when the ship were launched. The USS Enterprise in our top photo has a displacement of about 75,000 tons unloaded or 95,000 tons with a full load, when it sits somewhat lower in the water. Because freshwater is less dense than saltwater, the same ship will sit lower in a river (or an estuary—which has a mixture of freshwater and saltwater) than in the sea. Upthrust Unfortunately, none of this really explains why an aircraft carrier floats! So why does it? Where does that "magic" buoyant force actually come from? An aircraft carrier occupies a huge volume so its weight is spread across a wide area of ocean. Water is a fairly dense liquid that is virtually impossible to compress. Its high density (and therefore heavy weight) means it can exert a lot of pressure: it pushes outward in every direction (something you can easily feel swimming underwater, especially scuba diving). When an aircraft carrier sits on water, partly submerged, the water pressure is balanced in every direction except upward; in other words, there is a net force (called upthrust) supporting the boat from underneath. The boat sinks into the water, pulled down by its weight and pushed up by the upthrust. How low does it sink? The more it weighs (including the weight it carries), the lower it sinks: If the boat weighs less than the maximum volume of water it could ever push aside (displace), it floats. But it sinks into the water until its weight and the upthrust exactly balance.

The more load you add to a boat, the more it weighs, and the further it will have to sink for the upthrust to balance its weight. Why? Because the pressure of water increases with depth: the further into the water the boat sinks, without actually submerging, the more upthrust is created.

If the boat keeps on sinking until it disappears, it means it cannot produce enough upthrust. In other words, if the boat weighs more than the total volume of water it can push aside (displaces), it sinks.

Upthrust—made simple To get the idea of upthrust clear in your mind, think about what happens as you load a ship. With no load onboard, the ship sinks into the water by a certain amount. The amount of water it displaces (shaded area) weighs as much as the ship. The weight of the ship pulling down (red arrow) and the upthrust pushing up (blue arrow) are equal and opposite forces, so the ship floats. Now what if we start loading the ship? It sinks down further, displacing more water (bigger shaded area). The weight of the ship and its load pulling down (red arrow) and the upthrust pushing up (blue arrow) are still equal, but now both are bigger. Suppose we load the ship a bit more so that it just sinks beneath the surface but continues to float. Again, the weight pulling down and the upthrust pushing up are equal, even though both are bigger. But at this critical point, the ship is displacing as much water as it possibly can, so the upthrust cannot get any bigger. We didn't like the ship much anyway, so let's add a lead weight on top (a weight that's dense enough to sink all by itself). No matter how much weight we add, the ship cannot produce any more upthrust: once it's completely submerged, whatever depth it sinks to, it can only ever displace a certain amount of water and create a certain amount of upthrust. Now the weight of the ship is more than the maximum possible upthrust so it sinks to the bottom. Suppose we attached a giant weighing machine to the top of the ship at this point. The apparent weight of the ship plus its cargo would be much less than expected, by an amount equal to the weight of the displaced water (the size of the upthrust). In other words, if we wanted to raise the ship to the surface from the seabed, we'd need to use a lifting force equal to the difference between the weight and the upthrust (the red arrow minus the blue arrow). How do we know that the upthrust on something is equal to the weight of fluid it displaces? If you can bear a little bit of math, it's very easy to prove! We need to know two general bits of physics to do it. First, that pressure is defined as force per unit area (force divided by area), so the force on a given surface of the box is the pressure times the area of that surface. Second, that the pressure (P) at a given depth (h) in a fluid is equal to the depth times the density (ρ) times g (the acceleration due to gravity). Or P = h × ρ × g. Now take a look at this submerged box. How big is the upthrust? The water pressure at the top of the box is h1 × ρ × g while the pressure at the bottom is h2 × ρ × g. The difference in pressure is (h2−h1) × ρ × g = h × ρ × g. Because the area of the box is the same throughout, the difference in force is simply the difference in pressure times the area of the box: h × ρ × g × (w × l) But because (h × w × l) is the volume of the box, and ρ is its density (or its mass per unit volume), that's the same as saying the difference in force is equal to m × g, where m is the mass of the box. m × g is another way of writing the weight of the box. So we've very quickly proved that the upthrust is equal to the weight of the fluid the box displaces. In other words, the more fluid the box displaces (the bigger the box), the bigger the upthrust. And that's why bigger boats—ones spread out wider, to occupy more volume—can carry more stuff.

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