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1 Charity Collector Guy: The principles of zeppelin flight are quite simple, really.
2 Charity Collector Guy: Hydrogen displaces an equal volume of air, but weighs less, so provides buoyancy.
3 Yeti: I know the basics. We use them a lot to travel the world troubleshooting cryptozoological, occult, Fortean, and other anomalies.
4 Steve: Yetis on a Zepi! Crikey!
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Archimedes was a Greek mathematician, physicist, and  in the spirit of the times  pretty much everything else in related fields of scientific inquiry and invention. He lived in the third century BC and was one of the greatest natural philosophers of the classical period.
The best known story about Archimedes is the one concerning King Hiero II of Syracuse, who had ordered a crown made of solid gold from a goldsmith. When the smith delivered the crown, King Hiero suspected that the smith may have adulterated the gold by mixing in some silver, allowing him to pocket the replaced gold for a tidy profit. It was known at the time that silver was less dense than gold  weighing less for a lump of the same size. So the King's suspicions could have been tested by melting down the crown into a regular shape and comparing its weight to the weight of a lump of gold of exactly the same size. However he didn't want to melt down the crown on a suspicion, and its shape was so intricate that making a copy in solid gold of exactly the same size would have been far too difficult. So Hiero called in Archimedes, known as a very clever man, to solve the problem somehow.
To cut a long, and apocryphal, story short, Archimedes, in a eureka moment (in fact the ureurekamoment), realised while getting into his bath that an object submerged in water causes the water level to rise. His insight was that the amount by which the water level rises is related to the amount  or volume  of the object placed into it. In fact, the volume of water displaced is equal to the volume of the object submerged in it.
As a simple example, imagine filling a bucket with water, right to the top, so no more water can be added without spilling any. Now put a regular shaped block of stone  let's say marble  into the bucket. Obviously some water overflows and spills out of the bucket. But how much, exactly? If you collect the water that spills out, and you pour it into a container of the same length and width of the block of marble, you will find that the depth of spilled water in this container is exactly equal to the height of the block of marble. In other words, the volume of the water spilled from the bucket is the same as the volume of the block of stone placed into the bucket.
This is all very well, but it only applies to objects heavy enough, or rather dense enough, to sink in water. What happens if you remove the marble block, refill the bucket, and put a block of wood in instead? Well firstly the wood floats (unless it happens to be lignum vitae, or one of a handful of other incredibly dense woods that actually sink in water). Now if an object is floating in water, it's clearly not displacing its entire volume of water, since you can push it further under and it will spill more water. How much water does it displace?
If you repeat the above measurement for the spilled water, putting it into a container the length and width of the block of wood, you indeed find it doesn't match the height of the block of wood. As you may expect, the height of the water displaced (when in this container matching the wood's dimensions) is equal to the depth to which the bottom of the block of wood sinks into the water, which, since it's floating, is less than its full height. So the volume of water displaced is clearly not equal to the volume of the entire block of wood.
Now comes the tricky part. What if you weigh the displaced water? If you do this, you find that the weight of the water displaced equals the weight of the block of wood! What's happening here?
Well, the wood is less dense than water, so an equal weight of water takes up less volume. Once a weight of water equal to the weight of the wood is displaced, the weight of the wood can not sink any deeper into the water  to do so would be for the wood to exert more force on the water than its own weight, which is impossible. So the wood cannot sink completely into the water, and is held up, floating.
What is holding the wood up? Well, it's the water itself. More specifically, it is an upwards force that the water exerts on the wood, a force which we call buoyancy. The magnitude of this force is equal to the weight of the water displaced by the floating object  as if the water displaced wants to flow back down into the space occupied by the floating object, but of course the floating object is stopping it.
So when an object can displace at least its own weight in water  or in other words when the object is less dense than water  then it floats, because the weight of the displaced water acts upwards on the object to balance its weight. But when an object is denser than water, it displaces only its volume of water, which is not as heavy as the object itself. The buoyancy force is still there  but it is equal to the weight of a volume of water equal to the volume of the object, and in this case is not enough to stop the object from sinking.
This general result is known as Archimedes' principle: Any object in a fluid is supported by a buoyancy force equal to the weight of the displaced fluid.
Archimedes' principle applies to any fluid, not just water. That's any fluid, not just any liquid. A gas is also a fluid, in that it flows  that's the definition of "fluid".
So we can apply Archimedes' principle to air. You displace a volume of air equal to your volume, so therefore you feel a buoyancy caused by the air equal to the weight of the same volume of air. It turns out this isn't very much. You can actually work it out yourself. The density of a human being is, to a good first approximation, the same as that of water. This is obvious if you consider that a human can just float on fresh water, but sinks if all the air is expelled from the lungs. So, given that water occupies a volume of 1 cubic metre per 1000 kg, so does a human. So if you are 100 kg, then your volume is 0.1 cubic metres, if you're 50 kg, your volume is 0.05 cubic metres, and so on. Now, air weighs about 1.2 kg per cubic metre. So for a 100 kg person, displacing 0.1 cubic metres of air, the buoyancy of the air is 0.12 kg. Not very much. Enough to make a tiny difference on the bathroom scales  but then only if you compare your weight by weighing yourself in a vacuum, which people don't generally do.
But now consider a balloon full of hydrogen. Let's make it an even cubic metre for ease. Hydrogen has a density of 0.09 kg per cubic metre. So our balloon displaces a cubic metre of air and therefore has a buoyancy force of 1.2 kg, but only weighs 0.09 kg! The result, as you probably know, is that a balloon full of hydrogen rises  it floats in air. In fact, a cubic metre of hydrogen provides enough buoyancy to lift just over 1.1 kg of stuff.
Now an airship holds a lot of hydrogen. The Graf Zeppelin held 105,000 cubic metres, which gave it a lifting capacity of about 115 metric tonnes. The Hindenburg held almost twice as much, allowing it to lift over 230 tonnes. Much of that weight was taken up by the structure of the airship itself, but enough was left over to give them useful passenger and cargo capacity.
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