Was I right about Archimedes?
October 17, 2006 8:51 PM   Subscribe

Does a 'water bridge' (like the one in Magdeburg, Germany) have to be strong enough to carry both the water and the ship travelling upon it? Or, thanks to displacement, does it only need to be strong enough to carry the water?

A friend of mine sent me an email about the Magdeburg Water Bridge along with a little riddle 'for the engineers out there'

Did that bridge have to be designed to withstand the additional weight of ship and barge traffic, or just the weight of the water?

It goes on to say that the bridge only has to be able to carry 'just the weight of the water because A ship always displaces an amount of water that weighs the same as the ship, regardless of how heavily a ship may be loaded"

But I was under the impression that the volume of displacement has nothing to do with mass... only relative buoyancy. For example a cubic foot of lead will displace the same amount of water as a cubic foot of styrofoam when fully immersed but they would obviously weigh vastly different

Am I mistaken? Is that he correct reading of Archimedes' Principle?

Discuss
posted by TheOtherGuy to Science & Nature (12 answers total) 1 user marked this as a favorite
 
Key question: where does the displaced water go?

If you drop a boat in a small, enclosed body of water, then the container supports the weight of the water plus the weight of the boat, because the displaced water is contained and rises around the boat.

But if the boat is in a channel, where the displaced water flows up and down the channel and the water level remains effectively the same, then there is no change in the weight supported by the structure beneath the area concerned.
posted by randomstriker at 8:57 PM on October 17, 2006


To expand: you do have to put weight on the styrofoam to immerse it; if left to float (as boats generally are) it would hardly displace any water at all.
posted by transient at 9:00 PM on October 17, 2006


Response by poster: I assume that the water is displaced at either end of the bridge so the level would remain the same
posted by TheOtherGuy at 9:01 PM on October 17, 2006


So, wait, does the bridge actually bear less load, because the water above has been displaced?!
posted by phrontist at 9:02 PM on October 17, 2006


Oh, wait, yeah, the weight is "replaced" exactly. Disregard previous post.
posted by phrontist at 9:03 PM on October 17, 2006


Image putting a half-full fishtank on a bathroom scale, noting the weight, then adding a toy boat. The boat displaces water, but the water stays in the tank. The additional weight of the boat would register on the scale.

Now do it again. But before you add the boat, fill the fishtank completely to the brim before noting the weight. The water displaced by the boat runs over the edges (and not on the scale). The weight would go up when you add the boat but quickly return to the previous mark as the water leaves.

Of course, in Magdeburg, you're not dropping boats out of the air. They're in one of the lakes (let's assume) connected by the bridge, already displacing water. Adding a boat to the system of bridge and two lakes results in a very slightly higher water level for the entire system. There's no change to the load on the bridge when the boat moves across.
posted by hydrophonic at 9:20 PM on October 17, 2006


As a practical matter, civil engineers tend to overdesign by a factor of between 3 and 5. They never design for exactly the expected load. But as others have said, a boat going across such a bridge doesn't change the loading on the bridge at all.
posted by Steven C. Den Beste at 9:29 PM on October 17, 2006


Water that would otherwise be on the bridge canal is displaced by the boat. That displaced water is spread over the surface of the lake, so goes unnoticed. The amount of water displaced weighs exactly the same as the boat, provided the boat floats. If you then get on the boat, it displaces more water--exactly your weight in additional water. Still no extra weight on the bridge. If too many people get on the boat, it sinks. If the boat sinks, it displaces less than its weight in water, and there is more weight on the bridge than there was with just the water. If it's a wooden boat and capsizes but doesn't sink, there's no extra weight on the bridge, provided everyone can swim or hang on to the boat.
posted by weapons-grade pandemonium at 12:24 AM on October 18, 2006


Re: Archimedes, if you want to find out if your blacksmith put styrofoam in your lead crown, immerse it in water and measure the volume of water displaced. You then compare the weight of that volume of pure lead with the weight of your crown. It should be the same. A very sneaky blacksmith could contaminate the lead with a mixture of styrofoam and depleted uranium.
posted by weapons-grade pandemonium at 12:42 AM on October 18, 2006


The lead versus foam argument doesn't work, because the lead would sink to the bottom and apply its weight directly to the solid surface. It would only remove the weight of one cubic foot's worth of water by displacing it, leading to a large net increase in load on the bridge.

Anything that floats, however, displaces its own mass in water and adds nothing to the load. A better comparison would be a cubic metre of foam versus a cubic metre of heavy but buoyant wood. The wood is heavier, but it displaces more water so they end up putting the same (i.e. zero) extra load on the bridge.

For the foam to be fully immersed, as you ask near the end of your question, it would have to have something pressing down on it. Unless the foam hits the bottom and starts transferring that downwards pressure through to the bridge, there is no way the load can increase... water is too flexible, and it just gets out of the way if you press down on it like this rather than conveying the pressure to the bridge. Wood and foam both work exactly the same for this purpose.
posted by A Thousand Baited Hooks at 3:32 AM on October 18, 2006


If you wanted to get really picky, the answer to your question would be that the load on the bridge actually does increase slightly with the boat, since the displaced water (hence the weight of the boat) is spread over the lake surface, which includes the elevated canal. So there is a microscopic increase in the water level all over, unless Luciano Pavarotti gets out of the water on the other side of the lake.
posted by weapons-grade pandemonium at 9:37 AM on October 18, 2006


The Archimedes approach was to compare a crown to an equal weight of known-pure gold and to see how much water each displaced. If the crown was made of pure gold they would displace the same. If it was gold mixed with silver, the volume would be greater (because lower density for constant mass would require more volume).

The fact that both sink to the bottom doesn't matter.
posted by Steven C. Den Beste at 7:39 PM on October 20, 2006


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