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Physics and the Magdeburg Water Bridge

Image via Wikipedia

This is the Magdeburg water bridge in Germany. Pretty cool, right? But how is this even possible? How could you make a bridge that supports all that water AND a boat?

The answer is simple. The supports on the bridge always have the same forces pushing on them. It doesn’t matter if there is a boat on the bridge or not as long as the boat is floating. Why?


First, you can probably consider this post the second part of the Scale at the Bottom of the Pool post from a few weeks ago.

I will still give a super-quick explanation of buoyancy. Let me start with a block of water floating in water. Yes, I know that is almost too crazy to believe. Here is a diagram.

Drawings.key 5

Consider very still water such that in a short time, the water inside the dotted-line stays inside. Since the block of water stays at rest (as a whole), the total force on that block must be the zero vector. This means that the upward buoyant force must be equal in magnitude to the downward gravitational force.

Why is there an upward buoyant force? Well, essentially this is the net interaction from all the water around it pushing on it.

Now suppose I replace that block of water with a block of steel of the exact same dimensions. Here is a diagram:

Drawings.key 6

Yes, since the steel block would be heavier than the water block the steel would sink. However, the buoyant force would be the same as it was on the water. Why? Because that outside-the-block water is still interacting with the steel in the same way it did with the water block.

Since the water block had the two forces equal, we can always say that the buoyant force is equal to the weight of the water displaced. I can write this as:

La te xi t 1 6

Notice that ρ (the density of the water) multiplied by the volume of the object gives the mass of the water displaced.

Back to the Water Bridge

I haven’t yet answered the question about the bridge. Also, if you think about the scale at the bottom of the pool problem, it seems that if something is floating in a pool the forces on the bottom of the pool would be larger. Well, they would. Yes, I know it looks I just contradicted myself.

Ok, time for an experiment. Here is a beaker of water filled to the top and sitting on a balance.

I Photo 2

You can see, the scale reads on the order of 329 grams. The beaker of water is like the water bridge. For my boat, I have a container with some weights in it (mass of 54 grams).

I Photo 3

When I put this “boat” on the bridge, some of the water is going to spill out (since it was full). So, here is what happens:

I Photo 4

Ok, so the scale changed. It is suppose to still read 329 grams, but this is close enough. Really, the problem is that all the water doesn’t fall out of the beaker because of surface tension. I think we can agree that this finally balance reading is much closer to the 329 grams than it is to (329 g + 54 g) 383 grams. Right?

What if I put a lighter boat on the bridge? Here is a container with a different mass (23 grams) in the beaker of water:

Drawings.key 7

Same thing happens. But why? Well, if I have a 53 gram “boat”, it will displace 53 grams of water. This means the balance now has to push up the exact same amount it did with just the water (since the buoyant force is the same as that 53 gram boat). But what about the water bridge? Shouldn’t the water level rise? Technically, it does. However, if the bridge is sufficiently long this increase in water level will be very difficult to detect. In a sense, the weight of the boat is spread over the whole length of the water-way (and not just the part that is on the bridge).