Question:

As the boat travels along the water bridge, which side of the bridge is under heavier loading? The side with the boat, or the side without the boat?

Answer:

Fact: Magnitude of upthrust is equal to the weight of fluid displaced.

When the cargo was in the boat:

Say the boat was on the right. The right side is heavier because of the additional weight of the boat and cargo but also lighter because of the missing weight of displaced water.

Guess what? The missing water weighs exactly as much as the boat and cargo!!! Because the boat floats, the upthrust (and thus weight of the displaced water) must be equal to the weight of boat and cargo. (Archimedes Principle).

So the missing water exactly compensates the weight of the boat and cargo. The weight is evenly distributed along the bridge.

When the cargo dropped out

When the cargo was sunk on the right, the right side again has the additional weight of the cargo but less water.

But this time round, the missing water does not weigh as much as the cargo. The fact that the cargo sank meant that the upthrust (and thus weight of the displaced water) is less than the weight of the cargo.

So the decrease in weight of missing water is not enough to compensate the increase in weight of the cargo. So the right side is heavier, and the bridge toppled.

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P.S. By the way, water bridges really exist. The Romans built them to transport water and boats. See aqueducts.

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