# Bridges and Archimedes' principle

I was reading some of my physics chapter and cam across an interesting question, which I am sure someon here can answer. It just interested me, so here it is.

A barge filled high with sand approaches a low bridge over the river and cannot quite pass under it. Should sand be added to, or removed from, the barge? (Hint: Consider Archimedes' principle)

My instinct is to say more sand...

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Tide
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Your "instinct" says more sand but what does your logic and reason say?

Oh, so it is not a trick question?

Of course logic says less sand.

Tide
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Okay, let's test that logic.

If you add sand, what happens to the height of that sand pile? :)

Tide said:
Okay, let's test that logic.
If you add sand, what happens to the height of that sand pile? :)
If the sand is dense enough it may sink the boat in the water more then it adds to the pile in height.

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Tide
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Good! There is one caveat, however. The answer depends on how the new sand is distributed so we need to compare the change in height of the sandpile with the change in "depth" of the barge. Suppose we distribute the new sand with uniform depth over some area, a, and the barges cross section at the water line is A then Archimedes tells us

$$\rho_s a \Delta h = \rho_w A \Delta D$$

If the barge is to pass under the bridge then we require

$$\Delta h < \Delta D$$

so that

$$\rho_s a > \rho_w A$$

This is clearly satisfied if the cross section over which you distribute the new sand is the same as that of the barge (which I think your textbook intends). However, if you distribute the sand over a sufficiently small area then the barge won't pass under the bridge.

Of course, if the barge sinks, then, by definition, it isn't passing under the bridge! :)

Sand has greater density than water but if would need to be added over a significant cross section of the barges surface. Pileing it up probably wouldn't help.

Oh yeah, don't forget about the Tide!

Tide
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Dmstifik8ion said:
Oh yeah, don't forget about the Tide!
Excellent point!

That makes sense. So sand should be removed.

From the river bottom

Yes, so the answer would be to remove some sand.

Tide
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jrd,

No, we didn't say that! The answer depends on HOW you do it! If it's spread over a large enough area then adding sand will work.

HallsofIvy