# Bridges and Archimedes' principle

1. Dec 5, 2005

### jrd007

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...

2. Dec 5, 2005

### Tide

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

3. Dec 5, 2005

### jrd007

Oh, so it is not a trick question?

Of course logic says less sand.

4. Dec 5, 2005

### Tide

Okay, let's test that logic.

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

5. Dec 6, 2005

### moo5003

If the sand is dense enough it may sink the boat in the water more then it adds to the pile in height.

Last edited: Dec 6, 2005
6. Dec 6, 2005

### Tide

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! :)

7. Dec 6, 2005

### Dmstifik8ion

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.

8. Dec 6, 2005

### Dmstifik8ion

Oh yeah, don't forget about the Tide!

9. Dec 6, 2005

### Tide

Excellent point!

10. Dec 6, 2005

### jrd007

That makes sense. So sand should be removed.

11. Dec 6, 2005

### Dmstifik8ion

From the river bottom

12. Dec 7, 2005

### jrd007

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

13. Dec 7, 2005

### Tide

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.

14. Dec 7, 2005

### HallsofIvy

Staff Emeritus
Your instincts are good. Archimede's principle says that an object will sink into water until it displaces its weight in water (that is, the water displaced will weigh as much as the object itself. The more weight, the more water must be displaced which the barge does by sinking lower into the water.

15. Dec 8, 2005

### Dmstifik8ion

The question here is which has greater mass density water or sand? Does sand sink in water or float? Which has greater surface area, the surface of the top of the barge or the footprint of the barge in the water?

The only stumbling block I see to coming to a decision here is whether the additional sand can be distrubuted evenly over the surface of the barge or would it need to be bagged up so that it would not spill over the sides. This may be considered an academic concern but I believe the point should be addressed in stating your solution to the problem.