If I have two connected cylinders filled with water, and 50kg stones at both the

Question

If I have two connected cylinders filled with water, and 50kg stones at both their ends (pressing both ends of the water) how much weight do I need to add on one side so that one of the stones reach the bottom?

Image of connected cylinders

So lets assume that the water weighs 1kg. And thus if there were no stones at all, adding a 1kg stone would displace all the water on one side and the stone would touch the bottom on the other. Since the 50kg weights cancel one another it seems like I could count as if there were no weights and adding 1 kg would be enough, But my instincts tell me I need to add 50kg. I don't know why.

Thank you very much!

Explanation / Answer

First, change your thinking away from weight to pressure. Pressure is weight per unit area. The easiest way to do that is to assume a certain cross-section throughout, say 100 square centimeters, or a square tube 10 centimeters on a side. That is a nice size, because 10 centimeters of that tube enclose 1 liter, or 1 kilogram of water.

So if you have a vertical tube that size, and you fill it with 10 cm of water, the pressure at the bottom is the weight of the water divided by the area of the bottom - 1 000 grams divided by 100 square centimeters is 10 grams resting on each square centimeter.

If you fill it to twice that height, then the pressure at the bottom is twice that, and so on.

So, to rephrase your question, if the 50kg stone is sitting at the bottom of one tube, what pressure is it exerting on the bottom of the tube? 50 000 grams over 100 sq cm, or 500 grams per square centimeter. So what height of water would it take on the other side to equal that? Since 10 cm gave you 10 grams/cm^2, 500 cm (5 meters) would give you 500 gm/cm^2.

Did I get it right? You check my math.

Then you can replace any of that excess water by a stone, as long as that stone weighs the same amount as the water you're getting rid of, and fits in the tube.

Now, what happens if you use a more narrow tube? What if you use a wider one? What if the two tubes are different sizes?

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