Ocean liner in bucket full of water

[Borek]

Where is the flaw in my logic?

Is there one? If I understand you correctly you are surprised by the fact that 1L volume can occupy 10x10x10 cube or hundred times longer 1x1x1000 cuboid.

 

[DaveC426913]

Is there one?

Yes. If my logic is not flawed, the inevitable conclusion is that the act of dropping a 10kg cinder block on top of the boat would cause the boat-and-water-level-together to rise in the container by 125cm.

That is a ridiculous outcome.

Question: when the cinder block pushes down on the boat, the boat displaces an additional 10L of water. Where does that displaced 10L of water go? If it simply seeps up the 4m x 2mm gap, it would have to seep upward by 125cm.

 

[AlephZero]

There is nothing wrong with the experiment, but your logic is wrong.

When you add the 10kg weight in the lake, then measured relative the the bottom of the lake, the boat sinks (1-x) cm and the water rises x cm, where x is very small.

In the tank, measured relative the bottom of the tank, the boat sinks x cm and the water rises (1-x) cm where x is very small.

The level of the water relative to the boat changes by the same amount (1cm) in both situations.

 

[DaveC426913]

The level of the water relative to the boat changes by the same amount in both situations.

Absolutely. Which I too state, and have no qualms with. Regardless of the water level in the tank, the boat will float 9cm above the water and 101cm below it.

My issue is with the water level in the tank.

measured relative the the bottom of the lake, the boat sinks (1-x) cm and the water rises x cm, where x is very small.

The displaced 10L is distributed throughout the lake. Lake water level rises a tiny amount. No prob.

In the tank, measured relative the bottom of the tank, the boat sinks x cm and the water rises (1-x) cm where x is very small.

The displaced 10L is distributed into only a tiny gap, resulting in a calculated water level rise in the tank of 125cm. (Gap is 4.016m x 2mm = 80cm^2. A 10L volume on an 80cm area must be 125cm in height.) The inevitable conclusion is that the water level in the tank rises 125cm.This is not just intuitively ridiculous, it actually results in a paradox. (How can the water level in the tank rise 125cm? That would mean the boat (which is happily remaining floating 9cm out of the water and 101cm into the water at all times) is now physically sitting 125cm higher above the bottom of the container than it was before - meaning there's room for 1,250L of water underneath it - meaning it can't sit that high.)

I know it's wrong, I just can't spot it.

 

[DaveC426913]

Reexaming what you said instead of my own flawed logic. I know you're right (because I know that's the way it it really does work), I just haven't made the leap of logic as to why yet.

When you add the 10kg weight in the lake, then measured relative the the bottom of the lake, the boat sinks (1-x) cm and the water rises x cm, where x is very small.

In the tank, measured relative the bottom of the tank, the boat sinks x cm and the water rises (1-x) cm where x is very small.

Why do you treat these scenarios differently? Why does the same math not apply?

Is 10L of water not displaced from under the boat? That's a specific volume that has to go somewhere.

 

[maimonides]

Displacement (as has been mentioned) does not mean actual water transported to some place, but refers to the volume of the floating body below the water line. In the experiment discussed a very small amount of water makes a big change in displacement.

 

[AlephZero]

Why do you treat these scenarios differently? Why does the same math not apply?

It is the same math. I just renamed the variables, so "x" was the small quantity in both cases. If you prefer, call the movement of the boat down and water surface up b and w, both measured relative to the bottom, and where where b+w = 1cm. The relative size of b and w depends on the relative surface areas of the boat and the water.

Is 10L of water not displaced from under the boat? That's a specific volume that has to go somewhere.

I think that is the key point of your mistake. "The displacement increases by 10L" doesn't say anything directly about where the water goes. What it means is "The volume of the boat which is below the water level increases by 10L". In the lake, you need to move very nearly 10L of water to make that happen. In the tank, you only need to move a small amount of water to get the same effect.

 

[Per Oni]

This causes the boat to sink into the pool by an additional 1cm,

Where is the flaw in my logic?

To add to what other are saying:

The water will rise by that amount. The boat doesn’t sink that amount.

2mm gap x 4 lengths of 1 meter (approx. container size ) is a surface area of 8x10^-3 m^2. The volume of the rising water is that amount times the height of rising water (1 cm) which gives 8x10^-5 m^3. This is the volume of water the boat has to displace. Therefore it needs to sink 8x10^-5m^3 / 1m^2 (bottom area) = 8x10^-5m. Your 2mm gap all round should be enough.

I hope this makes sense.

 

[DaveC426913]

I think that is the key point of your mistake. "The displacement increases by 10L" doesn't say anything directly about where the water goes. What it means is "The volume of the boat which is below the water level increases by 10L". In the lake, you need to move very nearly 10L of water to make that happen. In the tank, you only need to move a small amount of water to get the same effect.

Hm. I concede, though I do not quite yet get the logic.

Counter-intuitively the movement of a mere 4m x 2mm x 1cm of (80mL/80g) of water is enough to support a 10kg load.

 

[nasu]

This is maybe because the image of "water displaced" from Archimedes' law is very strong in our minds. It is not the displacement or the movement of water that matters. Water displaced is just an easy to remember calculation tool.
The increased pressure is what matters. To support the 10 kg (100 N) the pressure over the 1 m^2 area needs to increase by 100 Pa or 1 cm of water. The block goes down in water until the level of the water relative to the bottom of the block increases by 1 cm. In the given conditions this will not displace 10 L of water (the original block did not have to displace/move 1000 L of water to float).

 

[DaveC426913]

(the original block did not have to displace/move 1000 L of water to float).

You're right!

The solution to the paradox is ... the experiment itself!

Part A of the experiment just demonstrated that a paltry bucketful of water is all that needs to be displaced in order to float a 1T mass. Why did I suddenly forget that when it came to part B!

 

[DaveC426913]

So back to the engineering component.

Trying to figure out:

- best shape
- - rectangle? cylinder? cone?
- - - which is easiest to make (with appropriate precision)
- - - which is least likely to present technical problems during the experiment (eg. cylinder, rectangle could tilt, bind, cone is self-calibrating)

- best manufacturing procedure
- - make inner shell, cover in wax, paint on outer shell, remove wax (somehow)
- - make outer shell, cover in wax, paint on inner shell, remove wax (somehow)

- test procedure
- - put water in first then inner shell
- - put shells together then pour water into gap

- scale, cost
- - 1 cubic metre / 1 tonne mass (too small? too big?)
- - 1mm gap / 5mm gap (impossible precision versus doesn't prove the point)
- - cost (assuming I did this myself)
- - - I've got a winch and a barn, a good start

Mostly, I'm trying figure out what to build the inner shell out of so that it's easy to make but does not flex even 1 mm.

 

[chingel]

Thank you all for the answers.

Another problem I have is that if I apply a 1kg per cm2 pressure through a small piston to a closed tank, why does the same pressure get applied to every cm2 of the tank? Isn't the only way that the force can transfer because the water molecules bump into each other and apply the force. But if one water molecule pushes two water molecules, shouldn't the force be divided by the two of them?

In the case of a solid block, the force gets divided by the area of contact at the other end. What is different with a liquid? The water molecules themselves are solid, and shouldn't the force transfer like in the case of solid objects? I can understand why the force goes in all directions in a liquid, because water molecules aren't exactly on top of each other and they are allowed to roll out to the sides if something pushes them at a slight angle.

 

[DaveC426913]

Another problem I have

chingel, please start a new thread.

 

[AlephZero]

Hm. I concede, though I do not quite yet get the logic.

Counter-intuitively the movement of a mere 4m x 2mm x 1cm of (80mL/80g) of water is enough to support a 10kg load.

I agree, Pascal's law of hydrostatics (pressure varies with depth of fluid, independent of the shape of the container) is counterintuitive till you "get used to it".

Here's another scenario that might help. Suppose the boat is the lake and resting on the bottom, but just at the point of floating off. Now add 10kg to the weight of the boat. To bring it back to be just on the point of floating, you have to add enough water to raise the level of the whole lake by 1 cm.

In the tank, the physics of situation is exactly the same, but you need a much smaller amount of water to raise the level by 1 cm.

I think you could make a first version of the experiment with something as simple as a glass aquarium tank and a suitable sized watertight box (e.g. make a wooden box and cover the outside with plastic film). Weight the box with sand so it floats at a suitable depth. With a glass tank, you can (literally) see what is happening.

 

[Per Oni]

Counter-intuitively the movement of a mere 4m x 2mm x 1cm of (80mL/80g) of water is enough to support a 10kg load.

That’s the principle of a hydraulic ram.

In the end it all comes down to energy. For example a miserly 1 joule of energy can in principle lift a vast amount of mass. Not very high but it will lift. Of course pretty soon we get into practical troubles but in theory it can be done.

 

[nasu]

That’s the principle of a hydraulic ram.

Not really. The buoyancy is not involved in the functioning of the hydraulic press (if this is what you have in mind).

 

[DaveC426913]

I agree, Pascal's law of hydrostatics (pressure varies with depth of fluid, independent of the shape of the container) is counterintuitive till you "get used to it".

No, as an owner of swimming pools and fish tanks, I'm comfortable with the concept that pressure varies with depth, not with shape or area.

Here's another scenario that might help.

I got it with nasu's explanation. When the water level in the tank rises 1cm yet the boat does not, that means the boat is now displacing 10 more litres of water. As he points out, it is not essential for 10L of displaced water to actually go anywhere.

We just proved that in the part A of the experiment, where we started with naught but a bucket of water in the bottom of a giant vat, and dropped a cubic metre vessel into it. There was no cubic metre of water to move in the first place, so it's not like we had to actually move a cubic metre of water to get the boat to float. The whole point of the entire experiment is that we only need to move a tiny amount of water to effect* buoyancy. I had just forgotten that, by the time I got to Part B is all.
*No, I did not use the wrong word.

 

[DaveC426913]

I think you could make a first version of the experiment with something as simple as a glass aquarium tank and a suitable sized watertight box (e.g. make a wooden box and cover the outside with plastic film). Weight the box with sand so it floats at a suitable depth. With a glass tank, you can (literally) see what is happening.

This is what I want to do, except most ideas I have for the inner vessel will experience deformation under pressure. We're talking < mm tolerances here for the experiment to work.

 

[AlephZero]

If you want to everything to stay "rigid", you have two basic choices. Either you do a proper stress analysis, or you just overdesign everything.

For a small scale experiment, I'm assuming a glass aquarium tank isn't going to deform much under the water pressure, otherwise the glass would crack. You could check that assumption with a measuring rod cut to fit accurately inside the tank when it is empty, and check the amount of clearance when it is full.

I don't think an aquarium-sized wooden box made from say 1/2'' or 1'' thick wood will deform much either, so long as you keep the wood dry.

Designing experiments is often a matter of trial and error. Starting with something simple and cheap to find out what is really important in practice is often a good strategy. Often the biggest problems are caused by something that nobody had thought about.

 

[DaveC426913]

If you want to everything to stay "rigid", you have two basic choices. Either you do a proper stress analysis, or you just overdesign everything.

Definitely overdesign.

For a small scale experiment, I'm assuming a glass aquarium tank isn't going to deform much under the water pressure, otherwise the glass would crack. You could check that assumption with a measuring rod cut to fit accurately inside the tank when it is empty, and check the amount of clearance when it is full.

I'm pretty sure it would deform by at least a mm or two. It could be mitigated by some internal bracing.

Designing experiments is often a matter of trial and error. Starting with something simple and cheap to find out what is really important in practice is often a good strategy.

I think the danger is that what might work in a scale test will likely not scale up well. We're dealing with masses cubing as we scale. This will be an excellent demonstration of Galileo's square-cube law.

 

[AlephZero]

I'm pretty sure it would deform by at least a mm or two.

Don't speculate. Measure it. That's what experiments are for.

I think the danger is that what might work in a scale test will likely not scale up well. We're dealing with masses cubing as we scale. This will be an excellent demonstration of Galileo's square-cube law.

Of course. But you learn how to run by falling over while trying to walk

 

[cmb]

...you learn how to run by falling over while trying to walk

ermmmm... you can learn that way, but that is not necessarily the way to learn! How did you learn not to walk into the road with oncoming traffic?

 

[AlephZero]

How did you learn not to walk into the road with oncoming traffic?

I probably personally learned that mostly by trial and error, considering I grew up in a place where it was perfectly normal for 6-year-olds to walk a mile and more to school and back every day on their own, with no supervision.

But that's beside the point. Most likely nobody has done this particlear experiment before, you won't figure out how to do it just by asking "experts" for advice.

 

[tony_physic]

If we assume a hemispherical ship 1km diameter the volume is 4/3pi r^3
523598775.6 cubic metres
If the bath is 1cm bigger all round to allow bending etc then volume of bath is
523630182.1
The difference being the water in your bucket 31416.52 cubic metres, a big bucket.
Of course if the gap is smaller less water is required
The easiest way to visulise this working is by filling the bath and then lowering the boat into the water, allowing the bath to overflow until the boat is floating. The remove the boat and fill the bucket with the water that is left.

 

[Per Oni]

Not really. The buoyancy is not involved in the functioning of the hydraulic press (if this is what you have in mind).

No, I was thinking of Dave’s statement of 80 grams supporting the load of 10 kg.

 

[olivermsun]

Most likely nobody has done this particlear experiment before, you won't figure out how to do it just by asking "experts" for advice.

No, but people have floated scale models in buckets, and people have also floated ocean liners in locks (which are just big buckets as far as I can tell).

 

[DaveC426913]

Don't speculate. Measure it. That's what experiments are for.
Of course. But you learn how to run by falling over while trying to walk

Design well before building. We're still in the design phase.

We have to start with the assumption of a certain amount of existing knowledge, or we'd still be testing tree bark as a structural component.

As oliver's mum points out.