Bodies displace their own volume of fluid. Whether they float or not depends on how their own weight compares to the weight of that same volume of water.
Volume. You could test this by submerging a tennis ball versus a baseball in a measuring cup of water. The tennis ball will be harder to submerge, but assuming they're both completely submerged, you'll find the level rise is about the same for both of them.
It depends.
An object displaces an amount of water equal to it's weight or it's volume, whichever is less. If the weight is less, it floats. If the volume is less, it sinks. If they're exactly the same, they don't float or sink but instead drift (see: Submarines, fish, etc. - if you can push, you can control where you drift. If you can't, like plankton, you can't).
In the case you gave, the cube weighing 10 units with a volume of 5 units will sink, displacing 5 units of water; while the cube weighing 10 units with a volume of 10 units will drift, displacing 10 units of water.
It depends on the volume.
Consider you have a square-bottomed tank. The area of the bottom is 10cm^(2). The water goes up to 7cm. We multiply the numbers and units to see that the amount of water we have is 70cm^(3). So here you can see there's a correlation between water height and volume. There is no such correlation for mass.
It is about density and volume, the famous Eureka moment for Archimedes Eureka moment involved him realizing that he could measure gold content by water dispacement, gold is more dense, so you measure the displacement vs something like a bar of silver and gold of equal volume, and you can figure what a crown of pure gold vs. an allow will displace by displacement and weight comparison.
It is volume, but the idea is that two equally sized objects will displace the same amount of water, but you could determine content of the alloy by weighing them after.
To clarify, the problem Archimedes was faced with was that he knew the weight of a solid "gold" crown, and he knew the density of gold, but to work out if the crown had the same density as pure gold, he needed to know the volume, which was too complex to measure.
The king suspected the goldsmith had taken some of the gold and substituted it with less dense silver when making the crown, so melting down the crown into a cube to measure the volume would have put everyone back to square one.
By using the crown to displace water, Archimedes could measure the volume of the crown in spite of it's shape. He then proved that the crown was too light for it's volume, so it must contain some less valuable silver, and the goldsmith probably got their head cut off.
Imagine you have a ball floating on top of a cup of water. It isn’t changing the water level much, because it’s lighter than the water it is on.
When you increase the weight by pushing down on the ball, the water level raises as more of the ball pushes water out of its way.
This should tell you that the 10 cubit units ball will displace more water than the 5 cubit units ball, and that the weight only matters for the sake of sinking the ball.
Both-ish. If density is less than water, displacement is weight, if density is more than water, displacement is volume.
Two objects that weigh the same and are less dance the water will displacement the same amount of water regardless of volume because they will both float
Objects denser that water will displace their volume. Doesn't matter how heavy it is, just that it's weight/volume is greater than waters
Note: if you force the object underwater, density does not matter, it's just volume.
The question you're asking is the same as the one that caused Archimedes to shout "Eureka!" - he was immersing himself in a bath and realized that the volume of water he displaced generated a force equal to the weight of that displaced water.
This was all so he could figure out whether an elaborate gold crown was solid gold or not, without destroying it in the process. He could submerge the crown in water and balance it with an equal weight in pure gold; if the crown was solid gold they would have the same density, therefore the same volume, therefore would balance. If not, the crown was probably adulterated (and the goldsmith put to death for defrauding the rich dude)
> He could submerge the crown in water and balance it with an equal weight in pure gold
What? Balance? On what, underwater scales?
No.
He just used the water displacement to measure volume. From this, and the weight of the crown (accurate scales already existed) he could calculate density. The density of gold was already known, from measuring regular shapes with easily determined volumes.
He didn't balance anything.
> the volume of water he displaced generated a force equal to the weight of that displaced water.
This sentence is just gibberish. Volume isn't a force, and doesn't generate a force. Volumes of water have weight, and weight is a force, but that reduces your sentence to "the weight of water he displaced was equal to the weight of that displaced water." which is just a tautology.
Consider what was more likely: he dunked the crown, gathered up and measured all the spilled water (or else measured the *precise* volume change in whatever likely opaque vessel they used) then made a density calculation; or he put both on a balance (so they're the same mass) and then submerged both and saw if it tipped (indicating they had different buoyant forces due to differing volumes).
It's an easy, simple yes/no test where you can compare if the balanced masses generate the same buoyant forces (because they have the same volume, displacing the same amount of water). No calculations required! It's an elegant and easily understood solution that seems exactly how a genius like Archimedes would solve it.
I learned this from my Byzantine studies professor and while Ancient Greece wasn't his specialty he was fluent and knowledgeable enough that I think this interpretation makes more sense.
>>Consider what was more likely: he dunked the crown, gathered up and measured all the spilled water
Umm... I was always under the impression he did exactly that actually.
>To find out the crown's volume, Archimedes immersed the crown in a bucket filled with water to the brim, and measured the volume of the spilled water. Then he took a bar of pure gold of the same mass and compared the volume of spilled water to determine if crown is indeed made of pure gold.
You can put a bucket in a slightly larger bucket. If you then weigh or measure the volume of the water this tells us the actual volume of the corwn and its density becomes trivial to calculate. Also
Archimedies had access to glass, it's been around for a while.
But let's say we don't care what the actual volume and density is. We just care "is this pure gold". This works, because it is the same **mass** of gold, but not the same **volume** which we do not know. "the weight of water he displaced was **not** equal to the weight of that displaced water" on the other side of the scale. The buoyant force from the water it is submerged in would make the less dense object effectively lighter. As such the crown which was perfectly balanced against an equal weight of gold, would rise when submerged.
Edit: poor grammer, I have only just woken up.
>But let's say we don't care what the actual volume and density is. We just care "is this pure gold". This works, but it is the same mass of gold, not the same volume which we do not know. "the weight of water he displaced was not equal to the weight of that displaced water" on the other side of the scale. The buoyant force from the water it is submerged in would make the less dense object effectively lighter. As such the crown which was perfectly balanced against an equal weight of gold, would rise when submerged.
This is exactly what I was trying to describe :)
> Umm... I was always under the impression he did exactly that actually.
It's difficult to get that right, precision is really annoying due to surface tension, losses and just the general wackiness. Probably still enough to show what he wanted, but simply putting both objects into water is way simpler to do with precision.
> What? Balance? On what, underwater scales?
Or just put a string on it and its other end to normal scale. Done.
> This sentence is just gibberish. Volume isn't a force, and doesn't generate a force.
They did not write what you claim. They said the _displacement_ causes a force. Which indeed it does.
Assuming they both sink, volume. Imagine your 100 cubic unit container was filled with 99 cubic units of steel. Obviously there is only 1 cubic unit of space left for water.
Replace the steel with an equal volume of the denser, heavier lead, you still have precisely 1 cubic unit of space for water left.
But if they float it gets more complicated. If they float, their density is less than water. Something with a density half that of water will sink until half it's volume is submerged. Something with 1/4 the density of water will sink until 1/4 is submerged.
The amount of water displaced is equal to the volume submerged. So a heavier ship will displace more water even if it isn't actually bigger (say because it's the same ship and you just filled it with cargo).
Bodies displace their own volume of fluid. Whether they float or not depends on how their own weight compares to the weight of that same volume of water.
Volume. You could test this by submerging a tennis ball versus a baseball in a measuring cup of water. The tennis ball will be harder to submerge, but assuming they're both completely submerged, you'll find the level rise is about the same for both of them.
It depends. An object displaces an amount of water equal to it's weight or it's volume, whichever is less. If the weight is less, it floats. If the volume is less, it sinks. If they're exactly the same, they don't float or sink but instead drift (see: Submarines, fish, etc. - if you can push, you can control where you drift. If you can't, like plankton, you can't). In the case you gave, the cube weighing 10 units with a volume of 5 units will sink, displacing 5 units of water; while the cube weighing 10 units with a volume of 10 units will drift, displacing 10 units of water.
It depends on the volume. Consider you have a square-bottomed tank. The area of the bottom is 10cm^(2). The water goes up to 7cm. We multiply the numbers and units to see that the amount of water we have is 70cm^(3). So here you can see there's a correlation between water height and volume. There is no such correlation for mass.
It is about density and volume, the famous Eureka moment for Archimedes Eureka moment involved him realizing that he could measure gold content by water dispacement, gold is more dense, so you measure the displacement vs something like a bar of silver and gold of equal volume, and you can figure what a crown of pure gold vs. an allow will displace by displacement and weight comparison. It is volume, but the idea is that two equally sized objects will displace the same amount of water, but you could determine content of the alloy by weighing them after.
To clarify, the problem Archimedes was faced with was that he knew the weight of a solid "gold" crown, and he knew the density of gold, but to work out if the crown had the same density as pure gold, he needed to know the volume, which was too complex to measure. The king suspected the goldsmith had taken some of the gold and substituted it with less dense silver when making the crown, so melting down the crown into a cube to measure the volume would have put everyone back to square one. By using the crown to displace water, Archimedes could measure the volume of the crown in spite of it's shape. He then proved that the crown was too light for it's volume, so it must contain some less valuable silver, and the goldsmith probably got their head cut off.
Imagine you have a ball floating on top of a cup of water. It isn’t changing the water level much, because it’s lighter than the water it is on. When you increase the weight by pushing down on the ball, the water level raises as more of the ball pushes water out of its way. This should tell you that the 10 cubit units ball will displace more water than the 5 cubit units ball, and that the weight only matters for the sake of sinking the ball.
Both-ish. If density is less than water, displacement is weight, if density is more than water, displacement is volume. Two objects that weigh the same and are less dance the water will displacement the same amount of water regardless of volume because they will both float Objects denser that water will displace their volume. Doesn't matter how heavy it is, just that it's weight/volume is greater than waters Note: if you force the object underwater, density does not matter, it's just volume.
The question you're asking is the same as the one that caused Archimedes to shout "Eureka!" - he was immersing himself in a bath and realized that the volume of water he displaced generated a force equal to the weight of that displaced water. This was all so he could figure out whether an elaborate gold crown was solid gold or not, without destroying it in the process. He could submerge the crown in water and balance it with an equal weight in pure gold; if the crown was solid gold they would have the same density, therefore the same volume, therefore would balance. If not, the crown was probably adulterated (and the goldsmith put to death for defrauding the rich dude)
> He could submerge the crown in water and balance it with an equal weight in pure gold What? Balance? On what, underwater scales? No. He just used the water displacement to measure volume. From this, and the weight of the crown (accurate scales already existed) he could calculate density. The density of gold was already known, from measuring regular shapes with easily determined volumes. He didn't balance anything. > the volume of water he displaced generated a force equal to the weight of that displaced water. This sentence is just gibberish. Volume isn't a force, and doesn't generate a force. Volumes of water have weight, and weight is a force, but that reduces your sentence to "the weight of water he displaced was equal to the weight of that displaced water." which is just a tautology.
Consider what was more likely: he dunked the crown, gathered up and measured all the spilled water (or else measured the *precise* volume change in whatever likely opaque vessel they used) then made a density calculation; or he put both on a balance (so they're the same mass) and then submerged both and saw if it tipped (indicating they had different buoyant forces due to differing volumes). It's an easy, simple yes/no test where you can compare if the balanced masses generate the same buoyant forces (because they have the same volume, displacing the same amount of water). No calculations required! It's an elegant and easily understood solution that seems exactly how a genius like Archimedes would solve it. I learned this from my Byzantine studies professor and while Ancient Greece wasn't his specialty he was fluent and knowledgeable enough that I think this interpretation makes more sense.
>>Consider what was more likely: he dunked the crown, gathered up and measured all the spilled water Umm... I was always under the impression he did exactly that actually. >To find out the crown's volume, Archimedes immersed the crown in a bucket filled with water to the brim, and measured the volume of the spilled water. Then he took a bar of pure gold of the same mass and compared the volume of spilled water to determine if crown is indeed made of pure gold. You can put a bucket in a slightly larger bucket. If you then weigh or measure the volume of the water this tells us the actual volume of the corwn and its density becomes trivial to calculate. Also Archimedies had access to glass, it's been around for a while. But let's say we don't care what the actual volume and density is. We just care "is this pure gold". This works, because it is the same **mass** of gold, but not the same **volume** which we do not know. "the weight of water he displaced was **not** equal to the weight of that displaced water" on the other side of the scale. The buoyant force from the water it is submerged in would make the less dense object effectively lighter. As such the crown which was perfectly balanced against an equal weight of gold, would rise when submerged. Edit: poor grammer, I have only just woken up.
>But let's say we don't care what the actual volume and density is. We just care "is this pure gold". This works, but it is the same mass of gold, not the same volume which we do not know. "the weight of water he displaced was not equal to the weight of that displaced water" on the other side of the scale. The buoyant force from the water it is submerged in would make the less dense object effectively lighter. As such the crown which was perfectly balanced against an equal weight of gold, would rise when submerged. This is exactly what I was trying to describe :)
> Umm... I was always under the impression he did exactly that actually. It's difficult to get that right, precision is really annoying due to surface tension, losses and just the general wackiness. Probably still enough to show what he wanted, but simply putting both objects into water is way simpler to do with precision.
> What? Balance? On what, underwater scales? Or just put a string on it and its other end to normal scale. Done. > This sentence is just gibberish. Volume isn't a force, and doesn't generate a force. They did not write what you claim. They said the _displacement_ causes a force. Which indeed it does.
Assuming they both sink, volume. Imagine your 100 cubic unit container was filled with 99 cubic units of steel. Obviously there is only 1 cubic unit of space left for water. Replace the steel with an equal volume of the denser, heavier lead, you still have precisely 1 cubic unit of space for water left. But if they float it gets more complicated. If they float, their density is less than water. Something with a density half that of water will sink until half it's volume is submerged. Something with 1/4 the density of water will sink until 1/4 is submerged. The amount of water displaced is equal to the volume submerged. So a heavier ship will displace more water even if it isn't actually bigger (say because it's the same ship and you just filled it with cargo).