# Solving the Mystery of a Floating Bottle: Density & Volume

• Callmelucky
In summary, the average density of the floating bottle is 400 kg/m3 and its volume is 1L, but the given answer in the textbook may be incorrect. The density of the bottle is not the same as the density of the material it is made of, as it is partially submerged in water. It is important to consider the principle of Archimedes when solving this type of problem.
Callmelucky
Homework Statement
If we throw empty but sealed glass bottle in water it will float with 60% of it's volume above surface of water.
What is average density of floating bottle and what is it's volume? Mass of bottle is 0.4 kg(mass of air inside of bottle is irelevant).
Relevant Equations
density = mass / volume
If we throw empty but sealed glass bottle in water it will float with 60% of it's volume above the surface of water.
What is average density of floating bottle and what is it's volume? Mass of bottle is 0.4 kg(mass of air inside of bottle is irelevant).

It's easy problem but I can't get right solution.

40% of bottle is under water, so density of bottle is 400 kg/m3. Therefore the volume of is 0.4 kg / 400 kg/m3 = 1dm3 = 1L.

But the answer at the end of textbook for density is 392 kg/m3.

Can someone please tell me where I am wrong?

Thank you.

Callmelucky said:
Homework Statement:: If we throw empty but sealed glass bottle in water it will float with 60% of it's volume above surface of water.
What is average density of floating bottle and what is it's volume? Mass of bottle is 0.4 kg(mass of air inside of bottle is irelevant).
Relevant Equations:: density = mass / volume

If we throw empty but sealed glass bottle in water it will float with 60% of it's volume above the surface of water.
What is average density of floating bottle and what is it's volume? Mass of bottle is 0.4 kg(mass of air inside of bottle is irelevant).

It's easy problem but I can't get right solution.

40% of bottle is under water, so density of bottle is 400 kg/m3. Therefore the volume of is 0.4 kg / 400 kg/m3 = 1dm3 = 1L.

But the answer at the end of textbook for density is 392 kg/m3.

Can someone please tell me where I am wrong?

Thank you.
You should be solving for the volume of the bottle first.

Is the denisty of water given in the text or you are supposed to use 1000 kg/m3?

erobz said:
You should be solving for the volume of the bottle first.
what do you mean?
I can use density of glass(2500 kg/m3) and get volume of bottle for the mass of 0.4 kg = 0.00016 m3. But I don't see how is that going to help me.

nasu said:
Is the denisty of water given in the text or you are supposed to use 1000 kg/m3?
Doesn't say. It's usually stated if water is salty or if it has different density, here it says nothing, so i suppose it's 1000kg/m3.

Callmelucky said:
what do you mean?
I can use density of glass(2500 kg/m3) and get volume of bottle for the mass of 0.4 kg = 0.00016 m3. But I don't see how is that going to help me.
I mean you should be applying Archimedes principle to solve for the volume of the bottle.

When they say find the average density they mean “of the bottle” i.e. the bottles mass per unit volume. That’s not the same as the density of the glass that makes the bottle.

@erobz How would you even find the density of the glass from the given data? All you can find is the average density. It does not even have to be made from glass. Can be anything. If it's submerged 40%, its average density is 40% of the density of the fluid. It can be a solid piece of wood or an empty container made from steel.

nasu said:
@erobz How would you even find the density of the glass from the given data? All you can find is the average density. It does not even have to be made from glass. Can be anything. If it's submerged 40%, its average density is 40% of the density of the fluid. It can be a solid piece of wood or an empty container made from steel.
at the end of textbook I have a chart with some densites, glas is 2500 kg/m3. But yeah, I still don't understand what @erobz is trying to explain, I just didn't want to ask more questions because I feel stupid lol

Callmelucky said:
40% of bottle is under water, so density of bottle is 400 kg/m3. Therefore the volume of is 0.4 kg / 400 kg/m3 = 1dm3 = 1L.

But the answer at the end of textbook for density is 392 kg/m3.
The density of pure water is a bit under 1000 kg/m3, but only 0.3%. The given answer is 2% below what you have calculated. That's about the difference between g and 10m/s2, but how that can have entered into it I have no idea.

erobz, nasu and Callmelucky
haruspex said:
The density of pure water is a bit under 1000 kg/m3, but only 0.3%. The given answer is 2% below what you have calculated. That's about the difference between g and 10m/s2, but how that can have entered into it I have no idea.
Then probably authors' mistake

nasu said:
@erobz How would you even find the density of the glass from the given data? All you can find is the average density. It does not even have to be made from glass. Can be anything. If it's submerged 40%, its average density is 40% of the density of the fluid. It can be a solid piece of wood or an empty container made from steel.
Sorry, when I saw them pulling numbers out of seemingly nowhere, I just assumed that's where the mistake was. I never bothered to commit that result to memory myself, so when I didn't see:

$$p\cancel{g} 0.4 V_T = m\cancel{g} \implies V_T = \frac{m}{0.4\rho }$$

I leaped to false a conclusion that a mistake had been made.

Again...Sorry.

hutchphd and Callmelucky
Callmelucky said:
at the end of textbook I have a chart with some densites, glas is 2500 kg/m3. But yeah, I still don't understand what @erobz is trying to explain, I just didn't want to ask more questions because I feel stupid lol
Don't feel stupid for being right(or wrong)! Stand your ground, and it will get sorted out. Anyone can make a mistake.

Callmelucky
erobz said:
Sorry, when I saw them pulling numbers out of seemingly nowhere, I just assumed that's where the mistake was. I never bothered to commit that result to memory myself, so when I didn't see:

$$p\cancel{g} 0.4 V_T = m\cancel{g} \implies V_T = \frac{m}{0.4\rho }$$

I leaped to false a conclusion that a mistake had been made.

Again...Sorry.
Thanks for trying to help.

erobz
Because the book gives the mass of the bottle as 0.4 kg and not 0.400kg or better still 4.00x10^(-1) ]kg their answer to three significant figures should be graded with a zero

erobz said:
Sorry, when I saw them pulling numbers out of seemingly nowhere, I just assumed that's where the mistake was. I never bothered to commit that result to memory myself, so when I didn't see:

$$p\cancel{g} 0.4 V_T = m\cancel{g} \implies V_T = \frac{m}{0.4\rho }$$

I leaped to false a conclusion that a mistake had been made.

Again...Sorry.
If it makes you feel better, I did not comit it to the memory either.
I just scratched on a piece of paper exactly wht you wrote here, before writing that post.

Last edited:
erobz
hutchphd said:
Because the book gives the mass of the bottle as 0.4 kg and not 0.400kg or better still 4.00x10^(-1) ]kg their answer to three significant figures should be graded with a zero
It's actually given in dag, so 40 dag.
I just converted it so that it "looks better"

hutchphd

## What is the principle behind a floating bottle?

The principle behind a floating bottle is based on the concept of buoyancy, which is governed by Archimedes' principle. This principle states that an object will float if the buoyant force (upward force) acting on it is equal to the weight of the fluid displaced by the object. For a bottle to float, its overall density must be less than the density of the fluid it is placed in.

## How does the density of a bottle determine whether it will float or sink?

The density of a bottle, which is its mass divided by its volume, determines whether it will float or sink in a fluid. If the density of the bottle is less than the density of the fluid, the bottle will float. Conversely, if the density of the bottle is greater than the density of the fluid, the bottle will sink.

## What role does the volume of the bottle play in its ability to float?

The volume of the bottle plays a crucial role in its ability to float because it determines the amount of fluid that is displaced when the bottle is submerged. A larger volume means more fluid is displaced, which increases the buoyant force acting on the bottle. If this buoyant force is equal to or greater than the weight of the bottle, it will float.

## Can the contents inside the bottle affect its floating behavior?

Yes, the contents inside the bottle can significantly affect its floating behavior. If the bottle is filled with a substance that has a high density, such as sand or water, the overall density of the bottle will increase, making it more likely to sink. Conversely, if the bottle is filled with a low-density substance, such as air or a light gas, it will be more likely to float.

## How can you experimentally determine the density of a floating bottle?

To experimentally determine the density of a floating bottle, you can measure its mass and volume. First, weigh the bottle to find its mass. Then, submerge the bottle in water and measure the volume of water displaced, which is equal to the volume of the bottle. Finally, calculate the density using the formula: Density = Mass / Volume. This will give you the overall density of the bottle.

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