Aluminium covered with gold submerged into water

[Artj0m]

Homework Statement

A piece of aluminum is completely covered with a gold shell to form an
object of weight W. When you suspend the object from a spring balance
and submerge it in water, the balance reads T. The density of water,
aluminum and gold is ρw, ρa, and ρg, respectively. The gravitational
acceleration is g.

What is the weight of the gold covering the object? Express
your answer in the given constants.

Homework Equations

m/V = Ro

The Attempt at a Solution

I don't know where to start

 

[Borg]

Artj0m, Welcome to PF! What formulas do you know that relate density to weight?

 

[Artj0m]

Artj0m, Welcome to PF! What formulas do you know that relate density to weight?

m/v = ρ

 

[Borg]

m/v = ρ

OK. Let's ignore the gold for now to make the problem simplier. If you put the aluminum in the water, what happens to the volumn of the water? Do you see a possible relation that you can work with?

 

[Artj0m]

OK. Let's ignore the gold for now to make the problem simplier. If you put the aluminum in the water, what happens to the volumn of the water? Do you see a possible relation that you can work with?

Yes, the buoyancy. I understand the concept, but the problem is that I eventually have to separate the two metals.

 

[Borg]

Yes, the buoyancy. I understand the concept, but the problem is that I eventually have to separate the two metals.

In order to solve the final equation, yes. Let's ignore W (and the gold) for now though.
When the aluminum is put into the water, the water level will rise. What do you think the relationship is for the increased water level w.r.t. the aluminum?

 

[Artj0m]

In order to solve the final equation, yes. Let's ignore W (and the gold) for now though.
When the aluminum is put into the water, the water level will rise. What do you think the relationship is for the increased water level w.r.t. the aluminum?

The amount of volume the water rises is equal to the volume of the object.
That I understand.
I'm just having troubles with writing a good equation.

 

[NascentOxygen]

So, what is the upward bouyancy force you can attribute to this volume, V, of displaced water? What is the formula?

 

[Borg]

The amount of volume the water rises is equal to the volume of the object.
That I understand.
I'm just having troubles with writing a good equation.

What is the relationship between the V_w increase and V_a? Can you apply your formula from post 3 to gain more info?

 

[Artj0m]

What is the relationship between the V_w increase and V_a? Can you apply your formula from post 3 to gain more info?

If I apply the formula, I know that V_w = m_w / ρ_w = m_alu / ρ_w

Because of the Buoyancy, B = W_alu = m_w * g = m_alu * g
So m_w = m_alu

 

[Borg]

m_alu / ρ_w

Are you sure about this part? Typo?

 

[Daan Janssen]

The piece of aluminum covered with a gold shell is in equilibrium so:

T=W-(ρw*V*g)

 

[Borg]

You might want to also review the Wiki article on Archimedes' principle.

 

[Borg]

@Daan Janssen - The goal of the homework section isn't to give people the answers. It's to guide them to the answer so that they understand how to solve it for themselves.

 

[Daan Janssen]

@Daan Janssen - The goal of the homework section isn't to give people the answers. It's to guide them to the answer so that they understand how to solve it for themselves.

Yes, sorry, I do understand that. I myself, just as Artj0m, don't know how to get to the answer. It is just there to insure we get the right answer out of it.

I've really been trying and I know I am very close to the right answer. I'm hoping you could guide us to it

 

[Borg]

Yes, sorry, I do understand that. I myself, just as Artj0m, don't know how to get to the answer. It is just there to insure we get the right answer out of it.

I've really been trying and I know I am very close to the right answer. I'm hoping you could guide us to it

OK. However, I think that your answer is wrong.

 

[Daan Janssen]

OK. However, I think that your answer is wrong.

The answer was taken out of a correction sheet from Eindhoven University. The Question was part of a final exam. I doubt that it would be wrong. It could still be ofcourse :p

 

[Borg]

The answer was taken out of a correction sheet from Eindhoven University. The Question was part of a final exam. I doubt that it would be wrong. It could still be ofcourse :p

It wouldn't be the first time that I was wrong. I'm having someone check my work before I guide you both down the wrong path.

 

[AlephNumbers]

Daan, although that is correct, that is not what the question is asking for. Artjom, I think you may have summed your forces incorrectly. Do you draw force diagrams? I find that drawing a diagram really helps me get the forces right.

 

[NascentOxygen]

Without the water, the spring balance will record the full weight of the metal. Dipping the metal into water reduces the balance reading. You should be able to come up with an equation involving expressions for these components.

 

[Artj0m]

You might want to also review the Wiki article on Archimedes' principle.

I've read it and I have come to this.
Like you said, I only started with the aluminium

ρ_a/ρ_w = W/(W-T)

This leads to

W - [(ρ_a/ρ_w) * (W - T)] = 0

This comes pretty close to the answer in my book:

W_g = (ρ_g / (ρ_g - ρ_alu)) * (W - (ρ_a/ρ_w) * (W - T))

 

[BvU]

Good.

If you repeat the experiment with iron, the expression would be ρ_iron/ρ_w = W/(W-T) with different W and T of course.
If you repeat the experiment with an alloy (e.g. Au and Al !) , the expression would be ρ_alloy/ρ_w = W/(W-T) with again different W and T of course.

All that remains for this exercise is to find ρ_alloy for W-W_gold kilograms of Al and W_gold kilograms of Au, and then work around to W_gold.

 

[Artj0m]

Good.

If you repeat the experiment with iron, the expression would be ρ_iron/ρ_w = W/(W-T) with different W and T of course.
If you repeat the experiment with an alloy (e.g. Au and Al !) , the expression would be ρ_alloy/ρ_w = W/(W-T) with again different W and T of course.

All that remains for this exercise is to find ρ_alloy for W-W_gold kilograms of Al and W_gold kilograms of Au, and then work around to W_gold.

I understand, I'm just struggling with the fact that I don't know where in the equation I have to place the W_gold.
Could you give a clue?

 

[Borg]

I understand, I'm just struggling with the fact that I don't know where in the equation I have to place the W_gold.
Could you give a clue?

Now that you've solved this for Al, add in the gold component. You should get an equation that uses the initial given values that you can then use to solve for W_g.

 

[BvU]

I understand, I'm just struggling with the fact that I don't know where in the equation I have to place the W_gold.
Could you give a clue?

Giving a clue boils down to to helping find a mixing rule for density, right ?
So if A kilograms of material with density $\rho_A$ and B kilograms of material with density $\rho_B$ are fused to A+B kilograms, what's the density $\rho_{A+B}$?

(Big clue: you may add up the volumes and you can use your own relevant equation)

 

[Artj0m]

Giving a clue boils down to to helping find a mixing rule for density, right ?
So if A kilograms of material with density $\rho_A$ and B kilograms of material with density $\rho_B$ are fused to A+B kilograms, what's the density $\rho_{A+B}$?

(Big clue: you may add up the volumes and you can use your own relevant equation)

So ρ_a/ρ_w = W - W_g / W - T
I don't understand how I should add this equation with the Aluminium one.

 

[BvU]

Try again: if A kilograms of material with density $\rho_A$ and B kilograms of material with density $\rho_B$ are fused to A+B kilograms, what's the volume ?

 

[Artj0m]

V_A+B = m_A / ρ_A + m_B / ρ_B = (m_A*ρ_B + m_B*ρ_A) / (ρ_A + ρ_B)

 

[BvU]

Yep ! Next step: what is the density of m_A+m_B kilograms of stuff that has a volume V_A+B ? :)

 

[Artj0m]

Yep ! Next step: what is the density of m_A+m_B kilograms of stuff that has a volume V_A+B ? :)

Just (ρ_A + ρ_B) right?

 

[BvU]

Nope. (m_A+m_B) / V_A+B. Check it out, do a numerical example and see that that isn't the same

 

[Artj0m]

Nope. (m_A+m_B) / V_A+B. Check it out, do a numerical example and see that that isn't the same

Oh I see! Because of ρ = M / V But how does this help me in the exercise?

 

[BvU]

You already had (I removed the a, since it is valid for iron too and for any stuff):

ρ/ρ_w = W/(W-T)

So ρ = ρ_w * W/(W-T).

Since then you set up an expression for ρ (namely (m_A+m_B) / V_A+B -- you still have to rewrite that to the given variables in the problem statement of this exercise).

And -- if we didn't make any mistakes -- equating this ρ to the earlier ρ leads to the book answer.

 

[Artj0m]

You already had (I removed the a, since it is valid for iron too and for any stuff):
So ρ = ρ_w * W/(W-T).

Since then you set up an expression for ρ (namely (m_A+m_B) / V_A+B -- you still have to rewrite that to the given variables in the problem statement of this exercise).

And -- if we didn't make any mistakes -- equating this ρ to the earlier ρ leads to the book answer.

For the ρ, I have:

ρ_{g + a} = [(W_G + W_A) / (W_G*ρ_A + W_A*ρ_G)] * (ρ_G + ρ_A)

So I will equate it with ρw * W/(W-T) and then I should get the correct answer right?

 

[BvU]

You still have to get rid of W_A .

 

[Artj0m]

You still have to get rid of W_A .

That's just W - W_G right?

 

[BvU]

Of course it is.

 

[BvU]

V_A+B = m_A / ρ_A + m_B / ρ_B = (m_A*ρ_B + m_B*ρ_A) / (ρ_A + ρ_B)

Before you go too far and get stuck anyway, I think there is something to be improved here. A sure sign of that is that the dimensions don't fit -- always a good check !

 

[Artj0m]

Before you go too far and get stuck anyway, I think there is something to be improved here. A sure sign of that is that the dimensions don't fit -- always a good check !

I'm stuck haha
What do you mean? I'm not sure I understand you correctly.

( I've changed the M into W, so it fits the equation ρ = ρw * W/(W-T))

 

[BvU]

What you found has a dimension mass. Should have been Volume

 

[Artj0m]

So the equation we made is not useful? Or did the equation had to be something like this?

m_A+B = ρ_A*V_A + ρ_B*V_B

 

[BvU]

V_A+B = m_A / ρ_A + m_B / ρ_B = (m_A*ρ_B + m_B*ρ_A) / (ρ_A + ρ_B)

Last + should have been a *

 

[mr166]

Here is how I would physically solve the problem.
1. Find the volume of the object.
2. Find the weight of the object.
3. Determine the density of the object.
4. Look up the density of that volume of AL and calculate the theoretical weight.
5. Subtract the theoretical weight from the true weight.
6. The difference is the weight of the gold.

Here is how I would do that.

1. Suspend the water tank from the spring scale.
2. Using a string of negligible volume and weight , suspend the object in the water without letting it touch the container. The change of weight in grams equals the volume of the object in CCs.
3. Let the object rest on the bottom of the container and measure the change in weight from reading in step one. This is the true weight of the object.

BTW buoyancy has nothing to do with the solution.

 

[BvU]

And with that we get the book answer ! Happily ever after (Have to admit it cost me a bit of effort too :) ) !

 

[Artj0m]

Thanks you guys! :)

 

[BvU]

Here is how I would physically solve the problem.
1. Find the volume of the object.
2. Find the weight of the object.
3. Determine the density of the object.
4. Look up the density of that volume of AL and calculate the theoretical weight.
5. Subtract the theoretical weight from the true weight.
6. The difference is the weight of the gold.

Here is how I would do that.

1. Suspend the water tank from the spring scale.
2. Using a string of negligible volume and weight , suspend the object in the water without letting it touch the container. The change of weight in grams equals the volume of the object in CCs.
3. Let the object rest on the bottom of the container and measure the change in weight. This is the true weight of the object.

BTW buoyancy has nothing to do with the solution.

Hello mr, welcome to PF :)

Nice to have you join in. So far I've been proceeeding on the hypothesis that post #1 is a homework exercise, not a lab practice (after all, gold is rather pricey :) !). And there isn't much physics in there, but a lot of juggling equations, something that can be pretty awkward (we needed over 40 posts already!).

I don't agree that buoyancy isn't involved. As far as I know buoyancy is our W-T !

But I do like your practical recipe. Do you agree that your 3. result minus your 1. result gives W, your 2. result minus your 1. result gives W-T. And with that we are back to the original exercise !

 

[mr166]

Hello mr, welcome to PF :)

Nice to have you join in. So far I've been proceeeding on the hypothesis that post #1 is a homework exercise, not a lab practice (after all, gold is rather pricey :) !). And there isn't much physics in there, but a lot of juggling equations, something that can be pretty awkward (we needed over 40 posts already!).

I don't agree that buoyancy isn't involved. As far as I know buoyancy is our W-T !

But I do like your practical recipe. Do you agree that your 3. result minus your 1. result gives W, your 2. result minus your 1. result gives W-T. And with that we are back to the original exercise !

After reviewing my reply, the the density of the object is not really relevant.

Only 1,2,4,5 and 6 are needed. Buoyancy is not a factor. Just the difference in weight between the empty, water filled, container and the container with the object resting on the bottom counts.

 

[Artj0m]

This has been a great help! Thanks! :)

 

[BvU]

After reviewing my reply, the the density of the object is not really relevant.

Only 1,2,4,5 and 6 are needed. Buoyancy is not a factor. Just the difference in weight between the empty, water filled, container and the container with the object resting on the bottom counts.

I really don't understand how you can say that. Buoyancy is W-T and it definitely is an actual factor in the answer. And of course the density of the object is relevant. We worked out this whole thing by equating the $\rho$ of the object as expressed in terms of W, W_Au, $\rho_{Al}$ and $\rho_{Au}$ on one hand and in terms of $\rho_{H_2 O}$, W, and W-T on the other.

 

[BvU]

This has been a great help! Thanks! :)

You're welcome. That's what PF is for.