Change in volume of sphere with change in temperature

[MatinSAR]

Homework Statement

Two spheres of same size are made of the same metal but one is hollow and the other is solid. They are heated to same temperature, then what will happen?

Relevant Equations

Thermal expansion.

I am sure that the radius of the two spheres changes equally.
But in answer of the question it said that their change in volume is the same. Is it correct?!

Link of website: https://www.toppr.com/ask/question/...are-made-of-the-same-metal-but-one-is-hollow/

I think it is incorrect because equal change in radius does not mean the same volume change.

 

[DaveC426913]

...equal change in radius does not mean the same volume change.

Explain your rationale.

If two spheres, both with radius r₁ double their radius to r₂,
what can we say about comparing their volumes?

 

[MatinSAR]

Explain your rationale.

If two spheres, both with radius r₁ double their radius to r₂,
what can we say about comparing their volumes?

$V_2=8V_1$

 

[DaveC426913]

$V_2=8V_1$

That's not what the question is saying.

Hollow Sphere _HS₁ has radius r₁.
Solid Sphere _SS₁ has radius r₁.
i.e. _HV₁ = _SV₁

As you said: "the radius of the two spheres changes equally"

So, after both spheres are heated, r₁ increases to r₂.What can we say about the relationship between _HV₂ = _SV₂?

 

[MatinSAR]

That's not what the question is saying.

Hollow Sphere _HS₁ has radius r₁.
Solid Sphere _SS₁ has radius r₁.
i.e. _HV₁ = _SV₁

As you said: "the radius of the two spheres changes equally"

So, after both spheres are heated, r₁ increases to r₂.What can we say about the relationship between _HV₂ = _SV₂?

We can say that they are equal.

But what do you mean by $r_1$?
outer radius of hallow sphere?

 

[DaveC426913]

Gotta say, I am a little concerned about the grammar on that website.

...its orientation will do not affect it's expansion...
...we can say both sphere will expand same...
...because material used for making both of them are same......

 

[DaveC426913]

We can say that they are equal.

So you acknowledge the text's answer then?
An equal final radius of both spheres results in an equal final volume.

But what do you mean by $r_1$?

No, r₁ is simply the initial radius of both spheres before expansion.
And r₂ is the radius of both spheres after expansion.

 

[MatinSAR]

So you acknowledge the text's answer then?
An equal final radius of both spheres results in an equal final volume

No beacuse :

$r_2-r_1$ is same for both spheres but $V_2-V_1$ is not.

 

[DaveC426913]

There is no mention in the problem about inner and outer radii.

A solid sphere of radius 1 has a volume of 4/3π.
A hollow sphere of radius 1 has a volume of 4/3π.
(If you sunk them both in a bathtub, they would both displace 4/3π units of water.)

 

[MatinSAR]

There is no mention in the problem about inner and outer radii.

A solid sphere of radius 1 has a volume of 4/3π.
A hollow sphere of radius 1 has a volume of 4/3π.

It said that they have same size.
I think It means that apparent volume of hallow sphere equals volume of solid sphere. But real volume of hallow sphere is smaller than volume of solid sphere.

 

[DaveC426913]

It said that they have same size.
I think It means that apparent volume of hallow sphere equals volume of solid sphere. But real volume of hallow sphere is smaller than volume of solid sphere.

There is no mention of apparent volume versus real volume in the problem.

You are overthinking it and what it is trying to teach you.

 

[MatinSAR]

There is no mention of apparent volume versus real volume in the problem.

You are overthinking it and what it is trying to teach you.

So... In your opinion the spheres have same volume change. Am i right?

 

[DaveC426913]

So... In your opinion the spheres have same volume change. Am i right?

It's not my opinion; it is what the problem intends.

As you are interpreting the problem, you have reached one technically correct solution. But the problem is not trying to teach you about apparent versus actual volume.

Volume - in this problem - is being considered simply as 4/3πr³. i.e. how much water it would displace if sunk in a bathtub.Part of learning is understanding the nature and scope of what, exactly, you are being asked to solve.

 

[MatinSAR]

It's not my opinion; it is what the problem intends.

As you are interpreting the problem, you have reached one technically correct solution. But the problem is not trying to teach you about apparent versus actual volume.

Volume - in this problem - is being considered simply as 4/3πr³. i.e. how much water it would displace if sunk in a bathtub.Part of learning is understanding the nature and scope of what, exactly, you are being asked to solve.

Thanks for your help.

 

[Cutter Ketch]

I agree with @DaveC426913 that what you were thinking about is not what the problem intended. However, what you were thinking about is also very interesting. Call it “volume of metal”.

You note that the change in volume of metal is not the same for the two spheres. However, the original volume of metal is not at all equal, so perhaps that isn’t too surprising. So maybe the question ought to be is the change in volume of metal proportional? Is $\frac {\Delta V} V$ the same in the two cases? You’ll pretty easily find that it is, and, in fact, that is a general principle for any shape.

 

[MatinSAR]

I agree with @DaveC426913 that what you were thinking about is not what the problem intended.

If we examine the question in this way, it will be too simple, that's why I thought that this is not the point.
These two spheres have the same initial volume and temperature change, so it is more than obvious that they have equal expansion.

You note that the change in volume of metal is not the same for the two spheres. However, the original volume of metal is not at all equal, so perhaps that isn’t too surprising. So maybe the question ought to be is the change in volume of metal proportional? Is $\frac {\Delta V} V$ the same in the two cases? You’ll pretty easily find that it is, and, in fact, that is a general principle for any shape.

Great idea!

 

[DaveC426913]

If we examine the question in this way, it will be too simple, that's why I thought that this is not the point.

Except you are given the answer. And the answer is the simple one.

Now you're saying you don't like the simplicity of the question, so you'll answer it wrong, for the fun of it?

These two spheres have the same initial volume and temperature change, so it is more than obvious that they have equal expansion.

Then why not move on to more challenging questions?

 

[MatinSAR]

Except you are given the answer. And the answer is the simple one.

Now you're saying you don't like the simplicity of the question, so you'll answer it wrong, for the fun of it?

I did not say that. I doubted the answer to the question for several reasons. And that's why I asked here.
I know that $\Delta V = \beta V_i \Delta T$
But still I am not sure that what is $V_i$. Is it real or apparent volume?

Then why not move on to more challenging questions?

Because I passed this lesson a year ago and Now I have harder lessons to pass.

 

[DaveC426913]

I did not say that. I doubted the answer to the question for several reasons. And that's why I asked here.

But .. you are told the answer they're looking for. They even show you their math.

It makes no sense to have both the answer and the rationale given to you by the article (and by us), and yet continue question if the article "really" meant what it is really telling you. But you do you.

 

[MatinSAR]

But .. you are told the answer they're looking for. They even show you their math.

It makes no sense to have both the answer and the rationale given to you by the article (and by us), and yet continue question if the article "really" meant what it is really telling you. But you do you.

I agree.
Thanks for your help and time.