How does the density of water affect a sphere dipped in it?

[Jahnavi]

Homework Statement

Homework Equations

The Attempt at a Solution

There can be two cases

1) The sphere might be a shell such that it floats on the water .When the temperature increases , even though density of water decreases , the force of buoyancy should remain same . Because force of buoyancy should be equal to the weight of the sphere at all times .

2) If it is a solid sphere , then it will sink and reach the bottom of the vessel. The volume of sphere will increase but the density of water decreases . So in the expression for force of buoyancy Vρg , V increases , but ρ decreases . So how can we conclusively say , which factor dominates ?

 

[haruspex]

The sphere might be a shell such that it floats on the water .

True, but it is not clear whether that is allowed in the question. "Dipped" suggests to me that it is fully submerged, whether by virtue of its own weight or by being held down.

2) If it is a solid sphere , then it will sink and reach the bottom of the vessel. The volume of sphere will increase but the density of water decreases . So in the expression for force of buoyancy Vρg , V increases , but ρ decreases . So how can we conclusively say , which factor dominates ?

I agree. It depends whether the water or the aluminium expands more. Maybe you are expected to look that up.

 

[Jahnavi]

True, but it is not clear whether that is allowed in the question. "Dipped" suggests to me that it is fully submerged, whether by virtue of its o

Fair enough

I will rule this out . But would you agree if it were a metallic shell floating on water , then on increasing temperature , the force of buyancy remains same ?

Would you also agree that in this case (metallic shell floating ) the rate of increase of volume of metal will be exactly equal to the rate of decrease of density of water such that force of buoyancy remains same ?

I agree. It depends whether the water or the aluminium expands more. Maybe you are expected to look that up.

Volumetric coefficient of aluminium is roughly 3 times less than that of water , which means water expands more . This means force of byuoyancy decreases on heating water i.e option b) . Do you agree ?

 

[haruspex]

would you agree if it were a metallic shell floating on water , then on increasing temperature , the force of buyancy remains same ?

Yes.

in this case (metallic shell floating ) the rate of increase of volume of metal will be exactly equal to the rate of decrease of density of water

Increase of submerged volume, yes.

Volumetric coefficient of aluminium is roughly 3 times less than that of water , which means water expands more . This means force of byuoyancy decreases on heating water i.e option b) . Do you agree ?

Yes.

 

[Jahnavi]

Increase of submerged volume,

I meant the same

But in case of the metallic shell , there are two types of volumes , volume of the shell i.s (4/3)πr³ AND volume of the metal . In calculating force of byuoyancy we use the former but while calculating thermal expansion we use the latter . Right ?

So in case of metallic shell , the rate of expansion of volume of metal will be such that the rate of expansion of volume of shell will be equal to the rate of decrease of volume of water . Right ?

 

[haruspex]

while calculating thermal expansion we use the latter

A uniform sphere expands uniformly. The sphere will expand to the same new radius whether solid or hollow.

 

[Jahnavi]

A uniform sphere expands uniformly. The sphere will expand to the same new radius whether solid or hollow.

OK . So in case of the shell , in the equation V_t = V₀(1+γ∆t) , volumes , V_t and V₀ are the volumes of the space occupied/enclosed by the shell and not the volume of the material(metal) ??

 

[haruspex]

OK . So in case of the shell , in the equation V_t = V₀(1+γ∆t) , volumes , V_t and V₀ are the volumes of the space occupied/enclosed by the shell and not the volume of the material(metal) ??

Right.

 

[Jahnavi]

Thanks !

 

[Jahnavi]

@haruspex , please see a similar problem .

I don't understand what is the need in the question to state that there is no change in the shape of the object . Could you explain this .

Even if there is a change in the volume of an object , since heat could exchange only between water and object , the temperature of object has to increase . Isn't it ?

Is it possible for an object to expand by absorbing heat but without increase in its temperature ? The answer is no . Then why is there a need for that statement .Irrespective of the change in volume of the object , if the temperature of water falls , then temperature of the object has to increase i.e option a) .

What do you think ?

 

[Jahnavi]

Please clear another doubt from an illustration given in the book . If you remember , I had asked a similar question earlier on the forum . Since this is an example , this has left me wondering if their logic is correct .

Don't you think , correct answer should be option b) ? Since more air is pumped in , the density of air increases which results in increased buoyant force due to air . Since weight of the block is same , this leads to a smaller buoyant force from the water .

Their assumption of P_atm both at the top of the block and at the surface of water is not correct .

Please let me know what you think .

 

[haruspex]

@haruspex , please see a similar problem .
View attachment 225267

I don't understand what is the need in the question to state that there is no change in the shape of the object . Could you explain this .

Even if there is a change in the volume of an object , since heat could exchange only between water and object , the temperature of object has to increase . Isn't it ?

Is it possible for an object to expand by absorbing heat but without increase in its temperature ? The answer is no . Then why is there a need for that statement .Irrespective of the change in volume of the object , if the temperature of water falls , then temperature of the object has to increase i.e option a) .

What do you think ?

It is not completely clear what changes are allowed. Can it be an object which is externally solid and constant in shape, but need not be so internally?
I note that options c and d are not mutually exclusive. If you have to pick one then they seem to be ruled out on that basis.

 

[haruspex]

Please clear another doubt from an illustration given in the book . If you remember , I had asked a similar question earlier on the forum . Since this is an example , this has left me wondering if their logic is correct .

View attachment 225269

Don't you think , correct answer should be option b) ? Since more air is pumped in , the density of air increases which results in increased buoyant force due to air . Since weight of the block is same , this leads to a smaller buoyant force from the water .

Their assumption of P_atm both at the top of the block and at the surface of water is not correct .

Please let me know what you think .

There are several awkwardnesses in the question. You are quite right that the increased air density will cause the wood to rise, if the wood is airtight and rigid. But what if air can seep in, raising the weight? Or if the wood compresses, raising the density?

 

[Jahnavi]

Can it be an object which is externally solid and constant in shape, but need not be so internally?

How does that matter ?

Since the temperature of water falls and there is no way the heat could be exchanged except the "solid" object , the temperature of the solid object has to increase irrespective of the change in its shape. No ?

 

[Jahnavi]

But what if air can seep in, raising the weight?

Where ? Air seep in wood ?

 

[haruspex]

Where ? Air seep in wood ?

Yes. Water can seep in slowly; maybe air can seep in faster.

 

[haruspex]

How does that matter ?

What is inside might not be solid at all times.

 

[Jahnavi]

What is inside might not be solid at all times.

You mean change of phase such that all heat is absorbed without increasing the temperature ?

 

[haruspex]

You mean change of phase such that all heat is absorbed without increasing the temperature ?

Yes.

 

[Jahnavi]

Yes.

OK . Assuming that is not the case (complete solid object at all times of heat transfer), do you agree that shape of the object is irrelevant and temperature of object has to increase ?

 

[haruspex]

OK . Assuming that is not the case (complete solid object at all times of heat transfer), do you agree that shape of the object is irrelevant and temperature of object has to increase ?

Yes.

 

[Jahnavi]

Thank you very much