A hollow sphere sinks some fraction f in water...

[Vriska]

Homework Statement

When completely hollow. To what fraction of volume should water be poured into it so that it is just enough to cause the sphere to sink. [/B]

Homework Equations

Archemedes : Buoyant fource equals weight of water displaced

The Attempt at a Solution

Let's first find the density of the metal and air part that goes into the water -

let d_s be density of the air metal combination that sinks. and d_w be that of water

d_s *V*g = d_w*f*V*g
with this we have found air metal density d_a.

annnd i don't think this step is correct so I won't proceed to get my incorrect answer. I'm not sure what else to do here, there's no finding out the volume of the shell because it's a shell, it has no volume. This is effectively the volume of the shell and air(or lack of) but this doesn't look to be constant honestly.

But my next step is let p be the fraction of water to be poured

d_w*p*V*g + d_a*V *g= (1-p)*V*g. V being the total volume of the sphere.

leads to a wrong answer.

 

[mjc123]

Archemedes : Buoyant fource equals weight of water displaced

And what is this also equal to, in the case of a floating body?

 

[Vriska]

And what is this also equal to, in the case of a floating body?

the weight of the body?

 

[Abhishek kumar]

the weight of the body?

If sphere sink completely then buoyancy force=weight of hollow sphere+weight of volume poured in hollow sphere

 

[Vriska]

If sphere sink completely then buoyancy force=weight of hollow sphere+weight of volume poured in hollow sphere

What's the weight of the hollow sphere?

 

[Abhishek kumar]

What's the weight of the hollow sphere?

Here nothing is given in question to calcute the weight

 

[Nik_2213]

A ship floats if mass of hull-displaced water exceeds the mass of ship hull + cargo.
{ Leaving aside free-board & safety margins. See submarines, FLIP-SHIP, also semi-submersible / whale-back ships on Great Lakes... }

Unless the spherical hull is very large and light compared to content, you'll need its mass. Consider the bathyscaphe, buoyed by a flimsy tank of kerosene...

 

[BvU]

What's the weight of the hollow sphere?

You have obscured from view the pertaining part of the problem statement: it has to do with the fraction $f$ in you thread title ...

 

[Vriska]

You have obscured from view the pertaining part of the problem statement: it has to do with the fraction $f$ in you thread title ...

Hm, I definitely can use that to get the average density of the hollow air + metal shell but I don't see that helping

 

[Vriska]

A ship floats if mass of hull-displaced water exceeds the mass of ship hull + cargo.
{ Leaving aside free-board & safety margins. See submarines, FLIP-SHIP, also semi-submersible / whale-back ships on Great Lakes... }

Unless the spherical hull is very large and light compared to content, you'll need its mass. Consider the bathyscaphe, buoyed by a flimsy tank of kerosene...

I'm not assuming anything about the mass of the shell. I got this from a problem that went like - a hollow spherical shell floats with half its volume submerged, to what fraction of volume should it be filled to make it sink. Answer by symmetry being V/2. I'm trying to get a general case of that here

 

[BvU]

Hm, I definitely can use that to get the average density of the hollow air + metal shell but I don't see that helping

You wanted the weight. See if you can ignore the mass of the air (both inside and outside of the sphere)

 

[Abhishek kumar]

I'm not assuming anything about the mass of the shell. I got this from a problem that went like - a hollow spherical shell floats with half its volume submerged, to what fraction of volume should it be filled to make it sink. Answer by symmetry being V/2. I'm trying to get a general case of that here

If problem state that what portion of volume submerg then you can find the density or weight of material but here nothing given about initial condition of sphere.

 

[haruspex]

a hollow spherical shell floats with half its volume submerged, to what fraction of volume should it be filled to make it sink. Answer by symmetry being V/2. I'm trying to get a general case of that here

Imagine replacing the shell with one of zero weight but partly filled with water, and with the same fraction submersed.

 

[Vriska]

Imagine replacing the shell with one of zero weight but partly filled with water, and with the same fraction submersed.

If the fraction f is filled with water in an invisible, weightless ball, then 1-f needs to be put to make it sink. But now let's assume that that f fraction of water were transformed into a hard shell, it's mass would be the same but since it's thin we can ignore the effect of volume so we need to had 1-f of water to make it sink.

amazing insight thank you so much!

 

[Vriska]

You wanted the weight. See if you can ignore the mass of the air (both inside and outside of the sphere)

how do I get the weight : (? the sphere is hollow, I can find the average density of the empty space and the metal by density_metal * V = f(fraction submerged) *V*d_water.

but what would i do with this?

 

[mjc123]

You know the average density and the volume, but you can't find the weight...?
You have already said that the weight is equal to the buoyant force, which is equal to the weight of water displaced.
So M*g = d_w*f*V*g
Now what would the mass have to be for the sphere to just sink (f=1)?

 

[Vriska]

You know the average density and the volume, but you can't find the weight...?
You have already said that the weight is equal to the buoyant force, which is equal to the weight of water displaced.
So M*g = d_w*f*V*g
Now what would the mass have to be for the sphere to just sink (f=1)?

oh yeah right that's how. I just have to do mg + d _w p(fraction of water desired) V = d_w *V*g?

taking mg = d_w *f*v*g so we get our expected 1-f! Thanks! I don't know what i was getting so confused over.