Submerged Fraction of Hollow Glass Sphere

[Fruityloop]

Homework Statement

A hollow, spherical glass has an inner radius of R and an outer radius of 1.2R. The density of the glass is d. What fraction of the shell is submerged when it floats in a liquid of density ρ = 1.5d (1.5 times the density of the glass)? (Assume the interior of the shell is a vacuum.)

Homework Equations

W=ρgh
Volume Sphere = (4/3)πr^3

The Attempt at a Solution

I attempted to set the weight of the water displaced equal to the weight of the shell. (Sum of forces = 0, Bouyont Force - Mass of Shell = 0, BF = Mass of Shell. For weight of shell I used W=ρgh with the volume being the volume of a sphere with r of 1.2R minus volume of a sphere with r of R. For the volume of water displaced I just used some variable fraction multiply by the volume of a sphere equation. ds and Rs then cancel out. I know the answer I'm supposed to get is 0.28, but I keep getting 0.48. Any insight towards a full solution would be hugely appreciated!

 

[ehild]

The fluid is not water. You have to take its density into account.

ehild

 

[Fruityloop]

I think I took that into account. I had it set up;

BF=Weight of Shell
ρgh (liquid) = ρgh (shell)

(1.5)dgh = dgh

d cancels?

 

[ehild]

What is h? You need to use the volume of the shell to get the weight of the shell and the volume of the outer sphere to get the weight of the displaced fluid.

ehild

 

[Fruityloop]

Oh sorry stupid typo. h is supposed to be V. I am fallowing you, that was my logic. I just don't know how to get from there to my answer >.<

 

[ehild]

The volume of the shell is not the same as the volume of the outer sphere.

ehild

 

[gneill]

Just to add a little noise into the relative peace and calm, when the problem statement asks for "What fraction of the shell is submerged", does it mean what fraction of the mass of the shell is submerged, what fraction of the volume of the shell (sphere!) is submerged, what fraction of the surface area of the shell is submerged, or what fraction of the diameter of the shell is submerged?

 

[ehild]

Just to add a little noise into the relative peace and calm, when the problem statement asks for "What fraction of the shell is submerged", does it mean what fraction of the mass of the shell is submerged, what fraction of the volume of the shell (sphere!) is submerged, what fraction of the surface area of the shell is submerged, or what fraction of the diameter of the shell is submerged?

It is fraction of the volume of the sphere, according to the "supposed" answer, but you are right, the question is not clear.

ehild

 

[Fruityloop]

Thanks for the help. I figured it out. @gneill, it is such an elusive question. I wasnt sure exactly what it was asking either.