Determining Optimal Density for Floating Hollow Spherical Ball in a Liquid

[DanicaK]

Hollow spherical ball with inner and outer radius R and r respectfully is made of a material with density 1. The ball is swimming in a liquid with density 2. How much should be the density of the substance that will fill the hollow ball so that it can float?

help pls

 

[LawrenceC]

In order to receive help, you must make an attempt at a solution.

 

[DanicaK]

I tried to find the volume that is immersed when the ball has no hollow and when it has, but this didn't help me because i thing the volume of the immersed part with the substance in it should be given. I am very confused. The solution in the book is ρ=ρ1+(ρ2-ρ1)R/r^3

 

[LawrenceC]

Hint:
Write an algebraic expression for the weight of the ball that is a function of densities and volumes of spheres. There will be an unknown density in your expression. Then think about Archimedes and how you know whether something will float.

 

[DanicaK]

ΟΚ, i did the following:
P=mg
P=g(ρ1+ρ)/(V1+V2)
F(arch)=ρ2gV(immersed)
P=F(arch)
g(ρ1+ρ)/(V1+V2)=ρ2gV(immersed)
(ρ1+ρ)/(V1+V2)=ρ2V(immersed)
Again i have two unknowns:V(immersed); ρ

 

[LawrenceC]

Not sure we are on the same page here. The volume of the entire ball is 4 pi r^3/3. The volume of the inside section is 4 pi (r^3 - R^3)/3, where R in inside radius and r is outside radius. Determine the weight of the ball by multiplying by the respective densities. If the weight of water displaced by the ball is equal to or greater than the weight of the ball, it floats.

 

[DanicaK]

Thanks a looot. :D

 

[bigfooted]

(edit: LawrenceC already gave more or less the same hint)

1.1 What is the mass m if the sphere R was made entirely of rho1?
1.2 What is the mass of the small sphere r?
1.3 What is the mass of your sphere?
There is a density where all forces will cancel each other out. The sphere will not sink to the bottom, nor will it float to the surface. It will just remain motionless, completely immersed in the fluid.
2. What is V(immersed)?

 

[LawrenceC]

ρ=ρ1+(ρ2-ρ1)R/r^3

The above posted answer cannot be correct.

 

[DanicaK]

But the condition is the ball to float.
V(immersed) is the volume of the ball that is under water or the displaced water. And when i did the equation: weight of water displaced by the ball is equal to the weight of the ball i need the volume of water displeased.

 

[LawrenceC]

The volume of water displaced is the volume of the whole ball. You want the weight of the ball to just equal the weight of the displaced water. The assumption here (in order to get a unique answer) is that the entire ball submerges and is suspended in the liquid.

 

[bigfooted]

ρ=ρ1+(ρ2-ρ1)R/r^3

The above posted answer cannot be correct.

Indeed, it's ρ=ρ1+(ρ2-ρ1)(R/r)^3

To be more precise, if ρ≤ρ1+(ρ2-ρ1)(R/r)^3 then the ball will float.