Determine whether the object is floating or not

[thumn]

I have this homework question in fluids.
A cylindrical object has one of its flat ends painted red and the other painted green. It has a radius R and height H and has a nonuniform density given by the function ρ = 6αH + 4αh where h is the distance measured from the red end, and α is a positive number with units of density per length.
(a) Will this object float in a container of liquid of density ρliq = 9αH or will it sink? (Make an argument using math.)
(b) If it floats, what orientation will it float in stable equilibrium, and how much of the cylinder will be submerged? If it sinks what is the normal force that the base of the container exerts on the cylinder? (Either way, draw a diagram.)

Here is my answer so far: i know that to determine if the object is float or sink, we compare the density of the object and density of the liquid.
My concern is that: how can I proceed when the density of object is not constant (nonuniform)?

 

[berkeman]

I have this homework question in fluids.
A cylindrical object has one of its flat ends painted red and the other painted green. It has a radius R and height H and has a nonuniform density given by the function ρ = 6αH + 4αh where h is the distance measured from the red end, and α is a positive number with units of density per length.
(a) Will this object float in a container of liquid of density ρliq = 9αH or will it sink? (Make an argument using math.)
(b) If it floats, what orientation will it float in stable equilibrium, and how much of the cylinder will be submerged? If it sinks what is the normal force that the base of the container exerts on the cylinder? (Either way, draw a diagram.)

Here is my answer so far: i know that to determine if the object is float or sink, we compare the density of the object and density of the liquid.
My concern is that: how can I proceed when the density of object is not constant (nonuniform)?

Use integration to figure out the total weight, and the volume is fixed by the rigid container's dimensions.

 

[SteamKing]

I have this homework question in fluids.
A cylindrical object has one of its flat ends painted red and the other painted green. It has a radius R and height H and has a nonuniform density given by the function ρ = 6αH + 4αh where h is the distance measured from the red end, and α is a positive number with units of density per length.
(a) Will this object float in a container of liquid of density ρliq = 9αH or will it sink? (Make an argument using math.)
(b) If it floats, what orientation will it float in stable equilibrium, and how much of the cylinder will be submerged? If it sinks what is the normal force that the base of the container exerts on the cylinder? (Either way, draw a diagram.)

Here is my answer so far: i know that to determine if the object is float or sink, we compare the density of the object and density of the liquid.
My concern is that: how can I proceed when the density of object is not constant (nonuniform)?

Sooner or later, you'll need to find the mass of the object. This will make it easier to determine if it is floating than by comparing the density of the object (which is non-uniform) with the density of the fluid.