Foam density required to float a certain weight

[Spinergy]

I have a polymer processing teacher that has decided to give us a final exam with a bunch of questions we have not covered the entire semester. I've figured out most of them but this particular one has me stumped.

I am given the dimensions for a part made of a known material and need to calculate the density required for a foam insert to be able to keep a certain weight afloat.

I know I need to calculate the weight of the part and add that to the total weight being supported but am not sure how to calculate the necessary foam density. Can anyone provide links to some resources that may be useful? My googling attempts haven't turned up anything so far.

 

[SteamKing]

Try googling 'Archimedes Principle'.

 

[DanPreLoaded]

Hey, this is to do with the buoyancy of the object. If you consider an object floating in the water, the forces pushing it up (buoyancy) are equal to those weighing it down (gravity). If it wasn't equal, the object would either be sinking (gravity overcoming buoyancy - a stone in water say) or rising from under water (buoyancy overcoming gravity - football under water).
So for the equations governing this:
We will use an object floating in water as an example.
Buoyancy force (acting up)
Fb = rho (water) x Volume (the submerged volume) x Gravitational acceleration

Now we have the downward force caused by gravity on the object (Fd)
Fd = mass (object) x gravity acceleration (9.81)

So since the object is floating (ie static), the forces acting both up and down are equal. They're in equilibrium.
So equating this

Fd = Fb
Mass (object) x g = rho (water) x (volume of submerged object) x g

Now if we tie this in with the relationship

Mass = rho (density) x Volume

And sub in this for the objects mass on the left hand side we get:

Rho (object) x Volume (object) x g = rho (water) x Volume (submerged - in contact with fluid) x g

Now just rearrange this for the density of the object and you've got it

 

[russ_watters]

Is the foam density calculated or given? Or is foam light enough compared to water that you can simply ignore it's density? Do you need an answer more than 98% accurate?

 

[Spinergy]

Is the foam density calculated or given? Or is foam light enough compared to water that you can simply ignore it's density? Do you need an answer more than 98% accurate?

The foam density is the unknown value being solved for. I know its volume from the specifications given for the part. The problem that I'm facing now is that I need to solve for a combined part+foam density to use the Archimedes principle. I plugged all of my known values into MathCad and I think it messed itself out of frustration while trying to convert units. The problem would be much easier if the total weight was assumed to be 500lbs instead of 500lbs+part+foam since the foam weight is unknown and dependent on the unknown density.

I'm a bit frustrated at how unreasonable this entire exam is at this point so I'm going to sleep on it. I appreciate the help so far

 

[Spinergy]

Hey, this is to do with the buoyancy of the object. If you consider an object floating in the water, the forces pushing it up (buoyancy) are equal to those weighing it down (gravity). If it wasn't equal, the object would either be sinking (gravity overcoming buoyancy - a stone in water say) or rising from under water (buoyancy overcoming gravity - football under water).
So for the equations governing this:
We will use an object floating in water as an example.
Buoyancy force (acting up)
Fb = rho (water) x Volume (the submerged volume) x Gravitational acceleration

Now we have the downward force caused by gravity on the object (Fd)
Fd = mass (object) x gravity acceleration (9.81)

So since the object is floating (ie static), the forces acting both up and down are equal. They're in equilibrium.
So equating this

Fd = Fb
Mass (object) x g = rho (water) x (volume of submerged object) x g

Now if we tie this in with the relationship

Mass = rho (density) x Volume

And sub in this for the objects mass on the left hand side we get:

Rho (object) x Volume (object) x g = rho (water) x Volume (submerged - in contact with fluid) x g

Now just rearrange this for the density of the object and you've got it

I understand all of that but having to solve for a combined object density (one polyethylene and assumed by me to be HDPE, the other of unknown mass but known volume) is really throwing me off.

Edit: I'll post my mathcad in the morning. Hopefully you guys can point out where I'm straying off-course.

 

[russ_watters]

The foam density is the unknown value being solved for. I know its volume from the specifications given for the part.

The problem that I'm facing now is that I need to solve for a combined part+foam density to use the Archimedes principle.

You've contradicted yourself: either you want to solve for the foam density or the combined foam+part density. If the teacher wants the foam density, then you are only solving for the foam density.

And you also know the mass or weight of the part based on its dimensions and density, right?

So you just need to write an equation for the required net force...of zero. Buoyancy is total volume times water's density and the downward forces are the weights of the foam and the part. Just add it all together and set it equal to zero. The equation is fairly straightforward: please give a try at writing it and we'll help point you to what, if anything is wrong with it.

I'm a bit frustrated at how unreasonable this entire exam is at this point so I'm going to sleep on it. I appreciate the help so far.

I'm sorry, but I don't see this as being unreasonable. Based on the title of the course it sounds like you are in college, but this strikes me as an early high school level algebra/physics problem.

Also, we have a homework help section and I'm moving this thread there.

 

[Spinergy]

You've contradicted yourself: either you want to solve for the foam density or the combined foam+part density. If the teacher wants the foam density, then you are only solving for the foam density.

And you also know the mass or weight of the part based on its dimensions and density, right?

So you just need to write an equation for the required net force...of zero. Buoyancy is total volume times water's density and the downward forces are the weights of the foam and the part. Just add it all together and set it equal to zero. The equation is fairly straightforward: please give a try at writing it and we'll help point you to what, if anything is wrong with it.

I'm sorry, but I don't see this as being unreasonable. Based on the title of the course it sounds like you are in college, but this strikes me as an early high school level algebra/physics problem.

Also, we have a homework help section and I'm moving this thread there.

I assumed I would need to solve for the part+foam density ((part mass + foam mass)/(part volume + foam volume)) needed to keep 500 additional pounds of dead weight afloat, and then pull out the foam density.

I see this as unreasonable considering that this is a polymer processing class. The objectives for the class are to be able to optimize an extruder, injection/blow molder, etc; not how to make a dock for the fish camp.

Thanks for moving the thread to the proper location.

Edit: I was given the material of the shell (polyethylene), the dimensions, wall thickness, and additional weight to be supported in water. So I know the volume of the foam, volume of the shell, and mass of the shell.

Original problem: https://www.dropbox.com/s/m533ou8ov2sodzb/Screenshot_2014-04-30-00-32-25.png

 

[Spinergy]

So I went ahead and assumed 500lbs was shell+foam+dead weight and came up with a reasonable foam density.

https://www.dropbox.com/s/hdeiscfhcjq7qb2/Prob2.xps