Determining Fluid Density Using Buoyancy and Simple Materials

[KDawgAtsu]

Homework Statement

In the laboratory, you are given a cylindrical beaker containing a fluid and you are asked to determine the density \rho of the fluid. You are to use a spring of negligible mass and unknown spring constant k attached to a stand. An irregularly shaped object of known mass m and density D > \rho hangs from the spring, so that the object (but non of the spring) is immersed in the fluid. You may also choose from among the following items to complete the task.

• A metric ruler
• A stopwatch
• String

Explain how you could experimentally determine the density of the fluid. Show explicitly, using equations, how you wuill use your measurements to calculate fluid density \rho. Start by identifying any symbols you use in your equations.

Homework Equations

F_buoyancy = \rho_fluidV_objectG
\rho = m/v
\rho_obj/\rho_fluid = V_fluid/V_obj

The Attempt at a Solution

I'm pretty much at a loss on this one. Seeing as I have the density of the object and its volume, I could plug that into the 3rd equation. However, I'm wondering if the third equation only applies to floating objects, as in this experiment, the object will sink. I can't figure out a way to obtain the fluid's mass, either. Thank you for the help.

 

[Delphi51]

Looks like you will have to work with the spring and its hanging mass.
What will happen to the spring when you slowly lift the mass out of the fluid?

 

[KDawgAtsu]

Will the spring extend to its maximum length, since there is no more buoyant force pushing it up any more?

 

[Delphi51]

Okay, you've got a start! You could measure the change in the extension with that ruler. Suggest you write some spring formulas and see if you can find anything from that.

Quite an interesting problem! I don't see my way through it yet!

 

[KDawgAtsu]

The force of the spring is -kx, so by measuring the change in the x and using that equation, I would get the buoyant force? And with that, I would be able to use F=pvg, to solve for p. Am I correct?

 

[Delphi51]

That sounds very promising. Remember, you don't have "x", you only have x2 - x1 or Δx so you need to play with F = kx (I hate that minus sign!) a little more. Don't abandon your other spring formulas just yet - make sure you write them all down on your scrap paper so you don't concentrate all your attention on this one.

 

[KDawgAtsu]

Hm, would it work if I simply measured the length of the spring of when the object is submerged and of when it is not, and find the difference in the forces, which would be the buoyant force?

 

[Delphi51]

Maybe. Start with F1 = kx1 and see if it works out.

 

[KDawgAtsu]

Well the units work out, and I don't see why it wouldn't work out conceptually. Thank you for your help!

 

[Delphi51]

So, do you know the value of k?

 

[KDawgAtsu]

Well there are no numbers involved in this FRQ, just variables. But yes, I can find k using the stopwatch and timing a period, then use T = 2\pi\sqrt{}m/k

 

[Delphi51]

Oh, good thinking!
But you don't know the mass . . .
If you knew the mass, you could find the buoyant force . . .
But still not the density because you don't know the volume.
More thinking still to do, but you have a good start.