Buoyancy question about styrofoam

[HadManySons]

Homework Statement

The first part of the question states this: If we want to buoy a 0.137 kg solid platinum object in water with an attached string from above, what is the tension required to support the piece of platinum?

The correct answer is: 1.28 Newtons. I got this by subtracting the buoyant force, which was

(.137kg/(21450kg/m^3)) * (9.8m/s^2) * (1000 kg/m^3) = .062593N
the volume of platinum g density of water

from the weight of the platinum: 1.3426 Newtons


What I'm trying to find now is this, "If the object is now attached to a piece of Styrofoam, what volume of Styrofoam (density rho_s = 95 kg/m3) should be used to have the Styrofoam and the metal floating just under the water surface?". But I've hit a stopping point and can't seem to get the right answer

Homework Equations

density = mass/volume

buoyant force = volume displaced * gravity * density of fluid

The Attempt at a Solution

I had set everything in equilibrium, meaning:
Buoyant force + Weight of Styrofoam + weight of the platinum = 0

But how do I know the volume of the water displaced if I don't already know the volume of Styrofoam that I need?

 

[Doc Al]

I had set everything in equilibrium, meaning:
Buoyant force + Weight of Styrofoam + weight of the platinum = 0

That's fine, but it might be easier to just look at forces on the Styrofoam piece. (Since you calculated the tension in the string, why not use it?)

But how do I know the volume of the water displaced if I don't already know the volume of Styrofoam that I need?

Express the forces in terms of the volume. Then you can solve for it.

 

[HadManySons]

Well I finally figured it out. The equation ended up being:

9.8 m/s² * 1000 kg/m³ * (Volume_Styrofoam + 5.67E-6_{The volume of platinum}) = 1.3426 N_{weight of platinum} + (9.8m/s^s * 95kg/m³_{density of Styrofoam} * Volume_Styrofoam)

Which is then:
9800V + .0559421= 1.3426 + 931V
8869V = 1.2870058
V = 1.45E-4 m³ of Styrofoam.

 

[Doc Al]

Good. Making use of the results from the first part (the tension), you could use:
ρ_wVg = ρ_sVg + T

Solving for V gives you the same result.

 

[HadManySons]

Good. Making use of the results from the first part (the tension), you could use:
ρ_wVg = ρ_sVg + T

Solving for V gives you the same result.

I had wondered how to go about that, since you had mentioned it earlier. Thanks for pointing that out.