Need Help finding radius of ball using bouyancy

[Roger Wilco]

Homework Statement

An iron ball is suspended by a thread of negligible mass from an upright cylinder that floats partially submerged in water. The cylinder has a height of 6.00cm, a face area A=12.0 cm^2 on the top and bottom, and a density $$\rho_c=.300\frac{g}{cm^3}$$, and 2.00 cm of its height is above the water. What is the radius of the ball?

Homework Equations

$$\sum F=0$$ $$F_b=\rho*V*g$$
Using Newton's Second and Archimedes' Principle I have used the following method. My concern comes at the point that I get the expression (V'-V)<-- this will yeild a NEGATIVE quantity and r cannot = negative. My problem is that the V'=volume of water displaced and V= volume of the cylinder. I think I need to assume that the CYLINDER IS HOLLOW in order to get a positive quantity. But whay is the volume of a hollow cylinder if I am not given an inner and outer radius??

The Attempt at a Solution

Subscript c is cylinder, b is the ball, and V' is the portion of the cylinder under water.

$$\sum F=0$$

$$\Rightarrow W_c+W_b-F_{buoyant}=0$$

$$\Rightarrow m_cg+m_bg-\rho_cV_c'g=0$$

$$\Rightarrow \rho_cV_c+\rho_bV_b=\rho_cV_c'$$

$$\Rightarrow V_b=\frac{\rho_c(V_c'-V_c)}{\rho_b}$$

I don't find it necessary to move any further than this last step as finding the r is easy enough from there. However it is in this last step that you can see that if I use
V=height*cross-sectional area...I will get a negative number for V'-V.

What should I be using for V? Shoud it be zero? I think that is a bold assumption, or is it?

Thank you,
RW

Edit: After looking at my diagram, I have encountered another problem: Do I need to consider the buoyant force on the ball, too?

I do not not see why I wouldn't.

Okay. I also noticed that for F_bouyant I should have used rho_water NOT of the cylinder.

Guess I need to re-work this. :( So I guess in re-working this my question is still: do I need to consider the buoyant force on the ball, too?

 

[Kurdt]

I think the problem is that you're multiplying V'_c by the density of the cylinder and not the density of water which is what you need to do for the buoyant force.

 

[Roger Wilco]

I think the problem is that you're multiplying V'_c by the density of the cylinder and not the density of water which is what you need to do for the buoyant force.

Yeah Kurdt, I just caught that. But tell me, should I be taking into account the buoyant Force of the ball?

See "Edits" in post #1.

 

[Kurdt]

Yeah Kurdt, I just caught that. But tell me, should I be taking into account the buoyant Force of the ball?

See "Edits" in post #1.

Thats a good thought, and yes I would do that. Its not that much harder.