Archimedes Principle Problem: Floating Object

[AMacias2008]

Homework Statement

A cylindrical log of uniform density and radius R=30.0cm floats so that the vertical distance from the water line to the top of the log is d = 12.0cm. What is the density of the log?

Homework Equations

F_bouyant=W_{waterdisplaced}

ρ_water * V _{displaced water} = ρ_log * V_log

The Attempt at a Solution

I started off by making a FBD. Buoyant force going up, mg coming down. No acceleration so I ended up getting to the second equation up above.

Solving for ρ_log, I get

ρ_log = ρ_water * (V_{displacedwater}/V_log) where V_{displacedwater} = ∏R²

I'm having trouble figuring out the volume of the displaced water. Since only part of the log is submerged, would the volume of the displaced water be the volume of the log that is submerged? And if so, how I would go about determining the ratio of the volumes?

I tried several things already to no avail. I tried taking the ratio of the radius of the log to the distance to the top of level (.12/.30) and used the answer of .4 to use into the R of V_{displacewater} but didn't get the answer.

I tried switching it and still got nothing. Some guidance in the right direction would be greatly appreciated. The answer is 858kg/m³.

 

[SteamKing]

I'm having trouble figuring out the volume of the displaced water. Since only part of the log is submerged, would the volume of the displaced water be the volume of the log that is submerged?

Obviously, yes.

And if so, how I would go about determining the ratio of the volumes?

You must use some subtle tricks called geometry to calculate the volume of displacement of the log. It always helps to draw a picture of the problem first.

http://en.wikipedia.org/wiki/Circular_segment

I assume you had no problem calculating the total area of the cross section of the log?

 

[AMacias2008]

You must use some subtle tricks called geometry to calculate the volume of displacement of the log. It always helps to draw a picture of the problem first.

http://en.wikipedia.org/wiki/Circular_segment

I assume you had no problem calculating the total area of the cross section of the log?

Thank you for your response SteamKing. Yeah I did.

Ah I see. Geometry is a cool little I should use more haha. Thank you for the help and I used that equation to find the area, took the answer minus the entire cross sectional area, did some more work with another helpful tool called algebra and got the correct answer. Thank you for the help and sorry for the silly question.