F = MA 2011 Exam # 8 (Fraction of an object above surface of water)

[SignaturePF]

Homework Statement

8. When a block of wood with a weight of 30 N is completely submerged under water the buoyant force on the block
of wood from the water is 50 N. When the block is released it floats at the surface. What fraction of the block will
then be visible above the surface of the water when the block is floating?
(A) 1/15
(B) 1/5
(C) 1/3
(D) 2/5
(E) 3/5

Homework Equations

F_app = F_g - F_b
For floating objects:
F_b = F_g

The Attempt at a Solution

So, to start off, we plug into the first equation to find the apparent weight underwater:
F_app = 30 - 50
F_app = - 20N
I can intuitively see why the answer is 2/5, but can't find the physics of it.

 

[dx]

If a fraction β of the block is above the water, then what is the buoyant force on the block?

Hint: The buoyant force is proportional the volume of the block that is underwater

 

[SignaturePF]

Did you mean β below the water? If so, then the buoyant force would be β(mg)

 

[dx]

Ok let's take β below water, but that's not correct. For example, you are told that if the block is fully submerged, then the buoyant force is 50N.

Turn the hint I gave before into an equation. If a fraction β is below water, then

Buoyant force = kβ

Now you know that the buoyant force is 50N when β = 1. Put these into the equation to find k.

 

[SignaturePF]

Ok, so:
F_b = F_g
β(50) = 30
β = 3/5
so 3/5 is below the water; this means 2/5 is above.
Is this reasoning sound?

 

[dx]

Yep, that's correct.

 

[SignaturePF]

Thanks, so in general on these types of problems, keep in mind that:
The buoyancy force is directly proportional to the fraction below the water, given it is of uniform density.
Right?

 

[dx]

The block doesn't have to be of uniform density though. The buoyant force is simply the weight of the displaced water, which is proportional to the volume of the block below water.