Understanding Archimedes' Principle: Solving for Density in a Buoyancy Problem

[brake4country]

Homework Statement

If an object is 500 N normally but has an apparent weight of 300 N when submerged in water, what is the density of the object?

Homework Equations

ρ_objV_objg = ρ_fluidV_objg

The Attempt at a Solution

So the correct answer is 2500 kg/m³ but when I set up the problem, it doesn't turn out correct.
ρ_objV_objg = ρ_fluidV_objg
Both volumes and gravity cancel resulting with:
ρ_fluid=ρ_obj
Is there an error in my set up? Thanks in advance!

 

[SteamKing]

Homework Statement

If an object is 500 N normally but has an apparent weight of 300 N when submerged in water, what is the density of the object?

Homework Equations

ρ_objV_objg = ρ_fluidV_objg

The Attempt at a Solution

So the correct answer is 2500 kg/m³ but when I set up the problem, it doesn't turn out correct.
ρ_objV_objg = ρ_fluidV_objg
Both volumes and gravity cancel resulting with:
ρ_fluid=ρ_obj
Is there an error in my set up? Thanks in advance!

You've assumed that the apparent weight is the product of some fictitious density and the volume of the object, neglecting Archimedes' Principle.

Remember that apparent weight of a submerged object = actual weight of the object in air - buoyant force of the water displaced by the object.

 

[brake4country]

Right, 500 N - 200 N = 300 N. Conceptually this makes sense but I am trying to work this out algebraically. Also, just to be clear, you are saying that I cannot use ρ_obj algebraically this way?

 

[SteamKing]

Right, 500 N - 200 N = 300 N. Conceptually this makes sense but I am trying to work this out algebraically. Also, just to be clear, you are saying that I cannot use ρ_obj algebraically this way?

It depends on what ρ_obj means. Is it derived from the mass of the object, as in ρ_obj = m_obj / V_obj ?

It's still not clear why you can't solve this problem algebraically by applying Archimedes' Principle.

 

[brake4country]

Can this even be solved algebraically even with an "apparent weight"? I understand apparent weight but in the formula written above: ρ_objV_objg=ρ_fluidV_objg; if I follow math rules, ρ_obj should be = to ρ_fluid. Perhaps I am assuming something but I cannot see it.

 

[SteamKing]

Can this even be solved algebraically even with an "apparent weight"? I understand apparent weight but in the formula written above: ρ_objV_objg=ρ_fluidV_objg; if I follow math rules, ρ_obj should be = to ρ_fluid. Perhaps I am assuming something but I cannot see it.

Your math is flawless, but it is based on a faulty assumption. Here, let me highlight it for you (again):

You've assumed that the apparent weight is the product of some fictitious density and the volume of the object, neglecting Archimedes' Principle.

As I explained in a previous post, that's not how the apparent weight of the object is defined.

 

[brake4country]

Oh ok. So, if apparent weight cannot be integrated in the algebra, can this be seen conceptually? Additionally, when ρ_obj=ρ_fluid, the object would be seen as submerged but won't sink, yes? I think I am getting the idea now.

 

[SteamKing]

Oh ok. So, if apparent weight cannot be integrated in the algebra, can this be seen conceptually?

This problem can be solved algebraically, just not in the way you are convinced it should be.

By ignoring what Archimedes principle is telling you about what the apparent weight of the object is when submerged, I really can't offer you any further guidance, except:

http://en.wikipedia.org/wiki/Archimedes'_principle

Additionally, when ρ_obj=ρ_fluid, the object would be seen as submerged but won't sink, yes? I think I am getting the idea now.

If the density of the object is the same as the density of the fluid in which it is immersed, the object is said to be neutrally buoyant, which is a fancy way of saying it won't float and it won't sink.

When submarines submerge, they take in just enough water as ballast to become neutrally buoyant. While submerged, the submarine changes depth by using its control planes to bring the bow up or down when the sub is going ahead. To surface, the submerged sub uses compressed air to remove water from the ballast tanks in order to restore positive buoyancy to the vessel, which then rises to the surface and floats naturally.