Buoyancy problem: Deriving an equation for volume of a floating object

[Hairy Biped]

Homework Statement

The problem consists of a ball almost completely submerged. The exact question is:
"An object with density ρO and mass m is floating in a pool of fluid with density ρF. Derive an expression for the volume of the object that is above the fluid."

Homework Equations

Density of object=mass/volume
Density of fluid= mass fluid/volume fluid

The Attempt at a Solution

I have no idea where to begin.
I know this looks like I'm just fishing for a free answer, but I am not. I tried setting the buoyant force to the (mass of the fluid displaced) * (g) but I don't know how to relate this to the other variables. Any help is greatly appreciated, thanks.

 

[Hairy Biped]

would the buoyant force equal the downward force? ie, would Fb=mg?

 

[Redbelly98]

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Welcome to Physics Forums

Homework Statement

The problem consists of a ball almost completely submerged. The exact question is:
"An object with density ρO and mass m is floating in a pool of fluid with density ρF. Derive an expression for the volume of the object that is above the fluid."

Homework Equations

Density of object=mass/volume
Density of fluid= mass fluid/volume fluid

The Attempt at a Solution

I have no idea where to begin.
I know this looks like I'm just fishing for a free answer, but I am not. I tried setting the buoyant force to the (mass of the fluid displaced) * (g) but I don't know how to relate this to the other variables. Any help is greatly appreciated, thanks.

would the buoyant force equal the downward force? ie, would Fb=mg?

Yes, F_b would have to equal mg if the ball is at rest (i.e. not accelerating). And, as you said, F_b is g times the mass of the displaced fluid.

 

[Hairy Biped]

so if bouyant force equals mg, then the mass of the displaced fluid equals the mass of the ball? If so then (density water)(Volume displaced)=(density ball)(volume ball)?

 

[rwisz]

To solve this problem, we can use the concept of buoyancy and the relationship between the densities of the object and fluid. First, let's define the variables:

ρO = density of the object
m = mass of the object
ρF = density of the fluid
V = volume of the object above the fluid (what we are trying to find)

We can start by writing the equation for buoyancy:

Buoyant force = Weight of the fluid displaced

The buoyant force is equal to the weight of the fluid displaced because the object is in equilibrium, meaning that the upward force of buoyancy is equal to the downward force of gravity on the object.

We can also write the weight of the fluid displaced as the product of its mass and acceleration due to gravity (g). The mass of the fluid displaced is equal to the volume of the object above the fluid (V) multiplied by the density of the fluid (ρF):

Weight of fluid displaced = ρF * V * g

Now, the buoyant force can also be expressed as the difference between the weight of the object and the weight of the fluid displaced:

Buoyant force = Weight of object - Weight of fluid displaced

The weight of the object can be written as the product of its mass and acceleration due to gravity (g). The mass of the object is equal to the volume of the object above the fluid (V) multiplied by the density of the object (ρO):

Weight of object = ρO * V * g

Substituting these equations into the original equation for buoyancy, we get:

ρO * V * g = ρO * V * g - ρF * V * g

Simplifying, we get:

ρF * V * g = 0

This equation tells us that the volume of the object above the fluid must be equal to zero for the object to be in equilibrium. However, we know that the object is not completely submerged, so the volume above the fluid cannot be zero. This means that the only way for this equation to hold true is if the buoyant force is also equal to zero.

To find the volume of the object above the fluid, we can set the buoyant force equal to zero and solve for V:

ρF * V * g = 0
V = 0 / (ρF * g)
V = 0

This tells us that the volume of the object above the