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

AI Thread Summary
The discussion centers on deriving an equation for the volume of a floating object partially submerged in a fluid. The key points include the relationship between buoyant force and the weight of the object, where the buoyant force (Fb) equals the weight of the displaced fluid (mass of fluid displaced times gravity). It is established that for the object to float at rest, the buoyant force must equal the object's weight (mg). The conversation also highlights the equation relating the densities and volumes of the object and the fluid, leading to the conclusion that the mass of the displaced fluid equals the mass of the object. Ultimately, the participants are working towards a formula that connects these variables to find the volume of the object above the fluid.
Hairy Biped
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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.
 
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would the buoyant force equal the downward force? ie, would Fb=mg?
 
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Hairy Biped said:

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

Yes, Fb would have to equal mg if the ball is at rest (i.e. not accelerating). And, as you said, Fb is g times the mass of the displaced fluid.
 
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)?
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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