Finding a buoyant force on an object

[SevenSeeds5]

Homework Statement

You are given a density of a fluid, volume of the rubber stopper and density of the rubber stopper. If a rubber stopper sinks to the bottom of the beaker filled with this fluid and stays there, find the buoyant exerted on the object by the fluid.

Homework Equations

Force of buoyancy = weight of the rubber stopper (?)

The Attempt at a Solution

I basically tried using the equation pVg and finding the force of buoyancy but I am not sure if that is 100% correct. Since the stopper is resting on the bottom of the beaker, maybe I need to take the normal force into consideration? Any tips would be great

 

[gneill]

Hi SevenSeeds5, Welcome to Physics Forums.

Buoyancy force is equal to the weight of the object only when the object floats at equilibrium (no external forces other than gravity are acting on the object, and it is just floating freely). After all, that's what allows an object to float: the buoyant force balances the force due to gravity.

Your thought of using the equation ρVg is good. The buoyant force for an object depends only on the weight of the volume of fluid displaced. If the object is fully submerged, then the volume of fluid displaced is equal to the volume of the object.

 

[SevenSeeds5]

What if the object is touching the bottom of the beaker? I am aware of archimedes' principle and such but I wasn't unsure how to understand the problem when the object is touching the bottom of the beaker. Does this simply mean that the gravitational force is much greater than the bouyant force? I was wondering if I should consider the normal force exerted by the bottom of the beaker also.

 

[gneill]

What if the object is touching the bottom of the beaker? I am aware of archimedes' principle and such but I wasn't unsure how to understand the problem when the object is touching the bottom of the beaker. Does this simply mean that the gravitational force is much greater than the bouyant force? I was wondering if I should consider the normal force exerted by the bottom of the beaker also.

The buoyant force is separate from other external forces. If the object sinks then its weight due to gravity exceeds the buoyant force. A neutrally buoyant object will neither rise nor sink because the two forces balance. If the object rises (or floats) then the buoyant force exceeds (or equals) its gravitational weight.

An object sitting on the bottom just means that its gravitational weight exceeds the buoyancy force. But the buoyancy force is STILL just equal to the weight of the fluid its volume displaces.

 

[SevenSeeds5]

Now I finally get it! So buoyant force will always equal the weight of the object itself when the object is completely submerged underwater?

 

[gneill]

Now I finally get it! So buoyant force will always equal the weight of the object itself when the object is completely submerged underwater?

No, when the object is free-floating then the buoyant force will equal the weight of the object. It's floating because the forces are in equilibrium.

When the object is submerged, the buoyant force will equal the weight of the volume of fluid displaced. The actual weight of the object is then irrelevant since only its volume determines the amount of fluid displaced.

 

[SevenSeeds5]

Oh that makes a lot of sense now.. So I should differentiate the calculations between the completely submerged object and an object in an equilibrium..

 

[haruspex]

Oh that makes a lot of sense now.. So I should differentiate the calculations between the completely submerged object and an object in an equilibrium..

Not quite. The distinction is between an object that is floating, i.e. it is in equlibrium while the only forces on it are gravity and buoyancy; one which has other forces acting on it (e.g. resting on the bottom); and one which is not in equilibrium. But the last two are not mutually exclusive. In principle, an object may be completely submerged yet floating.

 

[SevenSeeds5]

T.T Buoyancy is so complicated...
For the one that's resting on the bottom, other forces acting on it would be normal force exerted by the beaker then?

 

[HallsofIvy]

For a completely submerged object, the buoyancy force is equal to the weight of a volume of water equal to the volume of the object. That will, of course, be less than the actual weight of the object itself. The weight of the object minus the buoyancy force (the weight of an equivalent volume of water) is the "normal force" exerted by the bottom of the beaker.

 

[SevenSeeds5]

Ohhh... So instead of using the weight of the object, I should be using the volume of water instead..

So the equation would be buoyant force = (density of fluid)(volume of object)(g)