Should I Use the Total Length of an Object to Calculate Buoyancy?

[WherE mE weeD]

this is a float submerged in fluid the upthrust is greater than the weight. P=ρgh to calculate the pressure should I use the total length of this object as h?

Homework Equations

P=ρgh
Fb=PA (A=area)

The Attempt at a Solution

Curious to know whether I should use the total length of this object in the above equation I'm guessing yes but not too sure.

Which surface area should I input into the buoyancy force equation. The object being less dense than the fluid its submerged in is the reason for the positive buoyancy which makes me think it could be the whole surface area of the object. I'm also thinking the fluid could be compressing the object upwards from the bottom surface because of the difference in densities.

Any advice is appreciated.

 

[Daniel Gallimore]

View attachment 199434
this is a float submerged in fluid the upthrust is greater than the weight. P=ρgh to calculate the pressure should I use the total length of this object as h?

Homework Equations

P=ρgh
Fb=PA (A=area)

The Attempt at a Solution

Curious to know whether I should use the total length of this object in the above equation I'm guessing yes but not too sure.

Which surface area should I input into the buoyancy force equation. The object being less dense than the fluid its submerged in is the reason for the positive buoyancy which makes me think it could be the whole surface area of the object. I'm also thinking the fluid could be compressing the object upwards from the bottom surface because of the difference in densities.

Any advice is appreciated.

A simpler method for finding the buoyant force is to use the formula F_B=\rho Vg, where V is the volume submerged and \rho is the density of the fluid.

 

[WherE mE weeD]

A simpler method for finding the buoyant force is to use the formula F_B=\rho Vg, where V is the volume submerged and \rho is the density of the fluid.

Cheers for the reply Daniel.
This formula gives me the same result as when I calculate weight using mass x gravity (W=mg)

buoyant force = weight on object = upward thrust = weight of displaced fluid

I thought this would mean the object would be in a neutral buoyancy?

My idea to calculate the upward thrust was buoyant force - weight = resulting upward thrust force?

 

[haruspex]

buoyant force = weight on object = upward thrust = weight of displaced fluid

Writing that assumes neutral buoyancy. Without that assumption you can only write
buoyant force = upward thrust = weight of displaced fluid
which is what Daniel posted,

The reason your first approach had problems is that the pressure varies along the height of the object, and does not everywhere act vertically on it. You can get the answer by suitable integration, but Archimedes' principle allows you to circumvent that.

 

[WherE mE weeD]

Writing that assumes neutral buoyancy. Without that assumption you can only write
buoyant force = upward thrust = weight of displaced fluid
which is what Daniel posted,

The reason your first approach had problems is that the pressure varies along the height of the object, and does not everywhere act vertically on it. You can get the answer by suitable integration, but Archimedes' principle allows you to circumvent that.

I understand what you mean @haruspex with the pressure changing along the height of the object because of the shape of the cone.

1. So using the Archimedes principle "buoyant force = upward thrust = weight of displaced fluid" and "volume of water displaced by the float = volume of float" I can use m=ρxv (mass=rhoxvolume) and W=mg (weight=massxgravity) to work out the buoyancy force. knowing that the buoyancy force and weight are equal then the buoyancy would be neutral?

 

[haruspex]

So using the Archimedes principle "buoyant force = upward thrust = weight of displaced fluid" and "volume of water displaced by the float = volume of float" I can use m=ρxv (mass=rhoxvolume) and W=mg (weight=massxgravity) to work out the buoyancy force

Yes.

knowing that the buoyancy force and weight are equal then the buoyancy would be neutral?

If they are equal then the object will float. If they are equal and the object is fully submerged then buoyancy is neutral. Post #1 implies neither of those is true. It says the upthrust exceeds the weight of the object.

 

[WherE mE weeD]

Yes.

If they are equal then the object will float. If they are equal and the object is fully submerged then buoyancy is neutral. Post #1 implies neither of those is true. It says the upthrust exceeds the weight of the object.

"upthrust exceeds the weight" That was my interpretation of if a float rises in a fluid the upthrust force must be greater than the weight. But using Archimedes principle the weight is equal to the buoyancy force. which links into D'Alemberts principle of sum of forces =0. I have calculated the weight and buoyancy force both are equal the reason for the float to rise is the difference in density of the fluid to the float (float having lower density) Iam asked to calculate the upthrust of the float which would be the buoyancy force? for some reason I had it in my head the upthrust force would be greater than the weight and this would cause the acceleration upwards.

 

[haruspex]

upthrust exceeds the weight" That was my interpretation of if a float rises in a fluid the upthrust force must be greater than the weight.

Right.

But using Archimedes principle the weight is equal to the buoyancy force

Archimedes' principle says. The upthrust (buoyancy force) equals the weight of the fluid displaced. It does not say it equals the weight of the object.

 

[WherE mE weeD]

Right.

Archimedes' principle says. The upthrust (buoyancy force) equals the weight of the fluid displaced. It does not say it equals the weight of the object.

My notes use an example of a ship and say the mass of the ship = the mass of the fluid displaced by the ship. Its in there a couple of times mass of body = mass of fluids displaced.

So i need to find the objects mass to calculate the weight. I have fluid density (ρ), gravity(g) area and volume of body. mg=ρgha or mg=ρgv my problem is this calculates the weight of the fluid not the body?

 

[haruspex]

My notes use an example of a ship and say the mass of the ship = the mass of the fluid displaced by the ship.

That is an example of a floating object. As I wrote in post #6, that means the upthrust does equal the object's weight. But not all buoyancy problems involve floating objects. The one in this thread is not.

So i need to find the objects mass to calculate the weight.

If you need the object's weight, yes, but...

I have fluid density (ρ), gravity(g) area and volume of body.

... that is enough to find the upthrust. The weight of the object does not matter.

Here's how it works:
- The volume of fluid displaced, the density of the fluid and g determine the buoyant force** (=upthrust). Use Archimedes' principle for this.
- The buoyant force may be less than, greater than, or equal to the weight of the object.
- If the buoyant force is less than the weight of the object, and no other forces act on the object, then the net force on the object is down, so it sinks until it hits the bottom; when it hits the bottom, the normal force from that will create equilibrium.
- If the buoyant force exceeds the weight of the object, and no other forces act on the object, then the net force is up, so the object will rise until it either hits a barrier or breaks surface; if it breaks surface the displaced volume reduces, so the buoyant force reduces; these will continue to reduce until the buoyant force equals the weight of the object, as in the floating ship.

[**There is a special case, which I hesitate to mention for fear of confusing you. Archimedes' principle only applies if the fluid can reach all around the underside of the object. A suction cup can stay at the bottom of a water tank despite being lighter than the displaced water because the water cannot apply pressure to the underside.]

 

[WherE mE weeD]

That is an example of a floating object. As I wrote in post #6, that means the upthrust does equal the object's weight. But not all buoyancy problems involve floating objects. The one in this thread is not.

If you need the object's weight, yes, but...

... that is enough to find the upthrust. The weight of the object does not matter.

Here's how it works:
- The volume of fluid displaced, the density of the fluid and g determine the buoyant force** (=upthrust). Use Archimedes' principle for this.
- The buoyant force may be less than, greater than, or equal to the weight of the object.
- If the buoyant force is less than the weight of the object, and no other forces act on the object, then the net force on the object is down, so it sinks until it hits the bottom; when it hits the bottom, the normal force from that will create equilibrium.
- If the buoyant force exceeds the weight of the object, and no other forces act on the object, then the net force is up, so the object will rise until it either hits a barrier or breaks surface; if it breaks surface the displaced volume reduces, so the buoyant force reduces; these will continue to reduce until the buoyant force equals the weight of the object, as in the floating ship.

[**There is a special case, which I hesitate to mention for fear of confusing you. Archimedes' principle only applies if the fluid can reach all around the underside of the object. A suction cup can stay at the bottom of a water tank despite being lighter than the displaced water because the water cannot apply pressure to the underside.]

In the situation for my object in water the buoyancy force exceeds the weight of the object. thus the net force is up. and once the object breaks the surface the displaced volume reduces until neutral buoyancy is reached. Thanks I understand much more clearly.

As I do not have the objects mass I cannot work out the force net but this is not asked in the question, It feels like I am only going half way to find the force buoyancy and weight of water displaced when I want to complete the problem by calculating the resultant force but as the question does not ask for this its not something I should worry about. Earlier It was a backwards and forwards puzzle trying to figure out why I couldn't calculate the objects mass and I managed to get confused between weight of the water displaced with weight of the object in using (Weight of fluid displaced=mass of fluid displaced/gravity) leading me to wonder why my force buoyancy didn't exceed the weight. All is cleared now cheers.