Pressure Variance With Depth

[Ethan Godden]

It should be noted that this is not homework but rather practice for an exam. Is this the place I should be asking these types of questions?

Homework Statement

An apple is held completely submerged just below the surface of the water in a container. The apple is then moved to a deeper point in the water. Compared with the force needed to hold the apple just below the surface, what is the force needed to hold it at a deeper point? (a) larger (b) the same (c) smaller (d) impossible to determine

Homework Equations

P=P_o+ρgh

The Attempt at a Solution

I thought since we know pressure increases with depth that the answer would be (a), but apparently it is (b). I do realize it is in a container, but shouldn't the pressure inside the container vary with depth?

A good explanation would be greatly appreciated.

 

[haruspex]

It should be noted that this is not homework but rather practice for an exam. Is this the place I should be asking these types of questions?

Homework Statement

An apple is held completely submerged just below the surface of the water in a container. The apple is then moved to a deeper point in the water. Compared with the force needed to hold the apple just below the surface, what is the force needed to hold it at a deeper point? (a) larger (b) the same (c) smaller (d) impossible to determine

Homework Equations

P=P_o+ρgh

The Attempt at a Solution

I thought since we know pressure increases with depth that the answer would be (a), but apparently it is (b). I do realize it is in a container, but shouldn't the pressure inside the container vary with depth?

A good explanation would be greatly appreciated.

Once the object is fully submerged, pressure acts all around it. It acts more at the bottom than at the top because the pressure is greater there, and this provides the buoyancy. But if you increase the pressure, e.g. by taking the object to greater depth, you increase the pressure equally all round it, so there is no net change to the buoyancy.
Stick to Archimedes' principle. What does that tell you here?

But it is not clear from the question whether we should consider the apple and water as incompressible. Or if they are compressible, which is more readily compressed. How would that change your answer?

 

[Chestermiller]

Archimedes law, which is derived based on integrating the pressure force distribution over the surface of a submerged object, indicates that the buoyant force on the object is equal to the weight of the displaced volume of fluid. This doesn't change with depth.

 

[Ethan Godden]

Thank you, I think I understand now.

Archimedes principle tells us that the magnitude of the buoyant force on an object always equals the weight of the fluid displaced. I think the reason for this is even though the pressure on the top of the container and apple is less than the pressure at the bottom or the container and apple, they both increase by the same amount as the depth increases.

I would assume both are incompressible as that's what most other questions in my textbook assume. If they were compressible, however, the total volume displaced would shrink, and by Archimedes principle, the buoyant force would decrease meaning the force required to hold the container would decrease. Am I correct in my thinking?

 

[Chestermiller]

Not exactly. If the liquid got compressed, more mass of liquid could fit into the displaced space, and this would tend to increase the buoyant force. So you would have to determine whether the shrinking in volume of the object wins out over the increase in density of the liquid.

Chet