Does Shape Affect the Tension in Strings Suspended in Water?

[cy19861126]

Homework Statement

Two objects of the same mass and volume but different shape are suspended from strings in a tank of water as shown. Is there more tension force on the string by object A or B? Link to the picture can be found at http://students.washington.edu/cy1126/Buoyancy.JPG

Homework Equations

P = pgh, where p = density
F = PA
P = pghA
B = pgh1A1 - pgh2A2

The Attempt at a Solution

I think that object A is more buoyant than object B because object A has a greater difference of depth than object B therefore contributing to a greater buoyancy. For object B, although the force on top of it is greater due to larger area, the "difference" of area is almost the same as object A. I am not sure about my reasoning here. Can someone check my answer for me as well as giving me some clue if I am getting the wrong concept

 

[berkeman]

Well, at first glance, I'd think that the two bouyancies would be the same, since the density of the water does not change (with this small depth difference), so both objects displace the same mass of water, and hence get the same amount of lift.

Then I through about what makes the lift -- it's the pressure differential on the top surface versus the bottom surface. The side-pressure components don't contribute to the lift. So for two cylinders of different diameters and heights (but same mass), what can you say about the pressures and forces on the tops and bottoms, and what does that tell you about their comparative bouyancies?

And then finally, check your answer against this specific example where the shapes are stepped (cylinders or boxes -- it doesn't matter). Account for all the vertical forces on the various edges, and tell us if the bouyancies are the same or not.

 

[cy19861126]

Well, at first glance, I'd think that the two bouyancies would be the same, since the density of the water does not change (with this small depth difference), so both objects displace the same mass of water, and hence get the same amount of lift.

Then I through about what makes the lift -- it's the pressure differential on the top surface versus the bottom surface. The side-pressure components don't contribute to the lift. So for two cylinders of different diameters and heights (but same mass), what can you say about the pressures and forces on the tops and bottoms, and what does that tell you about their comparative bouyancies?

And then finally, check your answer against this specific example where the shapes are stepped (cylinders or boxes -- it doesn't matter). Account for all the vertical forces on the various edges, and tell us if the bouyancies are the same or not.

oh, I see now, it took me a while to think about it until you told me to draw boxes. So just to verify if my thinking is correct. Although object A has a greater depth difference, the pressure exerted on the vertical surface is less than that of object B because it has a smaller surface area. On the other hand, although there are more pressure exerting on object B because of its large surface area, it has a smaller depth difference. Therefore, essentially, the two buoyancies are the same? Sorry, it is really hard to word this, but I think I get the concept. Can you check if there's any flaw in my thinking? Thanks

 

[cy19861126]

C'mon I need the help~~

 

[berkeman]

I believe that you have the right idea. If it were me, I'd put some numbers in for some real objects -- especially the ones shown in the diagram. Just to convince myself that I was correct...