How would you answer this question

[anondukie]

“Balls A and B of equal mass are floating in a swimming pool, as shown below. Which will produce a greater buoyant force?




A). Ball A
B). Ball B
C). The forces will be equal.
D). It is impossible to know without knowing the volume of each ball.”

Excerpt From: Kaplan. “Kaplan MCAT Physics and Math Review: Created for MCAT 2015 (Kaplan Test Prep).” iBooks.

I would argue that D is the correct answer because the densities have not been specified and it is possible that one ball has a much greater density which would result in a greater percentage of its volume being submerged. Therefore, the ball with less volume overall could still have more volume submerged and this would result in a stronger buoyant force because the buoyant force depends on the volume of water displaced not the overall volume of the floating object. However, the answer provided makes the assumption that densities are equal without explicitly saying so. Is that fair?

Thanks!

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[anondukie]

^ Corrected!

 

[aldol16]

I had the same problem with this question when I did it. I don't think their explanation makes too much sense. An object will displace its weight in water. So without knowing the density of the object, it's not possible to know how much water the object will displace and thus impossible to determine the buoyant force. Now if both of those balls were completely submerged (without a force holding them underwater), then A would have a larger buoyant force because it displaced more water.

Edit: I take this back. This applies only if the masses of the two objects are unknown/not equal. See my answer below.

 

[WanderingGuitarist]

Was the answer A?

 

[aldol16]

Actually, I take back what I said above. I didn't notice that the masses were equal. In this case, since the balls are floating, the only two vertical forces acting on the ball are the buoyant force and the gravitational force. Since the ball isn't accelerating, the forces balance and since the balls have the same mass, they must have the same buoyant force regardless of density. (Note that I do not agree with the Kaplan answer on this).

 

[aldol16]

Was the answer A?

According to Kaplan, yeah. But I don't think it's right (see my latest post above).

 

[WanderingGuitarist]

So this is how I arrived at the answer.

The masses are the same. Therefore, the weight of the objects would be the same.

So, how do you know which is gonna have the greater bouyant force?

It is related to the area of the object.

Imagine you have 2 ships which weigh the same but one has a much larger area than the other.

Which one is gonna sink?

The one that is gonna stay afloat is the one with the greater area because this allows more of the water to push up against the ship.

This problem is more of a conceptual problem than a formulaic one.

Maybe this picture will help also: http://images.slideplayer.com/10/2743629/slides/slide_31.jpg

 

[aldol16]

So this is how I arrived at the answer.

The masses are the same. Therefore, the weight of the objects would be the same.

So, how do you know which is gonna have the greater bouyant force?

It is related to the area of the object.

Imagine you have 2 ships which weigh the same but one has a much larger area than the other.

Which one is gonna sink?

The one that is gonna stay afloat is the one with the greater area because this allows more of the water to push up against the ship.

This problem is more of a conceptual problem than a formulaic one.

What you're talking about is which ship will be immersed deeper into the water. More area = more displaced water per unit height. The buoyant force of two ships with the same mass must be the same because buoyant force depends only on an object displacing its weight of water.

What you're talking about is how far it will immerse into the fluid. So comparing two things with equal mass that have different cross-sectional areas would result in the thing with larger cross-sectional area immersing less of its height into water than the other object. This is because, obviously, both objects will displace some weight of water equal to their own weight. This will be equal to some fixed volume of water. And since both objects weigh the same, they will displace the same mass of water. So then all you have to do is imagine how to find the volume of a given rectangular prism versus another rectangular prism of smaller cross-sectional area, as it depends on height. Given a volume of displacement, the rectangular prism of greater cross-sectional area would have a smaller "height" of displacement.

 

[theonlytycrane]

@aldol16

quick clarification question: since they have the same buoyant force, ball A should be submerged less in the pool than ball B such that they both displace equal amounts of water. Correct?

 

[aldol16]

quick clarification question: since they have the same buoyant force, ball A should be submerged less in the pool than ball B such that they both displace equal amounts of water. Correct?

Yes, that is correct, assuming the figure is drawn to scale.

 

[anondukie]

Aldol, I like your answer! The objects are in translational equilibrium. Therefore, buoyant force must equal weight (which is the same in both cases). I'm mad that I missed that! Thanks!

 

[aldol16]

Aldol, I like your answer! The objects are in translational equilibrium. Therefore, buoyant force must equal weight (which is the same in both cases). I'm mad that I missed that! Thanks!

Yes - this will always be true for something in which only the buoyant force and gravity are acting on the object. The object will displace its weight in water (since rho(water)*V(H20)*g = mass(H20)/V(H20)*V(H20)*g = mass(object)*g; thus, mass(object) = mass(H20) during translational equilibrium).