Buoyancy

[Oorham]

Hi, Could someone please explain why the answer to the following question is B.

An experiment is performed with a brick and a bucket of water. In which of the following cases does the brick displace the most water?

A. The brick as allowed to sink to the bottom of a bucket.
B. The brick is floated on a piece of massless styrofoam.
C. The brick is held just beneath the surface of the water.
D. In all cases the same amount of water is displaced.

I know that the bouyant force is equal to the weight of the displaced fluid, and that when an object is submerged the volume of the fluid displaced is equal to the volume of the object.
On the other hand, for an object thats floating the volume displaced is less than the volume of the object.

So, shouldn't the volume displaced be greater in case of an object that's submerged?

Thank you

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[PlsLetMeIn21]

The weight of the brick never changes, and the buoyant force is equal to the weight of the brick. So no matter what, the buoyant force (density of the brick x volume of water displaced x g) is the same. The density of the brick and gravity do not change, so the displaced volume of water will be the same whether it sinks or whether it is held below the surface. Choices A and C are both the same amount of displacement. When it sits on another floating object that is less dense, then that other object must displace more water to offset the weight of the brick than the brick does by itself. So case B is displacing more volume.

Where this question is from, because from what I understand and have seen in my study books and AAMC, they do not have choices that say "all of the above." They have questions with Roman numerals that have that as an option, but never as an individual answer choice. For me personally, I am only doing the most realistic questions I can find.

edited after reading AdeptPrep's great answer!

 

[AdaptPrep]

Hi, Oorham--

I think the answer is "B". For option "A" and "C", the buoyant force is less than the weight force (hence it sinks as in case A or is "held" in case C). In case A, there are three forces: buoyant force up, weight force down, and normal force up from the bucket. In case C, there are also three forces: buoyant force up, weight force down, and arm force up (or whatever "held" force is). However, in case B, there are only two forces: buoyant force up and weight force down (and they are equal). Only in B are the buoyant force and weight force equal. What happened? The massless Styrofoam in essence decreased the density of the brick. If the brick weighs 1 kg, then 1 kg of Styrofoam needs to be shoved into the water. Well, that will take more volume of water since it is less dense (it is massless, in fact). Regardless, for the buoyant force to equal the weight force, more water had to be displaced in case B.

When you asked, "On the other hand, for an object that's floating the volume displaced is less than the volume of the object." That is true, but this has a brick pushing down on a less-than-water-density Styrofoam. Put a truck on an iceberg, and the iceberg will go downward and displace a mass of water equal to the mass of the truck (but the iceberg is less dense than water). Does that make sense?

I hope that helps.

 

[PlsLetMeIn21]

Adaptprep, I must tip my cap. That was pretty awesome.

I looked at this again using some numbers and it helped. Let's just say the density of Styrofoam is 0.5 g/ml, water is 1.0 g/ml, and the brick is 2.0 g/ml. If the Styrofoam has a volume of 10 liters, then it will displace a volume of water equal to 5 liters. Let's say the brick is 1.0 liters. The brick-Styrofoam system has a density of 7/11 g/ml. In order for buoyant force to equal the weight, the system must displace 7 liters (as opposed to 5 like before).

weight = buoyant force
(5 kg Styrofoam + 2 kg brick) g = 1.0 kg/L x Vdisplaced x g
7 x g = Vdisplaced x g
7 = Vdisplaced

Adding the brick to the Styrofoam resulted in 2.0 L of displaced water while putting it in the water directly was only 1.0 L of displacement.

 

[Oorham]

Hi, Oorham--

I think the answer is "B". For option "A" and "C", the buoyant force is less than the weight force (hence it sinks as in case A or is "held" in case C). In case A, there are three forces: buoyant force up, weight force down, and normal force up from the bucket. In case C, there are also three forces: buoyant force up, weight force down, and arm force up (or whatever "held" force is). However, in case B, there are only two forces: buoyant force up and weight force down (and they are equal). Only in B are the buoyant force and weight force equal. What happened? The massless Styrofoam in essence decreased the density of the brick. If the brick weighs 1 kg, then 1 kg of Styrofoam needs to be shoved into the water. Well, that will take more volume of water since it is less dense (it is massless, in fact). Regardless, for the buoyant force to equal the weight force, more water had to be displaced in case B.

When you asked, "On the other hand, for an object that's floating the volume displaced is less than the volume of the object." That is true, but this has a brick pushing down on a less-than-water-density Styrofoam. Put a truck on an iceberg, and the iceberg will go downward and displace a mass of water equal to the mass of the truck (but the iceberg is less dense than water). Does that make sense?

I hope that helps.

thank you for answering my question. I've read it a few times but I think I have to read it a few times more to grasp the concept. The only thing that I can think of now is that when the object is at the bottom of the bucket, it weighs less (<mg) because of the buoyant force but I don't know if this has anything to do with the volume of the water displaced. Since the idea is that for an object that is submerged the volume of the fluid displaced is equal to the volume of the object, not its weight.

On the other hand, when it's floating on a piece of massless styrofoam, its wight won't be affected and it would be equal to mg. Here I know that when an object is floating it's displacing a volume of water equal to its weight.

 

[Oorham]

Adaptprep, I must tip my cap. That was pretty awesome.

I looked at this again using some numbers and it helped. Let's just say the density of Styrofoam is 0.5 g/ml, water is 1.0 g/ml, and the brick is 2.0 g/ml. If the Styrofoam has a volume of 10 liters, then it will displace a volume of water equal to 5 liters. Let's say the brick is 1.0 liters. The brick-Styrofoam system has a density of 7/11 g/ml. In order for buoyant force to equal the weight, the system must displace 7 liters (as opposed to 5 like before).

weight = buoyant force
(5 kg Styrofoam + 2 kg brick) g = 1.0 kg/L x Vdisplaced x g
7 x g = Vdisplaced x g
7 = Vdisplaced

Adding the brick to the Styrofoam resulted in 2.0 L of displaced water while putting it in the water directly was only 1.0 L of displacement.

This is from the old EK 1001.

Regarding to your explanation, it says the styrofoam is massless. So I'm not sure if we could add their densities together and end up with more volume of water being displaced.

 

[AdaptPrep]

Hi, Oorham--

When you stated, "Since the idea is that for an object that is submerged the volume of the fluid displaced is equal to the volume of the object, not its weight" and "On the other hand, when it's floating on a piece of massless Styrofoam, its weight won't be affected and it would be equal to mg. Here I know that when an object is floating it's displacing a volume of water equal to its weight" you just hit the main two points and answered it. The submerged brick displaces a volume of water equal to the volume of the brick. Whereas the floating brick will displace a volume of water with the same mass as the mass of the brick (the Styrofoam would sink downward...it is massless, but it has volume). Which is more water? You are correct in saying the floating one since the weight of the brick does not change.