AAMC Archimede's Principle problem (MCAT 2015 Sample test)

[neuropanic]

Can someone please explain the Archimedes problem on the AAMC sample test?
The one asking about the density of a human body calculated by the weight in air and its weight submerged in water?

Answer was: Weight in air / (Weight in air - Weight in water)
I just really do not understand their explanation / the derivation for this.

Thanks

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[NextStepTutor_3]

I don't have the test open next to me, but am familiar with the question! Hope this helps:

First, "weight in air - weight in water" is just another way to express the buoyant force. Imagine a force diagram where a body is submerged in a fluid. "mg" points downward, while buoyant force points upward, canceling out some of the person's original weight. Buoyant force, then, IS the difference between a person's weight in air and water (I can draw this if you're still confused). On a similar note, I've seen other questions refer to buoyant force as "apparent loss of weight" - as in, you feel lighter in water because that force is pushing upwards.

So, we can simplify this express to "weight in air" / "buoyant force." Well, what is buoyant force? It's simply the weight of the fluid displaced by an object. So, we can further express this quantity as "weight of an object in air" / "weight of the same volume of water."

mg (human) / mg (water) = m(human) / m(water) (since g, of course, is always constant)

Furthermore, since the volume of the human body and the volume of water displaced are the same (which is always true for an object that is completely submerged - after all, it could never displace a volume different from its own), the ratio of their masses is the same as the ratio of their densities. Here's the mathematical way to look at this:

m(human) / m(water) => [m (human) / volume] / [m (water) / volume], because volume cancels out. (m/V) gives us density, so this expression is equal to the ratio of the density of a human body to the density of water (its specific gravity).

Let me know if anything is still confusing, fluids can be tricky! And good luck!

 

[neuropanic]

I don't have the test open next to me, but am familiar with the question! Hope this helps:


Furthermore, since the volume of the human body and the volume of water displaced are the same (which is always true for an object that is completely submerged - after all, it could never displace a volume different from its own), the ratio of their masses is the same as the ratio of their densities. Here's the mathematical way to look at this:

m(human) / m(water) => [m (human) / volume] / [m (water) / volume], because volume cancels out. (m/V) gives us density, so this expression is equal to the ratio of the density of a human body to the density of water (its specific gravity).

This part makes sense to me, but I am still confused about where the denominator in their answer comes from. Why do you subtract the weight of the person in water from the weight in air? if using the ratio above and solving for density of human in air (what the question was specifically asking about), then I get [mass(human air) * density of water (=mass water/ vol water)] / [mass (water)] and then its like the mass of the water cancels from the top and bottom and then you are left with mass (human) / volume of water?

I understand where specific gravity comes from etc. I just really don't understand how to relate the mass in air to the mass in water and what's up with the subtraction?

Thanks for your help!

 

[NextStepTutor_3]

Great question and I agree - the subtraction is the most confusing part! I included a picture of a force diagram that can really help with this concept. See how mg (the regular weight of the person in air) points down and Fb (the buoyant force) points up? This means that the DIFFERENCE between the two, mg - Fb, will give us the weight of the person in water. For example, say a person weighs 1000 N in air, and the buoyant force pushing up on them is 200 N. They're going to weigh 800 N if they sat on a scale underwater, because part of their original weight is being "canceled out" by an upward force.

So we've established that mg (in air) - buoyant force = mg (in water). Moving the terms around gives us mg (in air) - mg (in water) = buoyant force. In other words, that subtraction expression is just a confusing way that the AAMC expressed a simple quantity: buoyant force.

I'm running to a session right now, but will answer your other question afterward! Just wanted to make sure you had an answer to the subtraction part right away 🙂

 

[neuropanic]

I now see where you had started talking about the subtraction expression in the beginning of your first post - sorry for making you re-explain. So am I correct in thinking that:
If solving for density of human (= m / v), the reason that the denominator is Fb is because the volume displaced is equal to the weight of the liquid displaced? So the Fb = (Weight air - Weight water) which can be another way of saying the volume?

 

[scuzum2u]

Also confused on this one. I actually found the first part most clear, and lost you after you set buoyant force = the difference in weight in air vs. water.

You said this: Fb = Winair - Winwater
Then "we can simplify this express to "weight in air" / "buoyant force." ---> Winair / Fb (how did you simplify to this step and what happened to Winwater?)

 

[JoeCaputo]

Also confused on this one. I actually found the first part most clear, and lost you after you set buoyant force = the difference in weight in air vs. water.

You said this: Fb = Winair - Winwater
Then "we can simplify this express to "weight in air" / "buoyant force." ---> Winair / Fb (how did you simplify to this step and what happened to Winwater?)

Fb = Wair - Wwater

Density = (Wair / Fb)*(density of water)

density of water will almost certainly be limited to 1 kg/L on a no calculator test (as opposed to seawater), so you can substitute 1 in its place to simplify the density equation.

 

[cmmc5]

This question also gave me a headache, especially since it's easy to overthink (several of the practice test questions were!). I got really hung up on whether the person was fully submerged but not sunk, which would influence the way you think of the free body diagram. Approaching it conceptually ended up being the easiest. An object that is submerged (assuming it's submerged, not sunk) will have its apparent weight in air (Fg of the object) equal to the weight of the fluid that it displaces (in other words, the buoyant force FB). A free body diagram for this would be two perfectly balanced forces, with the buoyant force pointing up and force of the object pointing down.

So mathematically, let's write that as FB = Fg. We can go a step further and break that down into FB = (mfluid displaced)(g) and likewise, Fg=(mobj)(g). Now how can we relate this to the densities of the fluid and the object? Let's say p= density because I don't have a rho key and also I'm going to stop typing the whole fluid displaced thing and just write "fluid."

pobj= mobj/Vobj
pfluid = mfluid/Vfluid

But let's solve these for the masses, since we want to relate the forces to the densities and this way we can substitute densities into the equations for the forces seen in the paragraph above. So:

mobj = pobj(Vobj)
mfluid = pfluid(Vfluid)

Now substitute into the equations for the forces and we get:

Fg = pobj (Vobj) (g)
Fb = pfluid (Vfluid) (g)

Okay! Now we have something relating forces to densities. The question asks what is the density of a human body proportional to, so let's set up a proportion using these equations, relating forces to densities.

Fg/Fb = (pobj) (Vobj) (g) / (pfluid) (Vfluid) (g)
which I'm going to write for clarity's sake as:

(pobj) (Vobj) (g) / (pfluid) (Vfluid) (g) = Fg/Fb

We can simplify this further. The (g) cancels and so do the volumes--that's because a fully submerged object MUST displace a volume of fluid that's equal to its own volume--conservation of space if you will 🙂 Therefore the volume of the object is equal to the volume of the fluid displaced. Now we've got a nice simple equation:

pobj / pfluid = Fg /Fb

This is actually a really useful equation to memorize, because it follows intuition (for example, let's say the density of the object is greater than the fluid, then the weight of the object is greater than the buoyant force and so the object will sink...etc). How does this relate to the problem? Well, the density of the fluid here is 1 because the fluid is water. Fg is the weight of the object (the person) in air and Fb is the buoyant force, which someone above has pointed out is Wair-Wwater. Plug it all in and you get the answer: density of the object = Wair/Wair-Wwater

Hope that helps!

 

[rafiki2016]

This question also gave me a headache, especially since it's easy to overthink (several of the practice test questions were!). I got really hung up on whether the person was fully submerged but not sunk, which would influence the way you think of the free body diagram. Approaching it conceptually ended up being the easiest. An object that is submerged (assuming it's submerged, not sunk) will have its apparent weight in air (Fg of the object) equal to the weight of the fluid that it displaces (in other words, the buoyant force FB). A free body diagram for this would be two perfectly balanced forces, with the buoyant force pointing up and force of the object pointing down.

So mathematically, let's write that as FB = Fg. We can go a step further and break that down into FB = (mfluid displaced)(g) and likewise, Fg=(mobj)(g). Now how can we relate this to the densities of the fluid and the object? Let's say p= density because I don't have a rho key and also I'm going to stop typing the whole fluid displaced thing and just write "fluid."

pobj= mobj/Vobj
pfluid = mfluid/Vfluid

But let's solve these for the masses, since we want to relate the forces to the densities and this way we can substitute densities into the equations for the forces seen in the paragraph above. So:

mobj = pobj(Vobj)
mfluid = pfluid(Vfluid)

Now substitute into the equations for the forces and we get:

Fg = pobj (Vobj) (g)
Fb = pfluid (Vfluid) (g)

Okay! Now we have something relating forces to densities. The question asks what is the density of a human body proportional to, so let's set up a proportion using these equations, relating forces to densities.

Fg/Fb = (pobj) (Vobj) (g) / (pfluid) (Vfluid) (g)
which I'm going to write for clarity's sake as:

(pobj) (Vobj) (g) / (pfluid) (Vfluid) (g) = Fg/Fb

We can simplify this further. The (g) cancels and so do the volumes--that's because a fully submerged object MUST displace a volume of fluid that's equal to its own volume--conservation of space if you will 🙂 Therefore the volume of the object is equal to the volume of the fluid displaced. Now we've got a nice simple equation:

pobj / pfluid = Fg /Fb

This is actually a really useful equation to memorize, because it follows intuition (for example, let's say the density of the object is greater than the fluid, then the weight of the object is greater than the buoyant force and so the object will sink...etc). How does this relate to the problem? Well, the density of the fluid here is 1 because the fluid is water. Fg is the weight of the object (the person) in air and Fb is the buoyant force, which someone above has pointed out is Wair-Wwater. Plug it all in and you get the answer: density of the object = Wair/Wair-Wwater

Hope that helps!

The volume should only cancel out if the object is fully submerged which was not well defined in the problem statement makes hard to relate right away

 

[NextStepTutor_3]

The volume should only cancel out if the object is fully submerged which was not well defined in the problem statement makes hard to relate right away

In MCAT fluids, they rarely say "fully submerged" (even if it would make more sense). So if they talk about an object that is "submerged," you can assume it is fully underwater. When any part of the object is above water, they typically use the term "floating."

Here, for example, the AAMC uses the phrase "while submersed in water," so we can assume that it's totally underwater.

 

[nenk]

This question also gave me a headache, especially since it's easy to overthink (several of the practice test questions were!). I got really hung up on whether the person was fully submerged but not sunk, which would influence the way you think of the free body diagram. Approaching it conceptually ended up being the easiest. An object that is submerged (assuming it's submerged, not sunk) will have its apparent weight in air (Fg of the object) equal to the weight of the fluid that it displaces (in other words, the buoyant force FB). A free body diagram for this would be two perfectly balanced forces, with the buoyant force pointing up and force of the object pointing down.

So mathematically, let's write that as FB = Fg. We can go a step further and break that down into FB = (mfluid displaced)(g) and likewise, Fg=(mobj)(g). Now how can we relate this to the densities of the fluid and the object? Let's say p= density because I don't have a rho key and also I'm going to stop typing the whole fluid displaced thing and just write "fluid."

pobj= mobj/Vobj
pfluid = mfluid/Vfluid

But let's solve these for the masses, since we want to relate the forces to the densities and this way we can substitute densities into the equations for the forces seen in the paragraph above. So:

mobj = pobj(Vobj)
mfluid = pfluid(Vfluid)

Now substitute into the equations for the forces and we get:

Fg = pobj (Vobj) (g)
Fb = pfluid (Vfluid) (g)

Okay! Now we have something relating forces to densities. The question asks what is the density of a human body proportional to, so let's set up a proportion using these equations, relating forces to densities.

Fg/Fb = (pobj) (Vobj) (g) / (pfluid) (Vfluid) (g)
which I'm going to write for clarity's sake as:

(pobj) (Vobj) (g) / (pfluid) (Vfluid) (g) = Fg/Fb

We can simplify this further. The (g) cancels and so do the volumes--that's because a fully submerged object MUST displace a volume of fluid that's equal to its own volume--conservation of space if you will 🙂 Therefore the volume of the object is equal to the volume of the fluid displaced. Now we've got a nice simple equation:

pobj / pfluid = Fg /Fb

This is actually a really useful equation to memorize, because it follows intuition (for example, let's say the density of the object is greater than the fluid, then the weight of the object is greater than the buoyant force and so the object will sink...etc). How does this relate to the problem? Well, the density of the fluid here is 1 because the fluid is water. Fg is the weight of the object (the person) in air and Fb is the buoyant force, which someone above has pointed out is Wair-Wwater. Plug it all in and you get the answer: density of the object = Wair/Wair-Wwater

Hope that helps!

 

[nenk]

This question also gave me a headache, especially since it's easy to overthink (several of the practice test questions were!). I got really hung up on whether the person was fully submerged but not sunk, which would influence the way you think of the free body diagram. Approaching it conceptually ended up being the easiest. An object that is submerged (assuming it's submerged, not sunk) will have its apparent weight in air (Fg of the object) equal to the weight of the fluid that it displaces (in other words, the buoyant force FB). A free body diagram for this would be two perfectly balanced forces, with the buoyant force pointing up and force of the object pointing down.

So mathematically, let's write that as FB = Fg. We can go a step further and break that down into FB = (mfluid displaced)(g) and likewise, Fg=(mobj)(g). Now how can we relate this to the densities of the fluid and the object? Let's say p= density because I don't have a rho key and also I'm going to stop typing the whole fluid displaced thing and just write "fluid."

pobj= mobj/Vobj
pfluid = mfluid/Vfluid

But let's solve these for the masses, since we want to relate the forces to the densities and this way we can substitute densities into the equations for the forces seen in the paragraph above. So:

mobj = pobj(Vobj)
mfluid = pfluid(Vfluid)

Now substitute into the equations for the forces and we get:

Fg = pobj (Vobj) (g)
Fb = pfluid (Vfluid) (g)

Okay! Now we have something relating forces to densities. The question asks what is the density of a human body proportional to, so let's set up a proportion using these equations, relating forces to densities.

Fg/Fb = (pobj) (Vobj) (g) / (pfluid) (Vfluid) (g)
which I'm going to write for clarity's sake as:

(pobj) (Vobj) (g) / (pfluid) (Vfluid) (g) = Fg/Fb

We can simplify this further. The (g) cancels and so do the volumes--that's because a fully submerged object MUST displace a volume of fluid that's equal to its own volume--conservation of space if you will 🙂 Therefore the volume of the object is equal to the volume of the fluid displaced. Now we've got a nice simple equation:

pobj / pfluid = Fg /Fb

This is actually a really useful equation to memorize, because it follows intuition (for example, let's say the density of the object is greater than the fluid, then the weight of the object is greater than the buoyant force and so the object will sink...etc). How does this relate to the problem? Well, the density of the fluid here is 1 because the fluid is water. Fg is the weight of the object (the person) in air and Fb is the buoyant force, which someone above has pointed out is Wair-Wwater. Plug it all in and you get the answer: density of the object = Wair/Wair-Wwater

Hope that helps!

That was really helpful thank you!
Just a quick question, I thought Fb=Fg only when an object is floating, regardless of being fully submerged or half submerged.

 

[aldol16]

Just a quick question, I thought Fb=Fg only when an object is floating, regardless of being fully submerged or half submerged.

Fb = Fg whenever the object is in water and it's not accelerating up or down.

 

[NimbleNavigator]

This question also gave me a headache, especially since it's easy to overthink (several of the practice test questions were!). I got really hung up on whether the person was fully submerged but not sunk, which would influence the way you think of the free body diagram. Approaching it conceptually ended up being the easiest. An object that is submerged (assuming it's submerged, not sunk) will have its apparent weight in air (Fg of the object) equal to the weight of the fluid that it displaces (in other words, the buoyant force FB). A free body diagram for this would be two perfectly balanced forces, with the buoyant force pointing up and force of the object pointing down.

So mathematically, let's write that as FB = Fg. We can go a step further and break that down into FB = (mfluid displaced)(g) and likewise, Fg=(mobj)(g). Now how can we relate this to the densities of the fluid and the object? Let's say p= density because I don't have a rho key and also I'm going to stop typing the whole fluid displaced thing and just write "fluid."

pobj= mobj/Vobj
pfluid = mfluid/Vfluid

But let's solve these for the masses, since we want to relate the forces to the densities and this way we can substitute densities into the equations for the forces seen in the paragraph above. So:

mobj = pobj(Vobj)
mfluid = pfluid(Vfluid)

Now substitute into the equations for the forces and we get:

Fg = pobj (Vobj) (g)
Fb = pfluid (Vfluid) (g)

Okay! Now we have something relating forces to densities. The question asks what is the density of a human body proportional to, so let's set up a proportion using these equations, relating forces to densities.

Fg/Fb = (pobj) (Vobj) (g) / (pfluid) (Vfluid) (g)
which I'm going to write for clarity's sake as:

(pobj) (Vobj) (g) / (pfluid) (Vfluid) (g) = Fg/Fb

We can simplify this further. The (g) cancels and so do the volumes--that's because a fully submerged object MUST displace a volume of fluid that's equal to its own volume--conservation of space if you will 🙂 Therefore the volume of the object is equal to the volume of the fluid displaced. Now we've got a nice simple equation:

pobj / pfluid = Fg /Fb

This is actually a really useful equation to memorize, because it follows intuition (for example, let's say the density of the object is greater than the fluid, then the weight of the object is greater than the buoyant force and so the object will sink...etc). How does this relate to the problem? Well, the density of the fluid here is 1 because the fluid is water. Fg is the weight of the object (the person) in air and Fb is the buoyant force, which someone above has pointed out is Wair-Wwater. Plug it all in and you get the answer: density of the object = Wair/Wair-Wwater

Hope that helps!

Mind = blown that makes so much sense now. Thank you thank you thank you thank you I was so lost before reading this!

 

[nenk]

Fb = Fg whenever the object is in water and it's not accelerating up or down.

I see, thank you

 

[Christian713]

thanks so much