Magnitude of buoyant force in fluids of different densities

[I_Try_Math]

Homework Statement

A penguin floats first in a fluid of density $\rho_0$, then in a fluid of density 0.95$\rho_0$, and then in a fluid of density 1.1$\rho_0$. (a) Rank the densities according to the magnitude of the buoyant force on the penguin, greatest first.

Relevant Equations

$\rho = \frac{m}{V}$
$p = \frac{F}{A}$

The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

 

[I_Try_Math]

I can see now that my reasoning doesn't work because it assumes the volume of water displaced is constant. I'll have to think about this some more.

 

[haruspex]

I can see now that my reasoning doesn't work because it assumes the volume of water displaced is constant. I'll have to think about this some more.

What did Archimedes realise about the quantity of fluid displaced by a floating object?

 

[I_Try_Math]

What did Archimedes realise about the quantity of fluid displaced by a floating object?

That the weight of the displaced fluid will be equal to the weight of the floating object, I believe. I think I understand why the answer in the textbook is correct.

 

[mjc123]

Remember: a floating body displaces its own weight of fluid;
A submerged body displaces its own volume of fluid.
Whether it floats or sinks depends on its own density relative to the fluid.

 

[haruspex]

I have always assumed that Archimedes figured this out by imagining replacing the submerged portion with an equal volume of water, exactly filling the void in the water that would be created by removing the submerged portion. Clearly that water would float, so the force exerted on it by the surrounding water must be equal to its weight.

 

[Herman Trivilino]

Homework Statement: A penguin floats first in a fluid of density $\rho_0$, then in a fluid of density 0.95$\rho_0$, and then in a fluid of density 1.1$\rho_0$. (a) Rank the densities according to the magnitude of the buoyant force on the penguin, greatest first.

That's not worded correctly. It should be asking you to rank the magnitudes of the buoyant forces. Clearly, the densities are not all equal.

 

[Herman Trivilino]

I have always assumed that Archimedes figured this out by imagining replacing the submerged portion with an equal volume of water, exactly filling the void in the water that would be created by removing the submerged portion.

That's the version that appears in the textbooks. I also assume it's true.

Clearly that water would float, so the force exerted on it by the surrounding water must be equal to its weight.

You mean that water would be in a state of neutral buoyancy.

 

[haruspex]

That's not worded correctly. It should be asking you to rank the magnitudes of the buoyant forces. Clearly, the densities are not all equal.

The wording is awkward but not actually wrong. It means rank the fluids according to the buoyant forces they are exerting, but since the only way the fluids have been assigned labels is by their density variable names, those are what have to be listed.

That's the version that appears in the textbooks. I also assume it's true.

You mean that water would be in a state of neutral buoyancy.

Yes, necessarily.

 

[Herman Trivilino]

The wording is awkward but not actually wrong. It means rank the fluids according to the buoyant forces they are exerting, but since the only way the fluids have been assigned labels is by their density variable names, those are what have to be listed.

If the authors meant rank the fluids that's what they should have written. Not "rank the densities". Density is a property of a fluid, and it's the fluids that are being ranked not their densities.

 

[haruspex]

If the authors meant rank the fluids that's what they should have written. Not "rank the densities". Density is a property of a fluid, and it's the fluids that are being ranked not their densities.

That would have been clearer, yes. But consider..
1. "Cars of velocities v1 > v2 > v3 are driven from A to B. Rank the cars in order of increasing time taken."
2. "A car is driven at velocities v1 > v2 > v3 from A to B. Rank the velocities in order of increasing time taken."

 

[Herman Trivilino]

That would have been clearer, yes. But consider..
1. "Cars of velocities v1 > v2 > v3 are driven from A to B. Rank the cars in order of increasing time taken."
2. "A car is driven at velocities v1 > v2 > v3 from A to B. Rank the velocities in order of increasing time taken."

Okay. The instructions in 1 prompt us to rank the cars, not the relevant property (time) of the car's journeys. Prompting us for a ranking of the car's journies would be an improvement.

2 is worse. It asks for a ranking of the velocities.

I realize the authors' meanings are clear from the context for an expert, but the wording could confuse a novice.

We have the same issue in a phrase like consider "a mass hanging from a spring". Mass is a property of the object hanging from the spring, it is not the object itself. For students struggling with concept of mass this can be confusing.

I remember being confused by a physics professor asking us to differentiate a function such as $y=2x^3$ with respect to $x$. Having just completed the introductory calculus sequence I found the wording confusing. I was used to "take the derivative". So after some deliberation I concluded he was asking for the differential. So I wrote $dy=6x^2 dx$. He said that was okay but derivatives are usually written as $dy/dx$.

Okay, I thought, but he didn't ask for the derivative. He asked for the differential?

 

[renormalize]

We have the same issue in a phrase like consider "a mass hanging from a spring". Mass is a property of the object hanging from the spring, it is not the object itself. For students struggling with concept of mass this can be confusing.

But elementary physics texts regularly use shorthand terms like "point mass" (not "point object with mass") and "point charge" (not "point object with charge") when discussing mechanics, gravity and electrostatics. So is "mass hanging from a spring" really more confusing to the average student than "object with mass hanging from a spring"?

I remember being confused by a physics professor asking us to differentiate a function such as $y=2x^3$ with respect to $x$. Having just completed the introductory calculus sequence I found the wording confusing. I was used to "take the derivative". So after some deliberation I concluded he was asking for the differential. So I wrote $dy=6x^2 dx$. He said that was okay but derivatives are usually written as $dy/dx$.

Okay, I thought, but he didn't ask for the derivative. He asked for the differential?

No he didn't per the dictionary definition of differentiate:

(https://www.merriam-webster.com/dictionary/differentiate)

 

[haruspex]

It asks for a ranking of the velocities.

Quite, because those velocities represent available choices and it is reasonable to rank choices based on the outcomes.
What makes the usage in post #1 awkward is that there are three fluids each with a fixed density, so it is more natural to think of choosing a fluid than of choosing a density. But if it were a matter of choosing the density of a salt solution, say, the wording would be perfectly reasonable.

 

[weirdoguy]

For students struggling with concept of mass this can be confusing.

But do you know someone for whom this was actually a problem?

 

[Herman Trivilino]

But do you know someone for whom this was actually a problem?

Yes. Anyone who has ever taught introductory physics has encountered students who have trouble grasping the concept of mass, and sorting out its relationship with weight. And I have encountered a few students for which this was actually problem. Of course, those are the only ones I'm aware of. I'm sure there are many others who are confused, and some or even most of them cannot clearly articulate the source of their confusion. The cognitive load is just too high.

I know many instructors will dismiss these students as unteachable, but I'm not one of them.

 

[haruspex]

and sorting out its relationship with weight.

That's a different matter; the discussion was confusion between mass as an object and mass as a property.
English is replete with puns, and having the same word for an object and a quantifiable attribute of the object is very common: attach a weight, carry a load, cut off a length, turn down a light, smell a smell… Those never seem to cause a problem.
What may be different about "mass" is that it we grow up with it meaning a large collection, which is not quite either of the physics meanings.

I feel there are more worrying puns in maths. In y=y(x), the y on the left is a variable while the y on the right is a function. Another is $\int_0^xx\cdot dx$. The "x" as a bound is unrelated to the other two.

But this is all rather off topic. How about we move it to another forum… "STEM educators and teaching"?

 

[Herman Trivilino]

That's a different matter; the discussion was confusion between mass as an object and mass as a property.

Yes, but to the uninitiated, learning about mass involves sorting out its relationship to weight. In medicine and commerce they are synonymous.

English is replete with puns, and having the same word for an object and a quantifiable attribute of the object is very common: attach a weight, carry a load, cut off a length, turn down a light, smell a smell… Those never seem to cause a problem.

No, because the issues are familiar and the meaning is clear from the context. Just as when we say a mass is hanging from a spring.

I feel there are more worrying puns in maths. In y=y(x), the y on the left is a variable while the y on the right is a function.

I remember in quantum class the prof said psi is a function of $x$ and wrote $\psi = \psi(x)$. I was very confused and couldn't articulate why. It wasn't until years later that a math prof explained that, as well as the issue you mentioned with limits of integration. He was careful and sedulously avoided those notational errors.