TBR: Pascal's Law Doesn't Apply in Ear?

[justadream]

TBR Bio I page 58 #73

"The tympanic membrane has an area greater than the oval window. A result of this design is that the total:"

Answer: Force/Unit_area acting on the oval window is increased.


TBR's explanation says that the force stays the same. Why is that? I thought this would be a situation in which you have to apply Pascal's Law (where if you have a bigger area, you get a bigger force - this is how hydraulics work).

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[Cawolf]

Pascal's Law is that the pressure in is equal to pressure out.

So if a large force is applied over a large area (tympanic) then the fluid wave travels to a smaller are (oval) - the pressure is constant so the force is larger.

 

[justadream]

@Cawolf

So you're saying that for the tympanic membrane (which has the LARGER area), the force should be GREATER?

TBR's answer says the force is the same.

 

[Cawolf]

The pressure should be constant.

 

[Cawolf]

Per wikipedia, the oval window is 20x smaller in area than the ear drum.

So let's say that some sound wave hits your eardrum with a force F over area A.

So P = F/A

This pressure wave will travel to the oval window with an A' = .05A

Since Pascal's Law states that P = P ', we can state than P = F/A = P' = F'/A'

F/A = F'/A'

FA'/A = F'

F' = F.05A/A = .05F

So the force experienced at the smaller area membrane (oval window) is directly proportional to the difference in area.

 

[justadream]

@Cawolf
Did you substitute incorrectly?

You have:
FA'/A = F'

And A' = .05A

Thus, F' = F(.05) / A

Here is my notecard for this scenario:


F1/A1= F2/A2

F2= (A2/A1) * F1
If A2>A1, F2will be > F1
So the output force will be greater than the input force if the output area is bigger.

 

[Cawolf]

Yes I did - I fixed it to match yours.

It's been a long day!

The pressure is constant so larger area gets larger force.

 

[justadream]

@Cawolf

So if pressure is constant and the larger area (typanic membrane) gets larger force, do you disagree with TBR answer?

TBR answer is that the pressure increased and force is constant...

 

[Cawolf]

Ohhhh I should've looked at the book from the start.

This whole time I was answering questions from the point of view of a hydraulic system as you mentioned.

This is a set force being passed through the bones from eardrum to oval window (I was thinking of eardrum to round window before).

Since the force is not being lost, D makes the most sense - the force is same but the area is smaller.

 

[justadream]

@Cawolf

Wait so why can't you apply Pascal's law? Is it that, unlike in a hydraulic system, the two areas in which the force are being applied (the tympanic membrane and the oval window) are directly touching each other?

 

[Cawolf]

Pascal's Law deals with ideal fluids that change in surface area in a pipe. That is not what is occurring here.

This is as you say, more like the objects are touching each other.

 

[justadream]

Pascal's Law deals with ideal fluids that change in surface area in a pipe. That is not what is occurring here.

This is as you say, more like the objects are touching each other.

Got it. Too many scenarios to think of...