TBR: Fluid Flow Pressure: Incomplete Answer Explanation

[justadream]

TBR Physics Book II page 98 #24

“How can the pressure exerted by the fluid BEST be described as it passes from Region 1 through the tube and ultimately through Region 3”?

Correct Answer: “From region 1 to 2, pressure decreases; from region 2 to 3, pressure increases. P1 > P3


TBR decides to not explain the middle part because the other choices can be eliminated without looking at it.

Upon looking more closely, however, I don’t understand how that statement is true.

Using Bernoulli’s Equation: P + .5(rho)v^2 + (rho)g * height = CONSTANT

Thus, P = CONSTANT – [.5(rho)v^2 + (rho)g * height ]

Let me make up a scenario and plug in some numbers:

Let CONSTANT = 50000. Assume rho = 1000. Assume g = 10. Assume height_1 = 1 and assume height_3 = 2. Also, let initial velocity in region_1 = 1.


Applying Benoulli’s Equation:

At Region_1:

P = 50000 – [ .5*(1000)(1^2) + 1000*10*1 ] = 39500

At Region_2 (note that velocity has doubled since the cross-sectional area is twice as big):

P = 50000 – [ .5*(1000)(2^2) + 1000*10*1 ] = 38000

At Region_3 (note that velocity is the same as in region_1 but now the height is doubled):

P = 50000 – [ .5*(1000)(1^2) + 1000*10*2 ] = 29500



Using the numbers I plugged in, it appears that pressure DECREASES from region 2 to region 3.



Can someone spot a flaw in my example/thinking and/or in TBR’s question?

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[Cawolf]

A2V2 = A3V3

V3 = A2V2/A3 = A2V2/2A2 = V2/2

We know that increasing velocity results in a decrease in pressure. Since V3 is a decreased velocity from V2 - we can say that the pressure must increase.

 

[Cawolf]

Sorry I didn't address the height difference.

If you solve bernoulli's equation for section 2 and 3 you get this:

P2 = P3 - rho(3/8 (V2)^2 - gY2)

This shows that the pressure is less due to velocity but more due to gravity.

 

[justadream]

@Cawolf

I thought that the conclusions about pressure only held for a flat surface. Here, region 2 is BELOW region 3.

In Bernoulli's equation there are THREE variables: velocity, cross-sectional area, height

If height were constant, I would agree with you conclusion. But height isn't constant. So just because velocity increases, that doesn't mean pressure decreased I think. That extra "energy" can be stored at potential energy (in the height).

If you look at the numbers I plugged in (which I made up), you can see that it works out so that the velocity in region 3 is > velocity in region 2 BUT the pressure is not.

 

[Cawolf]

See my post right before yours.

 

[Cawolf]

The solution I posted is unsolvable without numbers though as we can't tell the magnitude of the velocity or height. It simply shows that they both influence it. I think that is why this question is a process of elimination to answer as this solution is relatively inconclusive with the information we have.

 

[justadream]

@Cawolf

Do you mind posting a few intermediate steps in how you derived that?

"This shows that the pressure is less due to velocity but more due to gravity."

I'm not sure I'm understanding how you make this connect with the pressure being higher in region 3.

There may be some fault with the way I plugged in numbers but when I chose some random numbers to plug-in, I somehow got pressure being greater in region 2.

 

[Cawolf]

 

[Cawolf]

It just means that if the pressure in 2 is lower than in 3; the velocity factor is larger in magnitude than the height factor.

 

[Oh_Gee]

this chapter is draining my lifeforce at a very high Q

 

[justadream]

@Cawolf

Thanks for taking the time to do that!

I speculate, however, that if the ratios of the radius of region 2 and region 3 changed (either radius 3 becomes only slight larger vs. radius 3 being 50000 times largers), then velocity might not always be the more significant factor? If so, how would you be able to deduce when the "breakeven" point is (esp with regard to the timing concerns on the MCAT)?

 

[Cawolf]

With something like that I don't think you could tell. I would assume that unless basic numbers were given, solving this equation would not be required to answer the question. Like you say, you could not determine the point if the numbers were so close.

 

[DrknoSDN]

It just means that if the pressure in 2 is lower than in 3; the velocity factor is larger in magnitude than the height factor.

Does this mean that TBR gave another incorrect answer?
Saying that the answer is true "if" these conditions are met, also means that it is not true under different conditions. I really dislike ambiguous questions, or even asking a question when the real answer should be "Cannot be determined".

If the height change is very large then the answer would be 1>2 decreases and 2>3 also decreases correct?

 

[Cawolf]

He had said that the choices ruled in the correct answer without needing to the solve it as I posted. He was just curious as why the pressure differential would be as so.

 

[DrknoSDN]

@Cawolf Oh, I assumed based on the one choice given that the options were:
decrease, decrease
decrease, increase
increase, decrease
increase, increase

And that decrease,anything was viable. Ty, should have read the whole thread.

 

[Cawolf]

I actually got my TBR copies today, let me look exactly.

A. Increase in P from 1 to 2, uncertain P from 2 to 3. P3 > P1
(Can't be right because only change in decreased area in 2, so higher V in 2 = lower P in 2)

B. Decrease in P from 1 to 2, increased P from 2 to 3. P3 > P1
(Can't be right because the pressure at 1 must be greater than pressure at 3 because only difference in height and the lower side has higher pressure)

C. Increase in P from 1 to 2, uncertain P from 2 to 3. P1 > P3
(Same as choice A)

D. Decrease in P from 1 to 2, increased P from 2 to 3. P1 > P3
(Therefore this must be right even though we can't prove the change from 2 to 3)

 

[Cawolf]

Those are summarized answer choices with my reasoning below them.

 

[DrknoSDN]

He had said that the choices ruled in the correct answer without needing to the solve it as I posted. He was just curious as why the pressure differential would be as so.

Newest version: (Physics book 2, page 98, #24)

It still seems like the answer should read:
"From region 1 to 2, pressure decreases; from region 2, to 3, the pressure change is uncertain. P1 > P3"

That's not an option.

 

[Cawolf]

Yah I pulled out my book and looked. I agree, but choice D is still the most correct answer though.

 

[DrknoSDN]

D. Decrease in P from 1 to 2, increased P from 2 to 3. P1 > P3
(Therefore this must be right even though we can't prove the change from 2 to 3)

Gotcha, good process of elimination. Still, stating something in an answer that is not necessarily true seems bad practice imo. Almost used to it from TBR by now.

It seems like TBR regularly spews out ambiguous statements. It is the right answer yes, but it is also very misleading, confusing, and the very reason for the creation of this post. That indicates poor test prep material, unless the goal is actually to confuse you so much that you are forced to search online and in other materials for a more comprehensive and accurate understanding.. If that is the goal then bravo, but there still seems to be too many questions about TBR "wording" than is appropriate from such an experienced prep company.

@Cawolf Good of you to catch the discrepancy mathematically via Bernoulli's and post the pic. Props.

 

[justadream]

@Cawolf and @DrknoSDN

First, you two are literally my MCAT saviors!

Second, yeah you could determine the answer without checking the middle statement. Of course, TBR was too lazy to actually write an explanation for that statement (hence....this thread + all the confusion)... grr...

But to TBR's defense, they did provide the relative ratio of the height change and radius change in the 2 regions. So technically, can't you figure it out based on those ratios?