1/r = 1/r[sub]1[/sub] + 1/r[sub]2[/sub] + 1/r[sub]3[/sub]

[mehc012]

So, in my notes for Physio, where we learn the most simplified version possible of everything (which is very frustrating), is a slide with this equation:

1/R = 1/R₁ + 1/R₂ + 1/R₃

Below it, the prof wrote:
"Resistances in the circulation are arranged In parallel. As resistances are increased the total resistance decreases."

Now, this professor makes a habit of having typos, incorrect facts, and oversimplifications which extend to the point of blatant contradiction of the actual truth. The trick is to figure out which are which.

Here's my problem with this statement: It just doesn't work mathematically. If the parallel resistors (and keep in mind this is NOT electrical resistance, which makes finding an explanation that doesn't rely on voltage and whatnot difficult) are all 1, the total resistance would be 1/3. If they were all tripled, the total resistance would also be tripled, ending up at 1 (1/3+1/3+1/3). So I suspect that this is another instance where his summary of the situation is either insanely oversimplified or typoed, or SOMEthing, and I would like to see if I can make it make sense to me. So I've tried to come up with my own very general explanation, and I was hoping someone could let me know if it's vaguely correct:

The total resistance in the system is less than the sum of the individual parallel resistances, and areas of high resistance have a diminished effect on the total resistance because there is less flow through those areas.


aaaaand I can't change the thread title and didn't realize the HTML wouldn't work there. So sorry for the messiness.

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[enervate]

The total resistance in the system is less than the sum of the individual parallel resistances, and areas of high resistance have a diminished effect on the total resistance because there is less flow through those areas.

I'll agree with the first part but not the second part. A strong resistor still contributes more resistance to the overall circuit.

"Resistances in the circulation are arranged In parallel. As resistances are increased the total resistance decreases."

This should probably read: as the number of resistors in parallel are increased the total resistance decreases if the resistance of each resistor is equivalent. Meaning: as you add resistors in parallel, the net resistance decreases unless the resistance of the added resistor is large enough to offset the added division of current.

edit: have fun.

 

[mehc012]

I'll agree with the first part but not the second part. A strong resistor still contributes more resistance to the overall circuit.

This should probably read: as the number of resistors in parallel are increased the total resistance decreases if the resistance of each resistor is equivalent. Meaning: as you add resistors in parallel, the net resistance decreases unless the resistance of the added resistor is large enough to offset the added division of current.

I mean, this is in relation to the cardiovascular system, not electronics, so you're not going to be adding or subtracting resistors, just changing the resistance of each one.

As for the second part, which you disagree with, I think my wording may be unclear. Of course a larger resistor will increase the overall resistance of a circuit, but it is contributing less proportionally than would be expected if it were directly additive (than if it were R = R1 + R2 + R3).

 

[enervate]

I mean, this is in relation to the cardiovascular system, not electronics, so you're not going to be adding or subtracting resistors, just changing the resistance of each one.

Ah, I thought Physio was a typo. I noticed that you mentioned this was not concerning electrical resistances but I had no clue what that meant.

Nonetheless, the concept should be the same. Current is just a measure of the flow of something whether it be blood or electrons. If everything carries over as it should, that quoted statement is still incorrect as written.

As for the second part, which you disagree with, I think my wording may be unclear. Of course a larger resistor will increase the overall resistance of a circuit, but it is contributing less proportionally than would be expected if it were directly additive (than if it were R = R1 + R2 + R3).

I'm not exactly sure how to verify that relationship. It's fairly obvious to determine the relative contribution of R1 to a series of resistors (R1/R) but what about parallel resistors?

 

[mehc012]

Ah, I thought Physio was a typo. I noticed that you mentioned this was not concerning electrical resistances but I had no clue what that meant.

Nonetheless, the concept should be the same. Current is just a measure of the flow of something whether it be blood or electrons. If everything carries over as it should, that quoted statement is still incorrect as written.

I'm not exactly sure how to verify that relationship. It's fairly obvious to determine the relative contribution of R1 to a series of resistors (R1/R) but what about parallel resistors?

This was the big thing I was trying to verify...now to figure out whether I should learn it accurately or regurgitatively...aka whether I put the wrong answer on the test to try and avoid losing points for disagreeing with the prof!

 

[jjunior]

perhaps what he meant in that pharse is "Resistances in the circulation are arranged In parallel. As numbers of resistors are increased the total resistance decreases." that would be from a physics point of view.
perhaps we can think of resistors arranged in parallel are like providing alternative routes for blood to circulate, so if have more branches of blood vessels (ie. more resistors in parallel), then the total resistance would decrease. but if one of the branches is obstructed (ie.super high resistance), then less blood flow would go to that branch.

 

[mehc012]

Ah, I thought Physio was a typo. I noticed that you mentioned this was not concerning electrical resistances but I had no clue what that meant.

Nonetheless, the concept should be the same. Current is just a measure of the flow of something whether it be blood or electrons. If everything carries over as it should, that quoted statement is still incorrect as written.

I'm not exactly sure how to verify that relationship. It's fairly obvious to determine the relative contribution of R1 to a series of resistors (R1/R) but what about parallel resistors?

If they were simply additive (aka R = R1 + R2 + R3), then in our example, doubling R3 would increase R from 3 to 4 (R = 1 + 1 + 2), an increase of 33.3%

Since they're not, doubling R3 changes R from 0.333 to 0.4 (1/3 to 2/5), an increase of less than 33.3% (closer to 20%). Tripling R3 would increase R by ~29% (vs. 67% for the cumulative), and quadrupling would increase R by 33% (vs 100% cumulative).

So it does diminish the impact of increasing an individual resistor, though my explanation for why is, I admit, completely made up out of my skull.

perhaps what he meant in that pharse is "Resistances in the circulation are arranged In parallel. As numbers of resistors are increased the total resistance decreases." that would be from a physics point of view.
perhaps we can think of resistors arranged in parallel are like providing alternative routes for blood to circulate, so if have more branches of blood vessels (ie. more resistors in parallel), then the total resistance would decrease. but if one of the branches is obstructed (ie.super high resistance), then less blood flow would go to that branch.

Highly doubtful that he meant it that way, just from what I've seen of his background, understanding, and style...but very helpful for me anyway, thanks!