Potentially Stupid Question about Bike Weight



FlyingPancreas

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Aug 9, 2013
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Just curious. It seems obvious that in the most general sense, lighter will usually be better. If one is not a competitive cyclist, however, I'm wondering if a bit too much consideration is given to variances of a few pounds in bike weight owing to things like disc brakes vs. V brakes, etc.

Here's the part where I may be just stupid. I was always an awful science student. But it seems to me that the weight of the PEOPLE riding the bikes varies so much that I would imagine this negates any impact a few extra - or fewer - pounds of bike might otherwise have. If I am able to shave a few pounds off a bike by spending thousands more for an all carbon setup, wouldn't I wipe out the entire weight savings if I myself am 10 pounds heavier than the guy next to me on the identical bike?

In other words, if one is not a competitive cyclist, does bike weight really matter worth a rip if we're talking about differences in the approx. 5 -6 lb range (difference in weight between typical low-end cyclocross bike and hybrid with suspension)?
 
FWIW. If the real question is whether or not you should buy a low end CX bike or a Hybrid with a Suspension fork, the answer is you should consider buying a Hybrid with a Rigid fork ...

  • some-but-not-all inexpensive suspension forks are little more than two pogo sticks attached to a bike frame
  • a Hybrid with a Rigid fork will cost less than a comparable Hybrid with a dubious Suspension fork

IMO, you generally want the least amount of weight on the front of the bike as possible ... it's an inertia thing.
 
Originally Posted by FlyingPancreas .


In other words, if one is not a competitive cyclist, does bike weight really matter worth a rip if we're talking about differences in the approx. 5 -6 lb range (difference in weight between typical low-end cyclocross bike and hybrid with suspension)?
All depends how you define "worth a rip". The total weight of bike + rider matters when climbing hills and accelerating, because that determines how much energy (work) you have to perform to get over the hill or up to speed. On the flats, total weight impacts your rolling resistance. But as you say, it's the overall weight that matters.

So, if bike and rider weigh eg 200 lbs, saving 6 lbs means saving about 3% of total system weight. On a course with 10 minutes of steep climbing at slow speeds, applying the same power will get you to the top 3% faster, saving 18 seconds (600 x .03). That's a huge savings to a racer, maybe not so much to you or me.

And if you're 30 lbs over best race weight, as I am, the place to look for the most weight savings isn't the bike.
 
Originally Posted by alfeng .

FWIW. If the real question is whether or not you should buy a low end CX bike or a Hybrid with a Suspension fork, the answer is you should consider buying a Hybrid with a Rigid fork ...

  • some-but-not-all inexpensive suspension forks are little more than two pogo sticks attached to a bike frame
  • a Hybrid with a Rigid fork will cost less than a comparable Hybrid with a dubious Suspension fork

IMO, you generally want the least amount of weight on the front of the bike as possible ... it's an inertia thing.
Thanks. Makes sense. I was initially leaning very heavily toward a bike with front suspension, but at this point I'm glad I waited a bit because I'm now pretty certain I do not want any kind of suspension. For my purposes (commuting almost entirely on pavement), although the roads here can get somewhat rough, basically a flatbar road bike seems to make the most sense. Maybe something with slightly wider tires (32 to 35mm range), as the roads are a bit of a concern and I'm not obsessed with speed. I need to hit a few shops.
 
Originally Posted by dhk2 .

All depends how you define "worth a rip". The total weight of bike + rider matters when climbing hills and accelerating, because that determines how much energy (work) you have to perform to get over the hill or up to speed. On the flats, total weight impacts your rolling resistance. But as you say, it's the overall weight that matters.

So, if bike and rider weigh eg 200 lbs, saving 6 lbs means saving about 3% of total system weight. On a course with 10 minutes of steep climbing at slow speeds, applying the same power will get you to the top 3% faster, saving 18 seconds (600 x .03). That's a huge savings to a racer, maybe not so much to you or me.

And if you're 30 lbs over best race weight, as I am, the place to look for the most weight savings isn't the bike.
Yeah, essentially what I am asking is if it ultimately makes any difference whether the weight comes off the bike or the rider. I got curious because in poking around through various threads there is a LOT of reference to bike weight by people who do not race. While I totally understand why as a matter of preference one would generally want the lightest bike possible, I'd personally find it easier to lop 10 pounds off my gut than cough up a few k for an all-carbon setup. *NOT* that these things are mutually exclusive. I would always opt to buy the best bike I can afford... It just happens that at the moment, that does not include the high end. I would if I could...
 
It really doesn't matter whether the weight comes off the bike or the rider. Some will argue that a given amount of weight off rotating parts (wheels, cranks, pedals) has more influence than weight off of non-rotating bits (like the frame, the rider, and etc), but the reality is that the difference in performance between those two options is very small. It's almost always more cost effective to remove weight from the rider. As a bonus, as weight comes off the rider, the rider generally gets more fit and performs better. Another win-win.
 
These are all valid points, however, rotational mass has a profound effect on any gyroscopic device be it a car wheel, motorcycle wheel, drivetrain on any engine driven vehicle. This is why heavy wheels on bikes vs light wheels is so noticeable. Some will point out it is a better value to spend less money on certain features of a groupset and apply that money to a good set of wheels.
 
schmeg said:
These are all valid points, however, rotational mass has a profound effect on any gyroscopic device be it a car wheel, motorcycle wheel, drivetrain on any engine driven vehicle. This is why heavy wheels on bikes vs light wheels is so noticeable. Some will point out it is a better value to spend less money on certain features of a groupset and apply that money to a good set of wheels. 
The difference between "heavy" and "light" wheels might be felt throwing the bike side to side, but the difference in acceleration when climbing and sprinting is very small. Profound? No. Do the math.
 
Originally Posted by alienator .


The difference between "heavy" and "light" wheels might be felt throwing the bike side to side, but the difference in acceleration when climbing and sprinting is very small. Profound? No. Do the math.
More parroted information ...

Perhaps YOU can figure out why it wrong ... but, I doubt it since it would actually require being able to "do the math."
 
Here you go, Alfeng. There's no need to redo what is already done well, and Mark McM's model of bicycle, displayed at the Weight Weenies forum, is done well. In fact, Mark McM's model will actually overstate the moment of inertia for wheels, and it still shows that the resulting difference in acceleration at a given power output between two wheels of vastly different mass is still very small. Now, I know you don't trust science and you might claim something like, "that's not a real world model". Uhm, as stated his model will over state a wheel's MOI. Mark McM's Model of the Influence of Rotating Mass in the Form of Wheels By the way, the influence of crank arms, chain rings, pedals, and cassettes is much, much smaller. Oh, if you haven't taken a differential equations course and passed it, you likely will have no idea what a fourth order Runge-Kutta method is, so here's some help for you: http://en.wikipedia.org/wiki/Runge–Kutta_methods See, no parroting required. All that's needed is an ability to understand physical relationships and the equations that underly said relationships. That's how we gain an understanding of how the physical world works.
 
Originally Posted by alienator .

Here you go, Alfeng. There's no need to redo what is already done well, and Mark McM's model of bicycle, displayed at the Weight Weenies forum, is done well. In fact, Mark McM's model will actually overstate the moment of inertia for wheels, and it still shows that the resulting difference in acceleration at a given power output between two wheels of vastly different mass is still very small. Now, I know you don't trust science and you might claim something like, "that's not a real world model". Uhm, as stated his model will over state a wheel's MOI.

Mark McM's Model of the Influence of Rotating Mass in the Form of Wheels

By the way, the influence of crank arms, chain rings, pedals, and cassettes is much, much smaller.

Oh, if you haven't taken a differential equations course and passed it, you likely will have no idea what a fourth order Runge-Kutta method is, so here's some help for you:

http://en.wikipedia.org/wiki/Runge–Kutta_methods

See, no parroting required. All that's needed is an ability to understand physical relationships and the equations that underly said relationships. That's how we gain an understanding of how the physical world works.
Asking 100+ 1st Graders whether or not the World is flat doesn't mean that they will provide the correct answer.

Jiggering the data to "prove" Global Warming is a byproduct of CO2 in the atmosphere doesn't mean it is so ...

  • AND, FYI, the flaw in THAT pseudo-scientific finding which I immediately recognized when I heard [a PBS radio interview] that the "Keeling Curve" is (one of) the foundation(s) for Al Gore's argument is that Keeling's readings have been taken above-and-downwind from the most active volcano on Planet Earth!!! if a person can't see the flaw in that then s/he is truly one of the useful idiots
[*] other arguments fall short if the (primary?) premise is flawed

Simply citing incorrectly surmised calculations OR applications of irrelevant formulas in the search of validation of a pre-ordained conclusion IS mindless parroting ...

The bottom line is that the only formulation one needs is for "Work" (as you know-or-should-know, mass x distance x time x height) that is done and the rest is no more meaningful than the color of one's handlebar tape or streamers attached to one's handlebar grips (i.e., background noise) ...

  • REGARDLESS, while looking at isolated fragments can-or-could be interesting, they only take on meaning when they are accumulated over a meaningful time & distance to yield a meaningful assessment

Think about it, if you can.

  • BTW. I think that I know the incorrect retort you are probably going to make, but I'll let you make a fool of yourself, first.

---​

I used to have a 36" vertical leap. Really.

Does THAT mean that I can play (competitive) basketball?

Hardly.

I have what used to be considered a really nice bike which has excellent components which may still be better than 99% of the bikes which are currently being sold.

Does that mean I am qualified to be in the TdF?

Hardly.

Isolated this-or-that is little more than a lot hand waving ...

  • Again, the bottom line is that the only formulation one needs is for "Work" (as you know-or-should-know, mass x distance x time x height) that is done and the rest is no more meaningful than the color of one's handlebar tape or streamers attached to one's handlebar grips ([COLOR= rgb(255, 0, 0)]i.e.,[/COLOR] background noise) ...
 
You make no sense old man, and your rant about global warming is out of place as is expected for all red herrings.
 
Originally Posted by alienator .

You make no sense old man, and your rant about global warming is out of place as is expected for all red herrings.
Oh, Troll ....

THERE YOU GO AGAIN!

Deflecting ... and, with your usual lack of civility!

Based on your prior modus operandi when you are unable to respond in a meaningful way, I guess that your inability to consider the inconsequential nature of the Moment of Inertia AND your lack of willingness to comprehend OR perhaps your inability to comprehend that "Work" is the REAL consideration where the weight of wheels & tires is concerned then it was inevitable that you would offer yet another meaningless reply.

We know that Chris Juden's matrix is still too complex for you to fathom despite its straightforward presentation ...

SO, despite your allusions to various formulations which you have ferreted out which were performed by other people, should we suppose that the formula for "Work" is too complex for you, too?!?

BTW. The example of "Global Warming" was-and-is to illustrate that meaningless so-called scientific analysis is actually meaningless no matter how many people have been hoodwinked into believing it ...

  • I guess that prologue was too complex for you to comprehend ...
  • no matter ... YOU are not the only person reading this thread
  • and, I am sure that most of the other people reading this thread can comprehend why it was included
 
Actually, no Alfeng. You are wrong again. First, you have yet to refute the idea that MOI in terms of bicycles has very little effect, which was the point I made and a point I brought up with the OP just because enough people appear to live and die by the idea that wheel weight, crank weight, pedal weight, et al have significant influence on performance. Second you can analyze things with energy equations, equations of motion, or both. As it happens using the equation of motion is pretty damned convenient as it includes system losses, whereas calculating what you suggest--mass * distance * time * height--reveals nothing. You calculation gives a result with the following units: [m][t][d]2 . Granted that's an interesting choice, but it has noting to do with work/energy which has units of [m][d]2 / [t]2 . Calculating the actual work done (excluding extra work required by physiological inefficiencies) is a straightforward step by integrating the power function over time. You can just multiply power by time to get the work done, but that assumes a time invariant power function. You see, the work done is intrinsically related to the equation of motion. That relationship is shown in an equation well known to anyone who's taken a physics course or studied engineering:
1000
Here are some hints: the 'W' above stands for 'work'. The 'P' stands for 'power'. The 'F' stands for 'force', and the 'v' stands for velocity. Alas your point about global warming has no value since it's just your opinion. It certainly bears no relevance at all to the discussion at hand. As for comments about scientific analysis in general, they sadly show your science teachers failed you.
 
Originally Posted by alienator .

Actually, no Alfeng. You are wrong again. First, you have yet to refute the idea that MOI in terms of bicycles has very little effect, which was the point I made and a point I brought up with the OP just because enough people appear to live and die by the idea that wheel weight, crank weight, pedal weight, et al have significant influence on performance. Second you can analyze things with energy equations, equations of motion, or both. As it happens using the equation of motion is pretty damned convenient as it includes system losses, whereas calculating what you suggest--mass * distance * time * height--reveals nothing. You calculation gives a result with the following units: [m][t][d]2 . Granted that's an interesting choice, but it has noting to do with work/energy which has units of [m][d]2 / [t]2 . Calculating the actual work done (excluding extra work required by physiological inefficiencies) is a straightforward step by integrating the power function over time. You can just multiply power by time to get the work done, but that assumes a time invariant power function. You see, the work done is intrinsically related to the equation of motion. That relationship is shown in an equation well known to anyone who's taken a physics course or studied engineering:





Here are some hints: the 'W' above stands for 'work'. The 'P' stands for 'power'. The 'F' stands for 'force', and the 'v' stands for velocity.

Alas your point about global warming has no value since it's just your opinion. It certainly bears no relevance at all to the discussion at hand. As for comments about scientific analysis in general, they sadly show your science teachers failed you.
Oh, Troll ...

Look at YOU!!!

Again, demonstrating that you don't know the meaning of what you are clipping & pasting ...

BECAUSE, it IS well known to anyone who has taken a Physics course that 'P' stands for MOMENTUM.
 
This is really becoming unfair, alfeng, and I'm starting to feel bad for you. Let me direct your attention to page 161 of Physics For Scientists And Engineers, Second Edition, by Fishbane, Gasiorowicz, and Thornton:
1000
Read the first few lines of section 6-5, Power. What is the variable for power? Note that the variable for momentum is usually a lower case p. For example, in Louis de Broglie's equation for the wavelength of a matter wave, λ=h/p, lambda is the wavelength; h is Planck's constant; and p is the momentum of the matter in question. Lest you think that use of the lowercase p is one of the peculiarities of quantum physics nomenclature, let me direct you to p. 200 of the same text as mentioned above as seen below. Kindly read lines two and three about linear momentum. For further reference, check out equation 8-2, p=mv. Now, what is the variable for linear momentum?
AppleMark
Now, back to that whole power thing. Note equation 6-23 in the first image above: P = dW/dt, which states that power is the first derivative with respect to time of work. If you integrate both sides with respect to time you get: ∫ Pdt = W. Hmmm. Now, that looks familiar, doesn't it? I've got several other texts on general Physics and classical mechanics if you need further reference.
 
Quote: Originally Posted by alfeng .

Oh, Troll ...

Look at YOU!!!

Again, demonstrating that you don't know the meaning of what you are clipping & pasting ...

BECAUSE, it IS well known to anyone who has taken a Physics course that 'P' stands for MOMENTUM.

alfeng, this one takes the cake. I've know for a while now that you have no understanding of math or physics, and you've again demonstrated it in spades. From basic algebra, you should have learned that in algebraic formulas, letters are defined to mean whatever you want them to mean. If alienator's equations had used K for power, or a greek letter, they would be equally valid, as long as he defines what they mean. Sure, for the universal concepts as he displayed above, there are conventional letters used, and that's what is used.

Besides, I can assure you that I've taken a physics course or two, and it's not known to me at all the P stands for Momentum. Not sure where you came up with that one. Besides, do really you think Alienator hasn't taken a HS or freshman Physics course....and passed it?

What he clipped out is nothing more than the definition of work, written differential-equation form. If the P (power) is constant, then the work performed (in joules) is just the product of the power (watts, or joules/second) times the interval in seconds from t (0) to t (1). EG, if I ride at a steady 100 joules/sec (watts) power for 3600 seconds (1 hr), I'll have done 360,000 joules of work, or as usually written, 360 kj. You should really try to understand this much.

There is no need to feel inferior or defensive just because you didn't have to master this stuff in school or in your work. Lots of people don't understand math and physics, and that doesn't mean they are inferior or without talent. I have a sister who is a professional classical musician with a fine career and loads of intelligence, but if I asked her to define "Work", see would no more be able to recite back "Force times Distance" than I could sit in front of the orchestra and play a piano concerto.
 
Quote: Originally Posted by alienator .

This is really becoming unfair, alfeng, and I'm starting to feel bad for you.

Let me direct your attention to page 161 of Physics For Scientists And Engineers, Second Edition, by Fishbane, Gasiorowicz, and Thornton:

--

Read the first few lines of section 6-5, Power. What is the variable for power? Note that the variable for momentum is usually a lower case p. For example, in Louis de Broglie's equation for the wavelength of a matter wave, λ=h/p, lambda is the wavelength; h is Planck's constant; and p is the momentum of the matter in question. Lest you think that use of the lowercase p is one of the peculiarities of quantum physics nomenclature, let me direct you to p. 200 of the same text as mentioned above as seen below. Kindly read lines two and three about linear momentum. For further reference, check out equation 8-2, p=mv. Now, what is the variable for linear momentum?

--

Now, back to that whole power thing. Note equation 6-23 in the first image above: P = dW/dt, which states that power is the first derivative with respect to time of work. If you integrate both sides with respect to time you get:

∫ Pdt = W. Hmmm. Now, that looks familiar, doesn't it?

I've got several other texts on general Physics and classical mechanics if you need further reference.


Well, yes ...

Life is sometimes unfair ...

Otherwise, you wouldn't be wrong about the potential significance of wheel weight ... let ME qualify that by saying wheel weight matters when ascending ...

BUT, yes, I am indeed old AND it has been over 40 years since I took a course in Physics (yes, believe it or not, I did pass it) ... so, since I do not have my text book any longer, I have to accept that YOU are correct that I am in error that 'p' (lower case) is the accepted abbreviation for Momentum and that 'P' (as I understand the situation) may be acceptable if it is part of a slightly more complex formulation (e.g., P0 = P1 + P2) for clarity where 'p1' presumably lacks visual clarity.

You also 'win' because the formulation which you cited unintentionally & effectively shifted the thrust of this thread to an exchange on the notation for Momentum rather than the issue of bike weight (which, as I have stated numerous times, I agree is not particularly relevant unless a person lives in a third floor walk up apartment).

Wheel weight, however CAN make a difference ...

Now, before explaining WHY, my recollection is that the current conventional wisdom (which I think you echoed at least once in the past) is essentially summed up by saying that what-goes-up-comes-down ...

Is THAT a fair summation?

Certainly, THAT may be true when riding on a perfectly level surface ...

BUT, it is not true that wheel weight doesn't matter when ascending ...

  • begin by visualizing an upright wheel where the valve stem (an arbitrary point on the wheel for this example) starts at 6 o'clock
  • as the wheel rotates & the valve stem will move up to the 12 o'clock position the "Work" done can be estimated using the effective diameter of the wheel
  • BUT, when it returns to the 6 o'clock position it has not descended to the effective diameter of the wheel ​ if the slope is 45º, then the effective diameter is approximately halved
  • so, in THAT instance, the amount of the "Work" done per 360º rotation exists & is not offset as others have suggested in the past by those promulgating that the wheel weight does not matter
[*] so, even a slope of 0.0001º will result in measurable "Work" because a measurable vertical displacement will occur

[*] and, the weight of the wheel will affect the calculable amount


IF THIS EXAMPLE NEEDS EXPANSION FOR CLARITY, THEN LET ME KNOW, BECAUSE I AM MORE THAN WILLING REPEAT IT WITH MORE DETAIL

It doesn't matter how complex or simple the formulation for "Work" is that one uses ... some measurable "Work" will occur as the rear wheel rotates which is greater than if a sled-or-wagon-or-bike where being pushed-or-pulled up the same incline AND SO the weight of a rear wheel will matter to some extent.
 
Quote: Originally Posted by dhk2 .
alfeng, this one takes the cake. I've know for a while now that you have no understanding of math or physics, and you've again demonstrated it in spades. From basic algebra, you should have learned that in algebraic formulas, letters are defined to mean whatever you want them to mean. If alienator's equations had used K for power, or a greek letter, they would be equally valid, as long as he defines what they mean. Sure, for the universal concepts as he displayed above, there are conventional letters used, and that's what is used.

Besides, I can assure you that I've taken a physics course or two, and it's not known to me at all the P stands for Momentum. Not sure where you came up with that one. Besides, do really you think Alienator hasn't taken a HS or freshman Physics course....and passed it?

What he clipped out is nothing more than the definition of work, written differential-equation form. If the P (power) is constant, then the work performed (in joules) is just the product of the power (watts, or joules/second) times the interval in seconds from t (0) to t (1). EG, if I ride at a steady 100 joules/sec (watts) power for 3600 seconds (1 hr), I'll have done 360,000 joules of work, or as usually written, 360 kj. You should really try to understand this much.

There is no need to feel inferior or defensive just because you didn't have to master this stuff in school or in your work. Lots of people don't understand math and physics, and that doesn't mean they are inferior or without talent. I have a sister who is a professional classical musician with a fine career and loads of intelligence, but if I asked her to define "Work", see would no more be able to recite back "Force times Distance" than I could sit in front of the orchestra and play a piano concerto.


WHOA!

A very interesting post ...

How long have you harbored these thoughts?!?

Really, while I am mildly embarrassed for not remembering 'p' (lower case) is the correct abbreviation for "Momentum" I do not believe that I been "defensive" in regard to this topic ...

BTW. I now take it that YOU, too, cannot fathom Chris Juden's matrix ...

  • maybe, both you & alienator can put your minds together to figure it out

FYI. Apparently, 'p' (lower case as pointed out alienator) has indeed been used to represent "Momentum" for Centuries vs. MY MISTAKEN remembrance that 'P' (Upper Case) was the abbreviation for "Momentum" ...

Close, but no cigar!

But, I seem to recall that you didn't believe that "dwell" was a valid concept (word choice?) for the phenomenon which I declared was the reason for the dodgy shifting which Shimano shifters experienced when downshifting while the drivetrain was under a load (e.g., going uphill) even though the www.parktool.com site uses that term, too ...

  • WTF?!? Where did they get it?
[*] No, I do not think they got it from me or from my use of the term "dwell" in this Forum (but, who knows?) nor do I recall getting it from them ...
  • someone whom I cannot give attribution to used the term "dwell" before I did
My recollection is that you claimed that the only time YOU experienced a problem with your Dura Ace shifters was when you were fatigued ... but, THAT was-and-is the point!

It is very difficult to execute mis-shifts with Campagnolo shifters regardless of the circumstances ...

  • that is, regardless of the rider's skill-or-lack-of, Campagnolo shifters are not handicapped by the "dwell" which has been engineered into Shimano's shifters which can result in dodgy shifting and, it is the ease with which Camapgnolo shifters engage the Cog or Chainring EVEN WHEN mated to non-Campagnolo components that make them superior
  • that's not just an opinion but also a fact which can be easily reproduced by ANYONE plug-and-play (vs. a utilizing a "Rube Goldberg" configuration) is the operative, contemporary phrase that can be applied to incorporating Campagnolo shifters with an otherwise Shimano drivetrain
  • of course, some people's aesthetic sensibilities are too adversely affected to mix-and-match components THAT's certainly the success of marketing!

For anyone who may think that the preceding discourse on "dwell" is OT, it is not.

The above discourse is intended to illustrate that any belief that I am fabricating labels which suit my fancy for accepted concepts is an error.

 
Of course wheel weight can make a difference when climbing, but the work done rotating a wheel is very small compared to the work done lifting the mass of the bike and rider. The work done to rotate a wheel is W=∫Iαdθ, where I is the MOI, α is the angular acceleration, and dθ is the incremental change in the angle through which a wheel is rotated. If you look at Mark McM's equation and his results, again, you'll see that the difference in acceleration between wheels of vastly different mass (wheel masses were 0kg, 5kg, and 10kg) is very small. If the difference in linear acceleration is very small, then the difference in angular acceleration is very small. In fact, angular acceleration is even smaller given this definition: α=a/r, again where α is the angular acceleration, a is the linear acceleration, and r is the radius of the wheel. So in the expression for work done to accelerate a wheel, we've now found that not only is I small, but α is very small, too. Well, that means the work done accelerating the wheel is very small. So of all the work done riding up a hill, the vast majority of the work done is just done lifting all the mass which supports what I initially said, which is that MOI is overblown in terms of how it influences bike performance. The real benefit to a light wheel set is that there is just less mass to lift up. Now if you want to argue that MOI might make the difference in a photo finish at a race, that might be a worthy argument, but then you also have to consider other small differences, like the differences the pavement over which each rider passed or the very small differences in aerodynamics due to outside influences (i.e. not related to the bike or rider). You should also note what I originally said to the OP:
me said:
Some will argue that a given amount of weight off rotating parts (wheels, cranks, pedals) has more influence than weight off of non-rotating bits (like the frame, the rider, and etc), but the reality is that the difference in performance between those two options is very small.
Note that I referred to the difference in influence of mass on rotating parts vs non rotating parts. As established, the effect of a change of mass on rotating parts on a bike is very small (even smaller on cranks, pedals, cassettes.......than on wheels), so the dominant influence of mass anywhere on a bike is as mass in linear motion, not in rotational motion. Lastly, P is just power. It doesn't matter if there's only P0 or if there's also P1 , P2 , or Pi . Also a necessary skill when studying science, especially when that study becomes a major and/or leads to graduate work and beyond, is the ability to interpret equations correctly despite the variables used.