# Climbing and power to weight ratios.

#### Eldrack

##### New Member
Ok, first to set up an illustration of what I am trying to get at.

We have four riders, of weights 55, 60, 65 and 70kg. They each have a bike weighing 7kg (around the UCI minimum weight) to give total masses of 62, 67, 72, 77 kilograms. Now we find some hills with constant gradients of 3,6,9,12 and 15% and send them up, each maintaining the same speed. They will therefore be putting out different amounts of power and have different power to weight ratio's.

Sticking some numbers into analytic cycling gives a rough idea of how this works but the results are simple. For a given speed up a given grade a heavier rider needs to put out less watts per kilo to hold the pace. The absolute and percentage differences in the watts/kg decreases as the gradient gets steeper but the fact remains that a 70kg rider who can generate 6.0 watts/kg will always beat a 60kg rider who can generate 6.0 watts/kg (well, at least in the theoretical model).

This is at odds with the general perception which is that lighter riders make better climbers. So, what in essence I am asking is, is it easier for someone with a lower bodyweight to hit higher watts per kilo?

#### Steve_B

##### New Member
Eldrack said:
For a given speed up a given grade a heavier rider needs to put out less watts per kilo to hold the pace.
Ummm...no. Neglecting things like tire rolling resistance, air drag, and all that, climbing speed is goverened largely by Watts/kg. For given speed, a heavier rider will need to achieve the same Watts/kilogram to hold the same pace as a lighter rider. For a given % grade, there is a Watts/kg to match a particular speed. Think of Watts/kg and speed as interchangable (sort of) in this context. So a heavier rider has to put out more power to make up for the fact that his weight is higher and thus balance the fraction to the same value as the lighter rider.

Eldrack said:
The absolute and percentage differences in the watts/kg decreases as the gradient gets steeper but the fact remains that a 70kg rider who can generate 6.0 watts/kg will always beat a 60kg rider who can generate 6.0 watts/kg (well, at least in the theoretical model).
No. See above.

Eldrack said:
This is at odds with the general perception which is that lighter riders make better climbers. So, what in essence I am asking is, is it easier for someone with a lower bodyweight to hit higher watts per kilo?
Because the body's ability to gain power does not linearly track with the weight gain necessary to put out more power. Said in English: If a rider gains muscle and gains power, the power they gain probably will not offset the gain in weight (on a % basis).A heavier rider maybe more powerful but it is usually is not enough to tip the Watts/kilo equation in their favor.It's just the way it works out in nature.

I'm not sure what you were plugging into to acc.com to come up with this.... Time for bed. Going to race a FLAT crit tomorrow.... #### swampy1970

##### Well-Known Member
Eldrack said:
Ok, first to set up an illustration of what I am trying to get at.

We have four riders, of weights 55, 60, 65 and 70kg. They each have a bike weighing 7kg (around the UCI minimum weight) to give total masses of 62, 67, 72, 77 kilograms. Now we find some hills with constant gradients of 3,6,9,12 and 15% and send them up, each maintaining the same speed. They will therefore be putting out different amounts of power and have different power to weight ratio's.

Sticking some numbers into analytic cycling gives a rough idea of how this works but the results are simple. For a given speed up a given grade a heavier rider needs to put out less watts per kilo to hold the pace. The absolute and percentage differences in the watts/kg decreases as the gradient gets steeper but the fact remains that a 70kg rider who can generate 6.0 watts/kg will always beat a 60kg rider who can generate 6.0 watts/kg (well, at least in the theoretical model).

This is at odds with the general perception which is that lighter riders make better climbers. So, what in essence I am asking is, is it easier for someone with a lower bodyweight to hit higher watts per kilo?
Unless your name in Indurain or Armstrong then the domain of the mountains is that of the little people - the skinny sub 140lb folk. As the slopes hit 8% or steeper then weight, or lack thereof, becomes even more of a premium. At 8% you pretty much need to add 1 watt for every 1lb of additional weight. So if your buddy is 30lbs lighter and can pop out 370watts for that 20 minute climb be prepared for a world of hurt....

You need to also take into account the micro accelerations that occour every pedal stroke on the steeper grades which further work against the heavier rider.

I used to be one of those folk that were darned skinny. For the most part I was 145 but got down to 140 for the end of year hill climbs. Now I'm a few bikes heavier than that.... but that said, two years ago you could also tack on the weight of another few bikes too! I know how much harder extra weight makes things - now I feel kinda bad for giving my friends so much sh*t all those years ago about just needing to "push 'arder on t' pedals lad" - as they say in Northern England.

#### Alex Simmons

##### Active Member
Eldrack said:
Ok, first to set up an illustration of what I am trying to get at.

We have four riders, of weights 55, 60, 65 and 70kg. They each have a bike weighing 7kg (around the UCI minimum weight) to give total masses of 62, 67, 72, 77 kilograms. Now we find some hills with constant gradients of 3,6,9,12 and 15% and send them up, each maintaining the same speed. They will therefore be putting out different amounts of power and have different power to weight ratio's.

Sticking some numbers into analytic cycling gives a rough idea of how this works but the results are simple. For a given speed up a given grade a heavier rider needs to put out less watts per kilo to hold the pace. The absolute and percentage differences in the watts/kg decreases as the gradient gets steeper but the fact remains that a 70kg rider who can generate 6.0 watts/kg will always beat a 60kg rider who can generate 6.0 watts/kg (well, at least in the theoretical model).

This is at odds with the general perception which is that lighter riders make better climbers. So, what in essence I am asking is, is it easier for someone with a lower bodyweight to hit higher watts per kilo?
That's because, while the impact reduces as gradients get steeper, you haven't accounted for the fact that riders of different sizes/weights will also have a different CdA (and also slightly different rolling resistance).

You have assumed that riders varying in mass from 50kg to 70kg will have the same CdA. That just ain't the case. Put in an increasing CdA as the rider gets larger and you'll see the figures even out somewhat. Even though gravity is the major force to overcome when the road tilts upwards, it doesn't mean air and rolling resistance disappear entirely.

If you have two riders, one of 50kg and another of 70kg who have the same CdA and rolling resistance (a very unlikely scenario) then yes, what you observe would be the case.

#### Eldrack

##### New Member
Mass is proportional to volume, so CdA is proportional to the cube root of your mass. I took account of the different in rolling resistance, but remember you have to take into account the weight of the bike, which is independant of rider mass.

Variables:
v = rider velocity
d = coefficient of drag = am^(1/3)
g = gravity
b = bike mass
m = rider mass
r = coefficient of rolling resistance

Forces on the rider F, using small angle approximation:

F = dv^2 + r(m+b) + (0.01s)g(m+b)
P = FV = dv^3 + r(m+b)v + (0.01s)gv(m+b)

so the power to weight ratio P/m:

P/m = rv + (0.01s)gv + av^3/(m^(2/3)) + (1/m)(brv + (0.01s)gvb)

So, the first two terms, power form rolling resistance due to rider mass and power from overcoming gravity due to rider mass give a constant p/m. However the drag (which is negligable at low speeds (i.e steep gradients)) has a mass dependance (one over the mass to the power 2/3) and the terms due to the mass of the bike both have a rider mass dependance in the power/weight ratio equation.

The cost of pulling your bike up a climb is not insignificant, after all the thing will weigh 7kg or more. In fact for a 70kg rider and in the approximation that both rolling resistance and wind resistance are small it accounts for 10% of your power output. This I believe is the origin of the discrepancy I illustrated in my first post.

Edit: I must say I didn't believe this result when I first found it. I was expecting that over a certain gradient the lighter rider had the advantage. When this proved not to be the case I sat down and worked out the physics (well, I am a physicist so that's what I do) and was quite surprised with the result myself.

#### Alex Simmons

##### Active Member
Eldrack said:
The cost of pulling your bike up a climb is not insignificant, after all the thing will weigh 7kg or more. In fact for a 70kg rider and in the approximation that both rolling resistance and wind resistance are small it accounts for 10% of your power output. This I believe is the origin of the discrepancy I illustrated in my first post.
Correct - I missed that bit , mass of bike as proportion of overall mass is much greater for lighter riders, although at the lower level gradients, the other factors still play their part.

#### Alex Simmons

##### Active Member
Eldrack said:
Mass is proportional to volume, so CdA is proportional to the cube root of your mass. I took account of the different in rolling resistance, but remember you have to take into account the weight of the bike, which is independant of rider mass.

Variables:
v = rider velocity
d = coefficient of drag = am^(1/3)
g = gravity
b = bike mass
m = rider mass
r = coefficient of rolling resistance

Forces on the rider F, using small angle approximation:

F = dv^2 + r(m+b) + (0.01s)g(m+b)
P = FV = dv^3 + r(m+b)v + (0.01s)gv(m+b)

so the power to weight ratio P/m:

P/m = rv + (0.01s)gv + av^3/(m^(2/3)) + (1/m)(brv + (0.01s)gvb)

So, the first two terms, power form rolling resistance due to rider mass and power from overcoming gravity due to rider mass give a constant p/m. However the drag (which is negligable at low speeds (i.e steep gradients)) has a mass dependance (one over the mass to the power 2/3) and the terms due to the mass of the bike both have a rider mass dependance in the power/weight ratio equation.

The cost of pulling your bike up a climb is not insignificant, after all the thing will weigh 7kg or more. In fact for a 70kg rider and in the approximation that both rolling resistance and wind resistance are small it accounts for 10% of your power output. This I believe is the origin of the discrepancy I illustrated in my first post.

Edit: I must say I didn't believe this result when I first found it. I was expecting that over a certain gradient the lighter rider had the advantage. When this proved not to be the case I sat down and worked out the physics (well, I am a physicist so that's what I do) and was quite surprised with the result myself.
Not sure of your assumption about mass/volume and CdA. More to it than mass/volume alone. Actually I prefere this version: #### Eldrack

##### New Member
That looks like a fairly full analysis there. I assumed constant speed (so no acceleration term) and neglected the bearings in the wheel. In the end the result is basically the same. Thanks to the weight of the bike the heavier riders need less watts per kilo to go up a given gradient at the same speed as lighter rider. So back to the original question:

Is it easier for a lighter rider to put out higher watts per kilo? This question would fall in the realms of physiology rather than physics, so I'm completely clueless as to an explanation here...

#### acoggan

##### Member
Eldrack said:
Is it easier for a lighter rider to put out higher watts per kilo?

On average, yes. From a theoretical perspective, VO2max, and hence sustainable power all else being equal, should scale with mass^2/3. The exact exponent derived in cross-sectional experiments tends to be a bit larger than that, but is still significantly less than one.

#### gvanwagner

##### New Member
Also, a smaller rider also should produce a lot less heat and also be better able to deal with it, which can be a big factor as well when it comes to " can a smaller rider produce a better w/kg.

#### dhk2

##### Well-Known Member
acoggan said:
On average, yes. From a theoretical perspective, VO2max, and hence sustainable power all else being equal, should scale with mass^2/3. The exact exponent derived in cross-sectional experiments tends to be a bit larger than that, but is still significantly less than one.
Mass ^2/3 is indeed a significant scale factor. That means that a 84 kg rider (who is 20% heavier than a 70 kg rider) would on average produce only 13% more power.

Even worse off would be the 84 kg rider who should weigh 78 kg......wouldn't know anyone like that of course #### 11ring

##### New Member
Why does power to weight scale in this manner- what stops power scaling linear to mass.

acoggan said:
On average, yes. From a theoretical perspective, VO2max, and hence sustainable power all else being equal, should scale with mass^2/3. The exact exponent derived in cross-sectional experiments tends to be a bit larger than that, but is still significantly less than one.

#### frenchyge

##### New Member
11ring said:
what stops power scaling linear to mass.
Since only some of the mass present contributes to pushing the bike forward and other mass is simply along for the ride, I don't think it'd be reasonable to expect bicycle power to scale linearly with mass at *any* point.