bicycle physics
Charles Coldwell
There's been some off-list correspondence from people who would like a
more complete explanation of bicycle physics, so I thought I would post
something longer to try to explain it.
First of all, if you are really interesting in the subject, the best book
I can recommend is "Bicycling Science" by David Gordon Wilson, now in it's
third edition. He goes into some more esoteric subjects like how
balancing a bike works, examines some of the physiology of human power
generation, aerodynamics, transmission systems, etc. Really a great book.
I'm going to limit myself to some basic physics.
First: force, work, energy and power. Force is what you apply to the
pedals. It is measured in Newtons (metric) or pounds (imperial). Usually
you apply a force on the pedals that is less than or equal to your weight.
If you measure your mass in kilograms, you will have to multiply it by the
acceleration due to gravity (9.8 m/(s*s) at sea level) to convert it to a
force in Newtons. If you measure your weight in pounds, you're done with
the force calculation.
Work and energy are the same thing, and are measured in Joules (metric) or
calories (imperial, also sometimes BTU or foot-pound). Work is force
times distance (hence, "foot-pound"). Lifting the force of your 200 pound
weight (bicycle plus rider) up a 1,000 ft climb does an amount of work
equal to 200,000 ft-pounds or 271,200 Joules or 64,780 calories. The work
you did climbing the hill is readily transformed into the energy coming
down -- they are the same thing. Energy is conserved, it can neither be
created nor destroyed, although using it against wind resistance to heat
the atmosphere is the practical equivalent of destroying it since you just
increase the entropy of the universe and never get anything back for your
trouble. Using energy to climb a hill, on the other hand, does not
destroy it. The calorie content quoted on food labels is actually
kilocalories, so your 1,000 foot climb cost you about a third of a
PowerBar.
Power is measured in Watts (metric) or horsepower (imperial). Power is
force times velocity, or equivalently, energy per unit time. Since energy
is force times distance, energy divided by time is (force times (distance
divided by time)) which is force times velocity. A fit cyclist cruises at
120 - 200 Watts. A Watt is a Joule per second. A horsepower is 746
Watts. We are quarter-horse motors.
The power developed by a cyclist, then, is the total force of all the
resistances he is working against (wind, gravity, friction) multiplied by
his speed. Wind and friction are hard to quantify. Wind resistance
depends on your cross-section to the wind, the direction it is blowing,
etc. Rolling resistance depends on your components and how recently you
lubricated them, but is mostly negligible for speeds above a few MPH.
Gravity is easy to quantify, see the example in the paragraph above.
Let's neglect rolling resistance since we hope we're never moving that
slow long enough for it to matter much.
We can say a few useful things about wind. First, the force due to wind
resistance grows proportional to the velocity squared. If you don't
shift, you have to put four times as much force on the pedals to maintain
the same speed when the wind doubles. Alternatively, if you don't shift
and the wind stays constant, you have to apply four times as much force on
the pedals to go twice as fast (what matters is your velocity relative to
the wind).
However, you could always downshift and spin faster. If you double your
cadence in a gear that is half the size, you go at the same speed but
apply half the force. That means that the force isn't really the quantity
that is of interest -- by choosing gear and cadence we can vary the force
at will while keeping the speed constant.
The quantity of interest is the power, which is the product of force times
velocity. If you like, you can measure it at the crank. The torque on the
bottom bracket that equals the force on the pedal times the crank length.
Power measured at the crank is that torque times cadence.
Note that if you downshift and increase your cadence to keep your speed
constant, your power remains constant. So whether you are a masher or a
spinner doesn't really matter, you can get the same speed either way.
If you follow it through the drivetrain, that power that you computed as
the torque times the cadence just becomes the speed of the bicycle times
the resisting force.
On level ground the dominant resisting force is the wind, and grows as the
velocity squared. But we've seen that the resisting force is not the
quantity of interest, the power which is that force times our velocity is.
So there's one more factor of velocity giving power proportional to
velocity cubed. That's bad news: develop twice as much power on level
ground and your speed goes up by a mere 26%.
On hills, the dominant resisting force is your weight, and doesn't depend
on the velocity at all. So if the power (force times velocity) doubles,
but the force remains constant, then the velocity must double. That's
good news: develop twice as much power on hills and you'll go twice as
fast.
This is basically the point that Jan Heine was making in his post.
Quoting:
> Going from 120W to 200W on a 5% uphill, your speed increases by 60%. On
> the flats, the same power increase makes you only 23% faster. On a steep
> 7% downhill, you gain about 3%. On that downhill, you actually are
> faster in an aero tuck than pedaling at 200W. The reason is air
> resistance - the faster you go, the more power you need for every mph of
> additional speed increase.
Going from 120W to 200W is a power increase of 67%, therefore my model
predicts a speed increase of 67%, Jan Heine quotes 60%. That's the same
ballpark -- it means that wind resistance is not completely negligible
when Jan climbs Washington Pass.
On the flats where wind resistance dominates, the 67% power increase
should make you 19% faster according to my model (the cube root of 1.67 is
about 1.186), Jan Heine claims 23%. We're still in the same ballpark.
But the news about uphills is even better than that. The energy you use
to climb is independent of the speed at which you climb (but the power
required is proportional to the speed at which you climb), so you won't
have to choke down any more PowerBars for going up the hill at 12 MPH than
for going up the hill at 10 MPH. But as I pointed out in a previous post,
if you go up one mile at 12 MPH (takes 5 minutes) and then down it at 30
MPH (takes 2 minutes), the elapsed time of 7 minutes is still less than if
you went up at 10 MPH (takes 6 minutes) and then down it at 50 MPH (takes
1:12, for a total of 7:12). Kent Peterson and Nick Bull chimed in on this
point -- I especially liked Nick's proof that your overall average speed
on an up-and-down trip is never more than twice your average speed going
up.
The important point is that if you want to raise your overall average
speed the best way to do it is to raise your lowest speed. And if your
lowest speed is going uphill, the good news is it won't cost you any more
energy to go up it faster. It will require you to develop more power, but
if you develop twice as much power you get to the top in half the time so
the energy consumed is the same.
Getting back to PBP: my experience in 2003 was that it is everywhere hilly
and nowhere steep. Except for a short stretch just outside the San
Quentin en Yvelines, you are on perpetual rollers. If you want to finish
well, train for hills. If you don't live in hilly country, don't worry:
training for hills is exactly the same thing as training for power. Try
to improve your power output, whatever your favorite cadence might be.
I apologize for the long post. I hope it was interesting for some of you.
Chip
--
Charles M. Coldwell
Somerville, Massachusetts, New England
Charles Coldwell
>
> We can say a few useful things about wind. First, the force due to wind
> resistance grows proportional to the velocity squared. If you don't
> shift, you have to put four times as much force on the pedals to maintain
> the same speed when the wind doubles. Alternatively, if you don't shift
> and the wind stays constant, you have to apply four times as much force on
> the pedals to go twice as fast (what matters is your velocity relative to
> the wind).
Sigh. You'd think I'd get it right after all this posting. The last
sentence above should read:
"Alternatively, if you shift to maintain a constant cadence and the wind
stays constant, you have to apply four times as much force on the pedals
to go twice as fast."
I think that corresponds to most people's experience on the bike.
Sorry for all the noise,
Charles Coldwell
>
> On Tue, 3 Apr 2007, Charles Coldwell wrote:
> >
> > We can say a few useful things about wind. First, the force due to wind
> > resistance grows proportional to the velocity squared. If you don't
> > shift, you have to put four times as much force on the pedals to maintain
> > the same speed when the wind doubles. Alternatively, if you don't shift
> > and the wind stays constant, you have to apply four times as much force on
> > the pedals to go twice as fast (what matters is your velocity relative to
> > the wind).
>
> Sigh. You'd think I'd get it right after all this posting. The last
> sentence above should read:
>
> "Alternatively, if you shift to maintain a constant cadence and the wind
> stays constant, you have to apply four times as much force on the pedals
> to go twice as fast."
I take it back. I had it right the first time.
Dave Cramer
I was a physics major in college, and met Lennard Zinn my freshman
year--he worked for the physics department for a year after he
graduated. He built his first bike in the back of the lab...
Dave "I'd ride faster if my stem didn't flex so much" C.
:)
Emily O'Brien
I particularly like playing around with the gear charts, but it's got lots of great stuff!
Emily
> We can say a few useful things about wind. First, the force due to wind
> resistance grows proportional to the velocity squared. If you don't
> shift, you have to put four times as much force on the pedals to maintain
> the same speed when the wind doubles. Alternatively, if you don't shift
> and the wind stays constant, you have to apply four times as much force on
> the pedals to go twice as fast (what matters is your velocity relative to
> the wind).
>
Dave Cramer
L'Alpe D'Huez, and then calculating the time difference for adding
some huge amount of weight to the bike, like 5kg. I'm always amazed at
how little difference it makes.
Dave