Does rotational weight matter on a bike

My last post sparked some interesting discussion among some of my cycling friends: Not all weight is equal. The consensus was that weight on the wheels was more important (2x-7x more important) than weight of the frame.  I'm having a hard time buying this... but wanted to clearly write out the arguments here.

Acceleration and Rotational Inertia

Work is the change in kinetic energy (W=\Delta K). For a mass on your frame or wheels, there is work done when you accelerate from rest to your riding velocity (we'll say here 15 mph). We'll stick with the 10 grams from the previous post, and assume our riding velocity is around 15 mph.  That means for both wheels and frame, the work done to accelerate 10 grams to 15 mph is given by:

W=\frac{1}{2}mv^2

This ends up being around 5.4\times 10^{-5} Calories. The wheels have an additional component of work being done... they also have rotational energy:

k_{rot}=\frac{1}{2}I*\omega^2.

here, we'll take the radius of the wheel to be 350 mm*. This gives us a rotational velocity \omega=\frac{v}{2\pi r}\approx 3 \frac{rev}{sec}.  The rotational inertia is given by I=mr^2.  If we plug in both I and \omega, the radius ends up canceling out and we're left with:

k_{rot}=\frac{1}{2}m*(\frac{v}{2\pi})^2.

so the work done to bring that 10 gram wheel spinning up to 15 mph is an additional 1.4\times 10^{-6} Calories! This means it takes about 2.5% more energy to accelerate a mass on your wheel to 15 mph than it does on your frame.

Let's pause and think about this for a second...

  1. Wheel Radius doesn't matter. 29ers get a bad wrap for being slow accelerators. It's clear from a physics argument this acceleration penalty isn't due to the larger diameter of the wheels.  While the 29" wheels do have higher rotational inertia, they also spin slower and the effect cancels out. (I'd buy an argument that maybe 29" wheels are less stiff than 26" wheels; or, by default, 29" wheels are heavier than 26" wheels...)
  2. How often do you accelerate? While acceleration might be important if you're racing matched doubles on a track, trying to break away from a peloton, or win a sprint finish... it really doesn't seem that important for the casual rider. This is especially true on a road bike ride, where you're riding a long distance at a relatively constant velocity...

Turning and Stability

The other time I can see wheel mass to matter more than frame mass is in cornering. Bicycle dynamics and stability are a pretty complicated matter, but we can think about a few things:

  1. Conservation of Angular Momentum When cornering, you want to lean your bike into the turn. You do this for two reasons. First, if you lean a spinning wheel over, it will cause rotation to conserve angular momentum. Second, turning the
  2. Work done by steering the work done when turning a wheel is given by
    W=\tau \Delta \theta

    where turning our wheel is given by the change in angular momentum (L):

    \tau=\frac{dL}{dt}

    and L=Iw

*700c rims are normally 622mm in diameter. the actual diameter of the wheel depends on the tire size. 350 mm seems like a good round guess.