In short: It's not! Any extra rotating weight (or reduction of rotating weight) is, at the most, twice as important as non-rotating weight. And even then, this factor only applies to situations when you're accelerating.
The first piece of bike-related engineering analysis I ever did was back in 1994. I wanted to include it here, for posterity.
I had just started an internship, during the 3rd year of my Engineering degree. I was doing my internship at a British company called the DRA (the Defence Research Agency). This was a government funded organisation that performed aeronautical and other defence-related research on behalf of the UK government and industry. It was the UK's equivalent of Germany's DLR, France's ONERA, or America's NASA.
I wasn't into cycling much at this time, but I shared an office with a guy called Gerry, who was himself a keen cyclist in his 30s. Among the many conversations that we had, one time he got onto the subject of rotating weight on bicycles. He explained that any weight savings on rotating components, such as wheels and cranksets were "much much more important" than weight savings elsewhere on the bike.
Many people will recognise this is an assertion that cycling enthusiasts been pushing for many years. Thankfully this myth has now been largely debunked. However, back in 1994, upon hearing this claim I had no access to the internet or any means to check or disprove what Gerry was telling me.
I was suspicious about his assertion, though, it didn't feel right. So I decided to look at the equations governing rotating weight and energy, in order to decide for myself how important rotating weight is. I don’t have my notes from back then, unfortunately, but I’ve recreated them below.
The basis for this assertion about rotating weight importance is that for non-rotating components, any additional mass is only subject to linear (translational) acceleration, when the bike/rider system is accelerated. On the other hand, any additional mass on a rotating component, such as a wheel, is subject to both linear acceleration and rotational acceleration. Proponents of the “rotating weight is important” idea believe that the rotational energy needed to 'spin up' this additional mass is significant, several times more significant than the energy needed to linearly accelerate and create the purely translational kinetic energy of a non-rotating component.
We need to examine this idea of additional rotating mass having a significant amount of rotational energy, relative to a non rotating mass, which has only linear kinetic energy when it is accelerated.
The equations below show how the two scenarios compare: How the total energy compares between a situation where the additional mass is rotating, compared with a situation where the additional mass is non-rotating.
Equation #1 below describes the reference situation, defining the total energy of a bike and rider that has accelerated from rest to velocity V. Equation #2 describes the same bike and rider that has accelerated again to velocity V, but with an additional non-rotating weight dm. Finally, Equation #3 describes the same bike and rider, but this time the additional weight dm is on a wheel at a radius r from the hub. The radius of the wheel+tyre is R.
The incremental effect of the rotating component of the additional weight is therefore defined by the difference between equations #2 and #3. Instead of having an additional energy term of 0.5*V^2*dm for the non-rotating weight, when the extra weight is on the wheel the additional energy term is 0.5*V^2*dm + 0.5*V^2*dm*r^2/R^2. Therefore, the situation with the rotating weight has more energy than the non-rotating extra weight case by an amount equal to 0.5*V^2*dm*r^2/R^2.
The extra energy therefore depends on the location of the extra weight. If the extra weight is near the axle (r/R is zero or very small), the rotating element of the extra weight is negligible, because the moment of inertia associated with it is negligible. In the opposite case, if the extra weight is close to R, i.e. in the tyre, tube or rim, then the rotating element of the extra weight becomes 0.5*V^2*dm, because r/R=1. In other words, if the extra weight is in the tyre or rim it is twice as important as having the weight on a non-rotating component.
That's the maximum amount possible; twice as important. Remember also that this extra energy only makes a difference during an acceleration. At a steady speed instead, such as steady climb up a hill, there is no acceleration, therefore it doesn't matter whether the extra weight (or weight saving) is on a rotating component or a non-rotating component.
