Moment of Inertia

My friend and a co-worker of his over at a bike shop got into an argument about whether it would take less energy to go up a slow grade or whether a person would go straight up, my friend argued that the energy it took to go up a shallow grade would be less than to straight up the same height. The bike mechanic argued that if that were the case he could solve all the world's energy problems by creating a device that moves mass up a shallow grade and then dropping it down in a straight line and capturing that energy.

Yet when he was mulling it over with me, he made the statement that the weight dispersion in the bike should not make any difference in how the bike climbs, I knew that that wasn't true, so I tried to tell him about Moment of Inertia, how if you had a disc of a certain mass let's say 2 kg and you had a rim of the same outer diameter and mass, and were to put both these masses at the top of a ramp, the solid disc would get to the bottom faster. He didn't get it quite away, so I thought for a while about how I could convince him that this was the case, that even though the mass was the same, the work required to move the latter was greater, and I came up with the following explination,

say you have a disc, and it moves at velocity v. The rate of forward motion is equal to the rate at which the outer circumfrence is moving.

at each radius of the disc, the distance the mass at different radius varies according to the circumfrence of that radius.

So for a rigid disc, where the rpm is the same for all radius, the distance travels proportional to the distance from the center.

In order to start moving at a certain velocity, mass at every point in the disc must rotate around the center at the same rate, but the distance the outer mass has to travel to maintain the same rate of rotation is greater than that of mass closer to the center.

So by the equation F = MA, the outer mass must be accelerated at a faster rate because it travels a greater distance over the same interval of time. Mass at the outer part of the rim would thus have a greater force because the acceleration needed to move it at the same RPM would need to be greater than mass in closer towards the center.

So it would be possible to make a quicker wheel out of more mass, just by dispersing the mass closer towards the center of the disc.