2 bowls rolling on an irregular surface

[mtr]

Homework Statement

2 identical bowls are rolling on 2 different routes. The first one has a convexity and the second one has a concavity in their middle parts. The convexity fits perfectly the concavity. Both bowls had the same initial velocities. Bowls are rolling without slipping and jumping. Which one will reach the goal first? Why? What can you tell about their final velocities and kinetic energy of rotational motion? Consider 2 cases:
a) there are no wastes of energy
b) there appears friction. How many times deeper or wider should be the concavity so that both bowls reach the goal simultaneously?

Homework Equations

KE = (mV^2)/2 kinetic energy
E = (Iw^2)/2 kinetic energy of rotational motion
T=fN friction

The Attempt at a Solution

The first part I did in such a way - consider an average velocities of both bowls. When the first one reaches the goal its average velocity is less than the second, because when the first one climbed the convexity its velocity decreased and thus affectioned average velocity in a negative way. The second bowl behaved just in opposite. Energies are the same, because there are no wastes.
However, I have a big problem with b). Actually, I have no idea how to solve this one.

 

[Mindscrape]

The problem is mostly about inertia. You will need to know that rotational kinetic energy is \frac{1}{2}I\omega^2. Think about a couple simpler cases: dropping a spherical ball down a hill, dropping a disc down a hill, and dropping a wheel down a hill, where all have the same density. Which one will finish first, or will they finish first?