Which would be faster: A ball rolling down a ramp with no groove

[limonade123]

Homework Statement

Which would be faster: A ball rolling down a ramp with no groove, or a ball rolling down a ramp with a groove?
What effect does the radius have, if any? Disregarding resistance and whether or not the ball rolls in a straight line.

Homework Equations

The Attempt at a Solution

I thought that a decrease in rolling radius due to being in the groove might have an effect but apparently not. Any input appreciated.

 

[paisiello2]

Can you prove it analytically?