handle
柯 centers on a handle, especially the wooden handle of an axe or tool. It is a specific object name rather than a broad abstract concept.
柯 combines 木 (tree/wood) with 可, which likely contributes the sound. The character originally referred to a wooden handle or stalk.
The left side is a tree 木, and the right side 可 looks like a handle shape. Picture a sturdy wooden handle carved from a tree branch.
For カ, imagine a car with a wooden steering wheel handle: car -> カ, and you grip the wooden handle to drive.