Why do we need У in "Ножка у сту́ла сломалась"?

Question

In the sentence "Ножка у сту́ла сломалась" (the chair leg broke), why do we need the у? The genitive case on "стул" already lets you know that it's the *chair's* leg that's broken. More generally, when and why is у used to indicate posession in Russian rather than simply using the genitive case?By the way, I'm aware that "сту́ла" is already in the genitive, I'm asking why у is there as well.

Answer 1

"Ножка у сту́ла сломалась" is a compression of two sentences: "У стула есть ножка" and "Ножка сломалась".

"Ножка стула сломалась" expresses the same meaning