Does grant funded equipment result in authorship?
Authorship, no, not if the only contribution was a piece of equipment. Authorship at a minimum requires an intellectual contribution to the project.

Acknowledgment, perhaps. $1000 sounds like a fairly minimal investment for an acknowledgment even, but the standards for what is worthy of acknowledging are much more up to the authors of a given paper. If the work truly would not have been possible without the equipment, then it may certainly be appropriate. If you were just borrowing someone's unused computer, that doesn't sound acknowledgment-worthy to me.

On a brief note in the other direction: some people come here and ask authorship questions where they feel someone (usually an advisor) doesn't deserve authorship, but in reality they made a substantial intellectual contribution to the project. Norms for this vary by field, but make sure when you are assessing answers you get here that you are describing this PI's complete contribution accurately.