Switching Dremel with relay (CNC)

I’m going to let my CNC controller switch the Dremel spindle off and on. The controller has is a relay which seems to click at the expected times for spindle control.

The cnc cabinet already has 120V running to it to supply the switch power supply for the CNC controller. I want to extend these three wires to an outlet for a dremel.

It has an L, N and G and I believe I need to put the relay on the L. Does this seem right?