The cord is required to be marked on the jacket, along it's length. You want 14 gauge at MINIMUM.
The next question is the outlet. It's "probably" wired for 20 amps. That means the max for any given length of time (not instantaneous) is at or about 16 amps.
A 14 gauge extension cord usually says on the label it's good for 15 amps. At that amperage, over any length of time, that extension cord will get "comfortably warm". Obviously it's uncomfortable when electrical things heat up, but I hope that makes sense... It should get "comfortably warm" and STAY THAT WAY. That is wrong, unacceptable, but pretty well accepted and UL listed, and should be corrected, but doesn't warrant panic immediately. (Vacuum cleaner cords usually heat up too, while in use FWIW.... Heat should be generated no faster than it dissipates at "comfortably warm". If it continues and heats more and more over time, you're on the way to a thermal runaway, where heat goes in faster than it goes out. Hours or days, doesn't matter. If it doesn't stabilize, it doesn't stabilize and that's NOT gonna be OK in any way.
The other issue is the corresponding issue inside of the walls, or however else the garage circuits are wired in place. If those wires are the same gauge as the extension cord (with SOME allowance for the fact that most extension cords don't meet the ohms per foot expectation that one might have if they assumed that the gauge of the wire was related to the size and copper content of the wire.,..... But that's another subject), with some allowance, the same thing is happening inside of the garage electrical system. So IF (-IF-) the extension cord is 14 gauge, it bears watching careful to be certain the extension cord "stabilizes" at a temperature. The wiring to a 120v outlet should oughtta be 12 gauge by all rights (Yellow Romex if you have access to peek), but depending on the age and the installer and some old exceptions, and lots of DIY and "discount electricians" doing rental stuff... I'd want to peek, because if it's white romex (14 gauge), the "in the wall" heating will be (theoretically) equal to the extension cord heating, with LESS opportunity for heat dissipation.
Personal opinion- Were that me-
A, make sure the cord is "OK", as "rated for" is gonna be barely adequate. But adequate. 14ga minimum.
B, make sure the wires to the outlet are indeed 12 gauge. They "should" be anyhow, but "big deal". Take the cover plate off (don't stick your fingers in), and IN THEORY the jacket should be visible right there where the romex enters the box.
C. Keep the "on time" to a minimum and work your way up, INCLUDING monitoring the cord AND unplugging the cord now and then to make sure the prongs themselves (the flat ones, not the ground) are not heating beyond what the cord it's self is.
D. When you have your buddy there, judge carefully as you go along, make sure he knows some stuff, and not just knows where to hook things..... Have your buddy look around and see if there is a place you can get ahold of 240 volts out there. That'd be ideal. Or if it might (or might not) be practicable to make up an extension cord to a 240 volt outlet elswhere in the house. PITA but it's workable.
The bottom line is that this is "probably" a workable situation you have there, but that motor is a monster as far as a "regular" outlet is concerned.
Verifying the in wall wire gauge is a big deal. Monitoring the extension cord is a big deal. Both of those can be "trusted" with enough experience. Just like your vacuum cleaner cord.
That motor is quite capable of taking more than the circuit can deliver. The onus will always be on you to not let it do that. But "heavy cuts" type of issues are gonna be a tripped breaker issue. No worries there.
If it comes to this (It may well not come to this- Your situation may well end up being workable), but if it comes to this, that lathe came with a quarter horse motor. The cadillac motor was a half horse. I have that. half a horse will twist that lathe up into ways it should not bend to if you try hard enough. Sure, more is "nice", you don't have to think as much about your cuts and all, but half a horse WILL do what the lathe can do if it comes to it. If you had to knock that motor back to one or 3/4 horse... You'd still have a hot rodded lathe and the outlets would be happier. If you do anything more than upgrade the extension cord, check prices- It'd not be prudent to have more into a "band-aid" fix for a lack of power to the garage than it would to have simply tossed on a more appropriate motor for 120v residential electrical systems. Not saying either one will be the answer, just know them both. This stuff can add up FAST either way, and if you sort it out ahead of time, the direction a soluton takes you can often be surprising and unintuitive.
But to start, just go back to my bullet list there. A and B for sure, don't even need your buddy for that. From there just play it smart, there's an answer somewhere that "might" take some doing, but you'll get this going.
