How to ground old turntable with no grounding wire

I have an old Marantz turntable that I want to start using again. I have a terrible hum through my speakers when it is turned on--worse when the tone arm is on its rest, better when it's over or on a record. I'm guessing it needs to be grounded (although it was never grounded when I used it years ago). But, it has no grounding screw or wire that I can locate. Suggestions? Any ideas for other causes of this new hum? The other thing I'm wondering about is electrical polarity, because it has an old (non-polarized) plug, plugged into a polarized extension cord. Any and all suggestions welcome!
I would look for a screw on the motor and attach a wire to that and run it to either the ground lug on your pre-amp/receiver/Phono pre-amp if one of them has one has one. Or if not try using the center screw on the wall plate if you have 3 prong recepticles.
I would get a wire and touch one end to a metal spot on the preamp and the other end to any metal spots you can find on the turntable like the spindle. Experiment with various spots and see if the hum goes away. If it does then look for a way to permanently attach it. It it doesn't then the problem lies elsewhere.

Does it play? Maybe a wire broke off where it attaches to the cartridge?
Herman, Yes, it plays fine (too fast, actually, but I'm going to adjust that). The cartridge and stylus seem fine. The hum diminishes significantly (but doesn't disappear) when the tone arm is moved off of its rest and onto the record. I like the idea of testing with a wire. I'll try that this weekend.
So probably the cartridge is picking up hum from some other equipment in the vicinity. Try moving the table, rotate it 90 degrees or move it further from the other stuff.
If this is any chance a Grado cartridge, they are notorious for hum. If you post your Table and cart models, it would make it easier to give a good answer. I have several Marantz (6100 6200 etc) all have ground wires. I suspect it would be easy to locate the ground point inside the table and run a wire. When I am messing with them I use a long piece of 20awg with alligator clips on each end. Clip one to the ground on my amp or pre then poke around on various grounding possibilities in the table till I get the best results. Be careful not to short out anything hot though.