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You only have 2cm movement on the tailstock? Something wrong there I would have thought. Pete
I can’t advise you to do it but after starting the hole with the tailstock locked I’ve seen folks feeding the bit in by sliding the unlocked tailstock along the bed. Best done with the quill retracted I would have thought.
I can probably push it a touch more (5-6mm) but it become unstable and wants to rotate. ☹️
From what I’ve read other turners say, Wood turning lathes are notorious for having poor tail stock quills. I have an Axminster WW lathe and MW lathe and you have to wind the quill on the WW lathe much further out before the taper locks than on the MW lathe. Why do they make them like that?!