"How To" Replace Banjo 5th Peg
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Banjo 5th pegs are just a machine head like your guitar, except its pressed into place rather than bolted in.

The most common issue that happens with these 5th pegs, is the internals fail, and they start slipping.

Time to repair approx 1 hr
Using a puller tool, these are simply a steel bar with an anchor and a cap, any puller will do the job.
Wind the wing nut (arrow pointing too) all the way down so the puller sits firmly on the rubber.
Wind the wing nut until the machine head pops out.
The widest point of the old machine head is 9.28mm
The widest point of the new machine head is 9.76, if the machine head was loose, this would slightly oversize the hole and fix a loose machine head issue..

Since the machine head was tight we need to open the hole up a fraction.
We use a reamer with a 6 degree taper, and ever so slightly open the hole up.
Dis-assemble the machine head of the knob and associated washers.
Lightly push the machine head into the freshly reamed hole.
As the machine head is pressed in, we need to protect the surrounding wood when we pull the machine head out.

Here, I have a piece of rubber.
I use masking tape and tape one to each side of the 5th Peg.
I slip the recess base over the existing machine head.
Using a small socket, I place it over the centre spigot , so that it sits on the body of the machine head.
Using a small clamp, we press it home.
Refit the button and job finished.

String it up and away you go.