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What nail sizes can a nail puller pull?

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The size of nail that can be pulled will depend on the tool being used Unfortunately this is not a question that Wonkee can answer easily, as it depends on the nail puller you are using and its specifications. It will also depend on the style and condition of the nail itself, as well as the material you are removing the nail from.
traditional nail puller can grip the shaft or under the head of the nail, check specifications of nail removing tools Some traditional nail pullers have jaws which only need to grip the shaft of a nail.

This grip will tighten as the tool is pivoted over,so that it can be repositioned to pull a longer nail out a bit at a time by placing the jaws around the part of the nail shaft which is closest to the wood.

traditional nail puller can grip the shaft or under the head of the nail, check specifications of nail removing tools Other traditional nails pullers will need to use the head of the nail to grip under, it will depend on the specifications of the individual nail puller.
Nail pullers come in different sizes, the larger tools are made for pulling longer nails Usually the nail pullers with handles are designed to pull larger nails are they are larger tools and built for tougher jobs.
traditional nail pullers have jaws which can dig into the wood and reach underneath the nail to pull flush and sunken nails The advantage of a traditional nail puller is that it can reach under timber to pull out sunken nails.

Once the nail head has been exposed, another nail pulling tool can be used if necessary.

extra leverage can be achieved with longer tools or by increasing the height of the pivot point The longer the tool, the more leverage it will be able to apply to removing nails.

Heavy duty nails will probably require a heavy duty nail puller such as a large crow bar or prying bar.

coated nails have an adhesive that makes them difficult to pull out, this is also true of nails with shaped shafts  It is worth noting that ring shanked nails or square twisted nails, with twisted or ringed shanks, will be more difficult to remove than standard nails, as they are designed to be a more permanent fixture.
Coated nails will have an adhesive which holds them firmly in place Coated nails have a coating of adhesive on their shank, which is melted by the friction from the nail being hit into the material, this adhesive then sets to hold the nail firmly.

These types of nails will also be more difficult to remove.

thinner nails may be broken off by a nail puller Also, thinner more delicate nails and brads may be broken or snapped off by the sharp jaws of the nails puller.
 Wonkee Donkee says "Don't over strain your tools, make sure you choose a nail puller which is suitable for pulling the nail you want to remove".