Is "arm is tender" an idiom? I can't match the meaning of this expression with the meaning of the word "tender".
Good question but no, "arm is tender" is not an idiom. In this context, the word "tender" means "sore". This refers to the muscles being sore or painful after using them a lot. So in this case, he says his "arm is tender" because he was using it to grind polenta before. However, the word "sore" is generally more common to use in this context instead of "tender". Ex: My arms are sore from working out yesterday. Ex: My legs are still a little tender from the hike.
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