To poke or push someone gently (often repeatedly) to encourage them to do something, or to provoke them into action.
From 'prod,' which likely derives from Old English or Low German, possibly imitating the action itself. The word entered Middle English with the sense of poking with a pointed object and extended to mean urging someone.
Prodding is such a useful concept—it's not forceful, it's that gentle repeated poking that parents do to get kids moving in the morning. It shows how English has specific words for different levels of pushing!
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