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We apply a force on a point on an object such that the point of application moves on the object but the object itself doesn't move. Then is the work done zero or will it have some value?

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Work is done if point of application moves hence in this case too work is done. Your statement suggests that object does not move and point moves, so object is being deformed and our work done is stored eithet in form of elastic potentiel energy or used up in permanently deforming object.

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