Question: Given that they live down south in Mississippi and it's Summer, when the judge walks up to Atticus' porch, he's fanning himself and Atticus says, "It's rather warm, isn't it?" If so, why is Atticus wearing a suit (OK) and a sweater? I get the suit, it's his work clothes, but a sweater?
Question: Why did everybody in the courtroom go silent when Tom said that he did chores for Mayella because he felt sorry for her?
Answer: At that time in history, Tom, a black man, was considered inferior to Mayella, a white woman. When he says he "felt sorry" for her, it is interpreted as him thinking he is above her or better off in some way. Regardless of his good intentions, for him to think of himself as being in a superior position to help her was considered unacceptable because it was seen as a black person rising above their lower place in society.
Answer: The film is set in Maycomb, Georgia. There's no explanation, so any answer is speculation. However, Atticus usually wore a three-piece suit, so it was a vest, not a sweater. The suit was likely a lightweight material and was a typical style of the era. Atticus probably considered it a more professional look for a lawyer. Most likely this was a movie wardrobe decision, giving Atticus a scholarly appearance and persona throughout the film, even if in reality, it wasn't practical.
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