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But I want it to be pointing towards the center
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and that is then the force of gravity on the Moon
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or on the apple or any object.
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If I'm thinking of one fixed object, like the Earth,
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this is the force on the apple.
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If you're thinking of two objects of comparable size,
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you'll have to say the force on one is due to the other.
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When you write the formula,
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then you've got to make sure
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that the force on one is pointing to the other one,
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and the force on that guy is pointing towards this guy.
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So, there is no universal symbol for all those things.