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What is the tension in a string having two equal and opposite forces each of magnitude 10 N at its both ends?

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I think it's 10 N. Consider a string hanging from a doorway, hang a 10 N weight on it, the tension is 10 N. There is a force of 10 N at the bottom end, the top is pulling on the door frame with 10 N, so the door frame is pulling with 10 N force. You have a 10 N force at each end of the string, but 10 N tension.

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DukeCityMuscle

Without any more details about direction of gravity, pulleys, elasticity, etc. it would be 20 Newtons. With tension, equal forces at opposite directions add rather than cancel.

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