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the horizontal tensions acting on both sides of the string segment are given by

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From Newton's second law for the vertical component, the mass (which is the product of its linear density and length) of this piece times its acceleration, , will be equal to the net force on the piece:

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Dividing this expression by and substituting the first and second equations obtains (we can choose either the first or the second equation for , so we conveniently choose each one with the matching angle and )

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