Solved: In the vertical direction we have two forces.

Chapter 12, Problem 12.1.31

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In the vertical direction we have two forces. namely, the vertical components - T] sin ll' and T2 sin f3 of Tl and T2; here the minus sign appears because the component at P is directed downward. By Newton's second law the resultant of these two forces is equal to the mass p ~x of the portion times the acceleration a2 uICJ(2, evaluated at some point between x and x + ~x; here p is the mass of the un deflected string per unit length. and ~x is the length of the portion of the undeflected string. (~ is generally used to denote small quantities; this has nothing to do with the Laplacian V2, which is sometimes also denoted by ~.) Hence

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