Chapter 5, Problem 105CP

Hanging from the ceiling over a baby bed, well out of baby’s reach, is a string with plastic shapes, as shown here. The string is taut (there is no slack), as shown by the straight segments. Each plastic shape has the same mass $m$ , and they are equally spaced by a distance $d$ , as shown. The angles labeled $\theta$ describe the angle formed by the end of the string and the ceiling at each end. The center length of sting is horizontal. The remaining two segments each form an angle with the horizontal, labeled $\varphi$ . Let $T_1$ be the tension in the leftmost section of the string, $T_2$ , be the tension in the section adjacent to it, and $T_3$ be the tension in the horizontal segment. (a) Find an equation for the tension in each section of the string in terms of the variables $m,g$ , and $\theta$ . (b) Find the angle $\varphi$ in terms of the angle $\theta$ . (c) If $\theta =5.10°$ , what is the value of $\varphi$ ?(d) Find the distance $x$ between the endpoints in terms of $d$ and $\theta$ .