Find the tension in the clothesline in terms of W.
In this problem, the center of the clothesline is a point in equilibrium, i.e., the sum of all forces equals zero. There are two tension forces of the same magnitude T directed at 30° above the horizontal pointing left and right from the center, as well as the weight W pointing straight down. The vertical or y-component of either tension force is the opposite side in a right triangle whose hypotenuse is T and is therefore given by T sin(30). Then the sum of the forces in the vertical direction equals zero:
∑ Fy = 2Tsin(30) -W = 0
Since 2 sin(30) = 2 (1/2) = 1, we have
T-W=0, or T=W.
The tension in the clothesline equals the weight.