EXERCISE 4.6 Suppose the traffic light is hung so that the tensions T_(1) and T_(2) are both equal to 80.0N. Find the new angles they make with respect to the x-axis.(By symmetry, these angles will be the same.) ANSWER Bothangles are 38.7^circ .

Both angles are 38.7 degrees.

This problem is about vector decomposition. The traffic light creates two tensions, T1 and T2, which are equal in this case. Each tension can be decomposed into two components: one along the x-axis and the other along the y-axis. The x-components of T1 and T2 balance each other out, so the system is in equilibrium in the x-direction. The y-components, however, do not balance each other out, which means there is a net force in the y-direction. This net force causes the system to rotate counterclockwise. The angle that the resultant tension makes with the x-axis can be found using trigonometric principles. Since the tensions are equal, the angles they make with the x-axis are also equal. This is due to the symmetry of the problem. The angle can be found using the formula tan(theta) = Fy/Fx, where Fy is the net force in the y-direction and Fx is the force in the x-direction. In this case, Fy = 0 (since the system is in equilibrium in the x-direction) and Fx = 80N (the tension in the ropes). Therefore, tan(theta) = 0/80 = 0, which means theta = 38.7 degrees.