Problem 1: The T-shaped arm ABC shown in the figure lies in a vertical plane and pivots about a horizontal pin at A. The arm has constant cross-sectional area and total weight W. A vertical spring of stiffness k supports the arm at point B. Obtain a formula for the elongation of the spring due to the weight of the arm. A B C

A stoplight of mass m is hung from a massless wire that has segments to a horizontal pole and a vertical pole as shown in the figure. The stoplight is not moving. 162 T₂ 60° T₁ 60° The wire segments create angles 60° with respect to the horizontal pole and 60° with respect to the vertical pole as shown and the mass creates tensions in the segments of the wire of magnitude T₁, and T2, as labeled in the diagram. (A) Draw a free-body diagram for the hanging mass system showing (and labeling) all forces acting on the hanging stoplight mass. Then draw x and y axes for your free-body diagram. (B) Use Newton's-second law to write equations for forces acting on the stoplight in component forms (x and y, separately) in terms of: tension magnitudes T₁ and T2, the angles shown, the mass of the object m and the gravitational constant g. You do not need to solve the equations but simplify as possible. (C) The weight of the stoplight is determined to be 343 Newtons. Solve for magnitudes of T₁, T2.