Physics Tension Problem

Question

Kirun is leaning a ladder (mass m = 15 kg and length L = 5.0 m) against the side of his house. An ice storm has just passed, so the ground and the side of the house are frictionless. Kirun realizes that the ladder will not remain propped up in these conditions, so he connects a support rope horizontally between the bottom of the ladder and the side of his house. Kirun allows the bottom of the ladder to slide outward until the rope tightens up and the ladder remains steady. Now the ladder makes an angle of θ = 65° with the horizontal and the support rope is completely horizontal. What is the tension T in the support rope? The symbolic answer must be written in terms of the variables m, L, T, θ, and g, though not all of these variables may be needed. (Hint: there are two normal forces acting on the ladder, one from the ground and one from the side of the house.

Answer 1

1) Make a free body diagram with a weight vector pointward downward from the midpoint, a tension vector pointing to the left from the bottom point "O", the reactions at the top are Nw to the right and Fw to the left, the reactions at the bottom are Ng pointing up and Fg pointing down.

2) Write sum of forces vertical equation = 0 = -W+Ng-Fg (up is +)

3) Write sum of forces horizontal equation =0 = T-Fw-Nw (right is +)

4) Write sum of moments about point "O" =0 = WL/2cos65-(Fw-Nw)Lsin65

5) Solve for T using 3) & 4) ... = W/2tan65