Question

The 10-kg uniform rod is pinned at end

Answer 1

Supposing that the spring is un stretched when θ = 0, and has a toughness of k = 60 N/m.It seems that the spring has a roller support on the left end. This would make the spring force direction always to the left 
Sum moments about the pivot to zero. 
10.0(9.81)[(2sinθ)/2] + 50 - 60(2sinθ)[2cosθ] = 0 98.1sinθ + 50 - (120)2sinθcosθ = 0 98.1sinθ + 50 - (120)sin(2θ) = 0 
by iterative answer we discover that 
θ ≈ 0.465 radians 
θ ≈ 26.6º