Find T and θ for equilibrium

Reading of the figure: a 10 kg mass hangs from a junction. Rope T pulls up and to the left, making angle θ with the vertical. A 50 N force pulls up and to the right at 30° above the horizontal. Take g = 9.81 m/s², so W = 10 × 9.81 = 98.1 N.

Equilibrium of the junction

ΣFx = 0: T sin θ − 50 cos 30° = 0, so T sin θ = 43.30 N …(1) ΣFy = 0: T cos θ + 50 sin 30° − 98.1 = 0, so T cos θ = 73.10 N …(2)

Dividing (1) by (2): tan θ = 43.30 / 73.10 = 0.5924, so θ = 30.64° (from the vertical).

T = √(43.30² + 73.10²) = 84.96 N

Answer: T = 84.96 N (≈ 85 N), θ = 30.64° with the vertical.