A block of mass M hangs in equilibrium. the rope which is fastened to the wall is horizontal and has a tension of 38N. the rope which is fastened to the ceiling has a tension of 59N and makes an angle theta with the ceiling. what is the angle

50°

Explanation:

The tension in the rope fastened to the ceiling has horizontal component 59cosθ and its vertical component is 59sinθ. For equilibrium,

59cosθ = 38 N  (the tension in the rope attached to the wall) (1)and

59sinθ = Mg (the weight of the block of mass)  (2)

Dividing (2) by (1)

59sinθ/59cosθ = Mg/38

tanθ = Mg/38

θ = tan⁻¹(Mg/38)

Also squaring (1) and (2) and adding

(59cosθ)² = 38² N

(59sinθ)² = (Mg)²

(59cosθ)² + (59sinθ)² = 38² + (Mg)²

59²((cosθ)² + (sinθ)²) = 38² + (Mg)² . Since (cosθ)² + (sinθ)² = 1

59² = 38² + (Mg)²

Mg = √(59² - 38²) = √(3481 - 1444) = √2037 = 45.13 N

θ = tan⁻¹(Mg/38) = tan⁻¹(45.13/38) = tan⁻¹(1.1877) = 49.9° ≅ 50°