what is the tension in the rope

Question

A 50N body is supported by two ropes, one making 30° with the vertical and another making 45°

with the horizontal from the left attachment respectively. What is the Tension in the ropes?

Answer 1

Let

F₃₀ be the tension in the first rope,

F₄₅ be the tension in the second rope.

In general, tension F in a rope at an angle α can be decomposed

into a sum of two forces --

one horizontal: Fsin(α) and one vertical Fcos(α).

In our case, we know that

1) The body is not moving laterally and therefore the two horizontal components must be equal, but opposite:

F₃₀sin(30°) = F₄₅sin(45°)

2) The body is not moving vertically and therefore the two vertical components must add up to the body's weight:

F₃₀cos(30°) + F₄₅cos(45°) = 50N

Substituting values for the sin and cos:

F₃₀/2 = F₄₅√2/2

F₃₀√3/2 + F₄₅√2/2 = 50N

Solving:

F₃₀ = 100/(1+√3) = 36.6N

F₄₅ = F₃₀/√2 = 25.9N