Questions: Se desea sostener el cuerpo de 140 lb que se muestra en la figura. Diga qué tensión T deberá aplicarse para lograrlo y cuál debe ser el ángulo. (Sol. T=81.2 lb ; Θ=29.5°)

Se desea sostener el cuerpo de 140 lb que se muestra en la figura. Diga qué tensión T deberá aplicarse para lograrlo y cuál debe ser el ángulo.
(Sol. T=81.2 lb ; Θ=29.5°)

Transcript text: 6. Se desea sostener el cuerpo de 140 lb que se muestra en la figura. Diga qué tensión T deberá aplicarse para lograrlo y cuál debe ser el ángulo. (Sol. $\left.T=81.2 \mathrm{lb} ; \Theta=29.5^{\circ}\right)$

Solution

Solution Steps

Step 1: Identify the forces acting on the body

The body is subjected to three forces:

The weight of the body (140 lb) acting downward.
A force of 80 lb acting at an angle of 30° to the horizontal.
The tension $ T $ acting at an angle $ \theta $ to the horizontal.

Step 2: Resolve the forces into components

Resolve the 80 lb force into horizontal and vertical components:

Horizontal component: $ 80 \cos(30°) $
Vertical component: $ 80 \sin(30°) $

Step 3: Set up equilibrium equations

For the body to be in equilibrium, the sum of the forces in both the horizontal and vertical directions must be zero.

Horizontal direction:

\[ T \cos(\theta) = 80 \cos(30°) \]

Vertical direction:

\[ T \sin(\theta) + 80 \sin(30°) = 140 \]

Step 4: Solve for $ T $ and $ \theta $

First, calculate the components of the 80 lb force:

$ 80 \cos(30°) = 80 \times \frac{\sqrt{3}}{2} = 40\sqrt{3} \approx 69.28 $ lb
$ 80 \sin(30°) = 80 \times \frac{1}{2} = 40 $ lb

Using the horizontal equilibrium equation: \[ T \cos(\theta) = 69.28 \]

Using the vertical equilibrium equation: \[ T \sin(\theta) + 40 = 140 \] \[ T \sin(\theta) = 100 \]

Divide the vertical equation by the horizontal equation to find $ \theta $: \[ \tan(\theta) = \frac{100}{69.28} \] \[ \theta = \tan^{-1}\left(\frac{100}{69.28}\right) \approx 29.5° \]

Now, solve for $ T $ using $ \cos(\theta) $: \[ T = \frac{69.28}{\cos(29.5°)} \approx 81.2 \text{ lb} \]