Consider the 692 N weight held by two cables shown below. The left-hand cable had tension T and makes an angle of θ with the wall. The right-hand cable had tension 760 N and makes an angle of 40◦ with the ceiling. a) What is the tension T in the left-hand cable slanted at an angle of θ with respect to the wall? Answer in units of N. b) What is the angle θ which the left-hand cable makes with respect to the wall? Answer in units of ◦.

Question

Consider the 692 N weight held by two cables shown below. The left-hand cable had tension T and makes an angle of θ with the wall. The right-hand cable had tension 760 N and makes an angle of 40◦ with the ceiling. a) What is the tension T in the left-hand cable slanted at an angle of θ with respect to the wall? Answer in units of N. b) What is the angle θ which the left-hand cable makes with respect to the wall? Answer in units of ◦.

Answer 1

To find the tension T in the left-hand cable and the angle θ it makes with the wall, we can use the principles of vector addition and equilibrium. Let's break down the problem into its components.

a) Finding the tension T in the left-hand cable.
We will start by analyzing the vertical equilibrium. The weights of the object and the tension in the right-hand cable should balance out the vertical forces.

1. Resolve the weight into vertical and horizontal components:
The weight of the object is given as 692 N. Drawing a right-angled triangle, we can break it down into vertical and horizontal components:

Vertical component: 692 * sin(40°)
Horizontal component: 692 * cos(40°)

2. Establish equilibrium in the vertical direction:
The vertical components of the weights and the tension T should balance out. We can set up an equation:

T * sin(θ) = 692 * sin(40°) + 760

Now, we have an equation with T and θ as unknowns.

b) Finding the angle θ that the left-hand cable makes with the wall.
To find this angle, we need to consider the horizontal equilibrium. The horizontal components of the weights and the tension T should balance out.

1. Establish equilibrium in the horizontal direction:
The horizontal components of the weights and the tension T should cancel each other out. We can set up an equation:

T * cos(θ) = 692 * cos(40°)

2. Rearrange the equation to solve for θ:
Divide both sides of the equation by T and substitute the value of T from the previously derived equation:

(T * cos(θ)) / T = (692 * cos(40°)) / T

cos(θ) = (692 * cos(40°)) / (692 * sin(40°) + 760)

Now, we have an equation with θ as the unknown.

Solve these equations simultaneously to find the values of T and θ. The solution will provide you with the tension T in the left-hand cable (a) and the angle θ it makes with the wall (b).