Chapter 8.5, Problem 2CFU

To determine

To explain: the force exerted by cables in example 4.

Answer to Problem 2CFU

Tension increases

Explanation of Solution

Given information:

On increasing the angle between the cables, the force exerted by the cable is affected

The diagram representing the tip-to-tail vector of the situation is shown below.

  

Calculation:

The angle $40°$ shown in the diagram opposite to the base decreases as the angle of $140°$ is increased. The base of the isosceles triangle has not changed and the angle opposite to the base has decreased. This shows that the height and sides must increase.

Now divide the isosceles triangle into two right triangles such that they both have a common base $\left(y\right)$ , the height $\left(h\right)$ is constant and the hypotenuse $\left(x\right)$ representing the tension as shown below.

  

The relation between angle and the tension can be derived as follows:

  $\begin{array}{c}\sin \theta =\frac{h}{x}\\ \theta \propto \frac{1}{x}\end{array}$

Thus the tension increase as the angle between the cables is decreased.