Chapter 6, Problem 65CR

To determine

To calculate:

The tension in each rope.

Answer to Problem 65CR

The tension in each rope in the given diagram is $180lb$ .

Explanation of Solution

Given information:

Weight $=180pounds$

  

Calculation:

  

The figure shows the situation in the problem. The vector $\overrightarrow{OR}$ represents the weight.

Therefore, $\Vert \overrightarrow{OR}\Vert =180$

The vectors $\overrightarrow{OP}$ and $\overrightarrow{OQ}$ are the vectors representing the tensions $T$ in the two ropes. Both vectors $\overrightarrow{OP}$ and $\overrightarrow{OQ}$ make an angle of $60°$ with the vertical as shown in the above figure.

The triangles $OQS$ and $OPS$ are congruent; the lengths of $OQ$ and $OP$ are same,

that is,

  $\Vert \overrightarrow{OP}\Vert =\Vert \overrightarrow{OQ}\Vert =T$

From the figure,

  $\begin{array}{l}\Vert \overrightarrow{OP}\Vert \cos 60°+\Vert \overrightarrow{OQ}\Vert \cos 60°=\Vert \overrightarrow{OR}\Vert \\ T\cos 60°+T\cos 60°=180\\ 2T\cos 60°=180\\ T=\frac{90}{\cos 60°}\end{array}$

This gives, tension in each rope as,

  $\begin{array}{l}T=\frac{90}{\left(1/2\right)}\\ =180lb\end{array}$