Chapter 8.2, Problem 13CFU

To determine

To find: the magnitude of the combined forces exerted on the log.

Answer to Problem 13CFU

  $927\text{ N}$

Explanation of Solution

Given information:

Let the force exerted by Terrell and Walker be $\overrightarrow{T}$ and $\overrightarrow{W}$ respectively.

The figure representing $\overrightarrow{T}$ and $\overrightarrow{W}$ is shown below.

  $\begin{array}{l}\left|\overrightarrow{T}\right|=400\text{ N}\\ \left|\overrightarrow{W}\right|=600\text{ N}\end{array}$

The horizontal direction of $\overrightarrow{T}$ is $65°$ and that of $\overrightarrow{W}$ is $110°$ .

Calculation:

The ordered pair of $\overrightarrow{T}$ is evaluated as follows:

  $\begin{array}{c}\text{Component}=\left(\text{magnitude}\right)\left(\text{direction}\right)\\ \left|{\overrightarrow{T}}_x\right|=400\cos 65°\\ =169.05\text{ N}\\ \left|{\overrightarrow{T}}_y\right|=400\sin 65°\\ =362.52\text{ N }\end{array}$

The ordered pair of $\overrightarrow{T}$ is $\langle 169.05,362.52\rangle$ .

Similarly the ordered pair of $\overrightarrow{W}$ is evaluated.

  $\begin{array}{c}\left|{\overrightarrow{W}}_x\right|=600\cos 110°\\ =-205.21\text{ N}\\ \left|{\overrightarrow{W}}_y\right|=600\sin 110°\\ =563.82\text{ N}\end{array}$

The ordered pair of $\overrightarrow{W}$ is $\langle -205.21,563.82\rangle$ .

The ordered pair for $\overrightarrow{T}+\overrightarrow{W}$ will be evaluated as follows:

  $\begin{array}{c}\overrightarrow{T}+\overrightarrow{W}=\langle 169.05,362.52\rangle +\langle -205.21,563.82\rangle \\ =\langle 169.05-205.21,362.52+563.82\rangle \\ =\langle -36.16,926.34\rangle \end{array}$

The magnitude of the combined forces exerted on the log will be evaluated as follows:

  $\begin{array}{c}\left|\overrightarrow{T}+\overrightarrow{W}\right|=\sqrt{x^2+y^2}\left[\text{Put ordered pairs of }\overrightarrow{T}+\overrightarrow{W}\right]\\ =\sqrt{{\left(-36.16\right)}^2+{\left(926.34\right)}^2}\\ =927\text{ N}\end{array}$

Thus the magnitude of the vector is 927 N.