Chapter 9.4, Problem 90AYU

a.

To determine

To explain: the farmer’s estimate of the needed 6-ton force of the given condition.

Answer to Problem 90AYU

  $\Vert F\Vert =11757.4$ .

Explanation of Solution

Given:

The estimate that remove a stump from a field by pulling out by the tractor is 6 ton or 12000 pound.

The capacity of the tractor is only 7000 pounds.

Now two tractors are attached to the stump with ${40}^0$ angles between the force as shown in the figure:

  

Calculation:

According to the question,

  $\begin{array}{l}{F}_1=7000i\\ F_2=5500\left[\cos \left({40}^0\right)i+\sin \left({40}^0\right)j\right]\\ F_2=5500\left[0.7660i+0.6427j\right]\\ F_2=4213.4i+3535.33j\end{array}$

  $\begin{array}{l}{F}_1+F_2=7000i+4213.4i+3535.33j\\ =11213.4i+3535.33j\\ \Vert F\Vert =\sqrt{{\left(11213.4\right)}^2+{\left(3535.33\right)}^2}=11757.4\end{array}$ .

Hence, the required force is $\Vert F\Vert =11757.4$ .

b.

To determine

To check: that the angle is ${25}^0$ instead of ${40}^0$ of the above condition in part (a).

Answer to Problem 90AYU

  $\Vert F\Vert =12208$ .

Explanation of Solution

Given:

The estimate that remove a stump from a field by pulling out by the tractor is 6 ton or 12000 pound.

The capacity of the tractor is only 7000 pounds.

Now two tractors are attached to the stump with ${40}^0$ angles between the force as shown in the figure:

  

Calculation:

According to the question,

  $\begin{array}{l}{F}_1=7000i\\ F_2=5500\left[\cos \left({25}^0\right)i+\sin \left({25}^0\right)j\right]\\ F_2=5500\left[0.906308i+0.422618j\right]\\ F_2=4984.69i+2324.4j\end{array}$

  $\begin{array}{l}{F}_1+F_2=7000i+4984.69i+2324.4j\\ =11984.69i+3535.33j\\ \Vert F\Vert =\sqrt{{\left(11984.69\right)}^2+{\left(3535.33\right)}^2}=12208\end{array}$ .

Hence, the required force is $\Vert F\Vert =12208$ .