Chapter 6.3, Problem 111E

a.

To determine

Find $\Vert F_1+F_2\Vert$ as a function of $\theta$

Answer to Problem 111E

  $=\boxed{\sqrt{125+100\cos \theta }}$

Explanation of Solution

Given information:

Consider two forces

  $F_1=\langle 10,0\rangle$ and $F_2=5\langle \cos \theta ,\sin \theta \rangle$

Calculation:

Consider the two forces as

  $F_1=\langle 10,0\rangle$

  $F_2=5\langle \cos \theta ,\sin \theta \rangle$

The sum of these forces is

  $\begin{array}{l}{F}_1+F_2=\langle 10,0\rangle +5\langle \cos \theta ,\sin \theta \rangle \\ =\langle 10,0\rangle +\langle 5\cos \theta ,5\sin \theta \rangle \\ =\langle 10+5\cos \theta ,5\sin \theta \rangle \end{array}$

The magnitude of the sum is,

  $\Vert F_1+F_2\Vert =\sqrt{{\left(10+5\cos \theta \right)}^2+{\left(5\sin \theta \right)}^2}$

  $\begin{array}{l}=\sqrt{{\left(10+5\cos \theta \right)}^2+5^2{\sin }^2\theta }\\ =\sqrt{100+25{\cos }^2\theta +100\cos \theta +25{\sin }^2\theta }\end{array}$

  $=\boxed{\sqrt{125+100\cos \theta }}$

b.

To determine

Use a graphing utility to graph the function in part (a) for $0\le \theta \le 2\pi$

Answer to Problem 111E

  

Explanation of Solution

Given information:

Consider two forces

  $F_1=\langle 10,0\rangle$ and $F_2=5\langle \cos \theta ,\sin \theta \rangle$

Calculation:

Consider the two forces as

  $F_1=\langle 10,0\rangle$

  $F_2=5\langle \cos \theta ,\sin \theta \rangle$

To plot the function using TI-83 graphing calculator, follow the below steps:

Press $Y=$ and enter the function

  

Then press the GRAPH button on the calculator and graph appear as shown below,

  

c.

To determine

Use the graph in part (b) to determine the range of the function.

Answer to Problem 111E

  $\boxed{\theta =0,2\pi }$

Explanation of Solution

Given information:

Consider two forces

  $F_1=\langle 10,0\rangle$ and $F_2=5\langle \cos \theta ,\sin \theta \rangle$

Calculation:

Consider the two forces as

  $F_1=\langle 10,0\rangle$

  $F_2=5\langle \cos \theta ,\sin \theta \rangle$

To find the maximum and minimum values of the function using the TI-83 graphing calculator follow the steps:

Press SHIFT + TRACE and select minimum from the menu as shown below,

  

Then press the TRACE button on the calculator and graph appear as shown below,

  

From the graph it is clear that the minimum value of the function is,

  $\Vert F_1+F_2\Vert =\boxed{5}$

And it occurs at,

  $\boxed{\theta =\pi }$

Similarly, the maximum value of the function is,

  $\Vert F_1+F_2\Vert =\boxed{25}$

And it occurs at,

  $\boxed{\theta =0,2\pi }$

d.

To determine

Explain that the magnitude of the resultant is never $0$ .

Answer to Problem 111E

The function $\Vert F_1+F_2\Vert$ never goes zero.

Explanation of Solution

Given information:

Consider two forces

  $F_1=\langle 10,0\rangle$ and $F_2=5\langle \cos \theta ,\sin \theta \rangle$

Calculation:

Consider the two forces as

  $F_1=\langle 10,0\rangle$

  $F_2=5\langle \cos \theta ,\sin \theta \rangle$

The magnitude of the resultant is,

  $\Vert F_1+F_2\Vert =\sqrt{125+100\cos \theta }$

The minimum value of the cosine function is $-1$ .

Hence, the function $\Vert F_1+F_2\Vert$ never goes zero.