<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="description">
    <name>
      <text>9.1.3.2 Resolving Components of Vectors</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<h4>Resultant of two forces acting upon a body<br></h4><p>A force \(\underline{F}_1\) of \(2\ \mathrm{N}\) acts vertically downwards, and a force \(\underline{F}_2\) of \(3\ \mathrm{N}\) acts horizontally to the right, upon the body shown below.</p>
<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/9-1-4ResultantVector.svg" alt="A rectangular object is acted on by two forces, shown as arrows. One acts vertically down and the other acts horizontally to the right. A dotted line marks the resultant of the two pointing to the bottom right." class="img-responsive atto_image_button_text-bottom" width="380" height="252">
</figure>
<p>We can use vector addition to find the combined effect or<b> resultant</b> of the two concurrent forces. (Concurrent means that the forces act through the same point.) Translating \(\underline{F}_2\) until its tail touches the head of \(\underline{F}_1\), we complete the triangle ABC as shown. The vector represented by the third side is the resultant, \(\underline{R}\). We write 
\[\underline{R} = \underline{F_1} + \underline{F_2}\]
and say that \(\underline{R}\) is the <b>vector sum</b> of \(\underline{F_2}\) and \(\underline{F_1}\). The resultant force acts at an angle of \(\theta\) below the horizontal where \(\tan\theta  = 2/3\), so that \(\theta = 33.7^\circ\), and has magnitude (given by Pythagoras’ theorem) \(\sqrt{13}\ \mathrm{N}\).</p><p><br></p>

<h4>Resolving a force into two perpendicular directions<br></h4>
<p>In the previous section we saw that two forces acting upon a body can be replaced by a single force which has the same effect. It is sometimes useful to reverse this process and consider a single force as equivalent to two forces acting at right-angles to each other. Consider the force \(\underline{F}\) inclined at an angle \(\theta\) to the horizontal as shown below.</p>
<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/9-1-4ResolvingComponents.svg" alt="An arrow represents a force acting at an angle theta to the horizontal plane.  Vertical and horizontal arrows show how it can be broken down into components" class="img-responsive atto_image_button_text-bottom" width="366" height="166">
</figure>

<p>\(\underline{F}\) can be replaced by two forces, one of magnitude \(F\cos\theta\) and one of magnitude \(F\sin\theta\) as shown. \(\underline{F}\) has been <span><strong>resolved into two perpendicular components</strong></span>. This is sensible because if we re-combine the two perpendicular forces of magnitudes \(F\cos\theta\) and \(F\sin\theta\) using the triangle law we find \(\underline{F}\) to be the resultant force.</p>
<hr>
<h4 class="HELM_example">Example<br></h4>

<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/9-1-4ResolveComponentsQuestion.svg" alt="Arrow whcih represents a force of 5N angled at 30 degrees to the horizontal.  A set of axes indicates that the x axis is horizontal and the y axis is vertical." class="img-responsive atto_image_button_text-bottom" width="255" height="111">
</figure>
<p>The figure shows a force of \(5\ \mathrm{N}\) acting at an angle of \(30^\circ\) to the \(x\) axis. Resolve this force into its components parallel and perpendicular to the \(x\) axis.</p>
<h4 class="HELM_solution">Solution</h4>
<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/9-1-4ResolveComponentsAnswer.svg" alt="The above image but now also showing horizontal and vertical arrows indicating the components of the force in the x and y directions" class="img-responsive atto_image_button_text-bottom" width="480" height="141">
</figure>
<p> The force can be resolved into two components, one directed along the \(x\) axis with magnitude \(5\cos30^\circ\) and one perpendicular to this of magnitude \(5\sin30^\circ\). Together, these two components have the same effect as the original force as shown in the diagram.</p>]]></text>
<file name="9-1-4ResolveComponentsAnswer.svg" path="/" encoding="base64">[encoded data, 36172 characters]</file>
<file name="9-1-4ResolveComponentsQuestion.svg" path="/" encoding="base64">[encoded data, 26472 characters]</file>
<file name="9-1-4ResolvingComponents.svg" path="/" encoding="base64">[encoded data, 21512 characters]</file>
<file name="9-1-4ResultantVector.svg" path="/" encoding="base64">[encoded data, 30744 characters]</file>
    </questiontext>
    <generalfeedback format="html">
      <text/>
    </generalfeedback>
    <defaultgrade>0</defaultgrade>
    <penalty>0</penalty>
    <hidden>0</hidden>
    <idnumber/>
  </question>
</quiz>
