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      <text>15.2.1.1 The centre of mass of a collection of point masses</text>
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      <text><![CDATA[<h3> The centre of mass of a collection of point masses</h3>
<p>Suppose we have a collection of masses located at a number of known points along a line. The <strong>centre of mass</strong> is the point where, for many purposes, all the mass can be assumed to be located.</p>
<p>For example, if two objects each of mass \(m\) are placed at distances 1 and 2 units from a point \(O\), as shown in part (a) of the Figure below, then the total mass, \(2m\), might be assumed to be concentrated at distance 1.5 units as shown in part
    (b). This is the point where we could imagine placing a pivot to achieve a perfectly balanced system.</p>
<figure class="HELM_figure"><img src="@@PLUGINFILE@@/15-2-fig-2-CoM.svg" alt="15-2-fig-2.svg" width="600" height="230" class="img-responsive atto_image_button_text-bottom"><br>
    <figcaption>Equivalent position of the centre of mass of the objects in (a) is shown in (b)</figcaption>
</figure>
<p>To think of this another way, if a pivot is placed at the origin \(O\), as on a see-saw, then the two masses at \(x = 1\) and \(x = 2\) together have the same <strong>turning effect</strong> or <strong>moment</strong> as a single mass \(2m\) located at
    \(x = 1.5\). This is illustrated in the Figure below.</p>
<figure class="HELM_figure"><img src="@@PLUGINFILE@@/15-2-fig-3-CoM.svg" alt="15-2-fig-3.svg" width="600" height="262" class="img-responsive atto_image_button_text-bottom"><br>
    <figcaption>The single object of mass \(2m\) has the same turning effect as the two objects each of mass \(m\)</figcaption>
</figure>
<p>Before we can calculate the position of the centre of mass of a collection of masses it is important to define the term ‘moment’ more precisely. Given a mass \(M\) located a distance \(d\) from \(O\), as shown in the Figure below, its moment about \(O\)
    is defined to be <strong>moment&nbsp;</strong>\( = M\times d\)</p>
<figure class="HELM_figure"><img src="@@PLUGINFILE@@/15-2-fig-4-CoM.svg" alt="15-2-fig-4-.svg" width="600" height="125" class="img-responsive atto_image_button_text-bottom"><br>
    <figcaption>The moment of the mass \(M\) about \(O\) is \(M \times d\)</figcaption>
</figure>
<p>In words, the moment of the mass about \(O\), is the mass multiplied by its distance from \(O\). The units of moment will therefore be kg m if the mass is measured in kilogrammes and the distance in metres. (N.B. Unless specified otherwise these will
    be the units we shall always use.)</p>]]></text>
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