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      <text>15.2.2.6 COM of semicircular lamina (old copy)</text>
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      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p></p><p>Find the position of the centre of mass of a uniform semi-circular lamina of radius \(a\), shown below.<br></p><p><img src="@@PLUGINFILE@@/15.2.2%20Exercise%203.png" alt="" role="presentation" class="img-responsive atto_image_button_text-bottom" width="400" height="193"><br></p><p>The equation of a circle centre the origin, and of radius \(a\) is \(x^2 + y^2 = a^2\).
By symmetry \(\bar{x} = 0\). However it is necessary to calculate \(\bar{y}\).
</p><p><br></p><ol class="HELM_parts">
<li><p>The lamina is divided into a number of horizontal strips and a typical strip is shown. Assume
that each strip is rectangular. Writing the mass per unit area as \(\rho\), state the area and the mass of
the strip: </p><p> Note: write \(\delta\) and \(\rho\) as 'delta' and 'rho'</p><p>Area: [[input:ans1]] [[validation:ans1]] [[feedback:prt1]]</p><p>Mass: [[input:ans2]] [[validation:ans2]] [[feedback:prt2]]</p></li>
<li><p>Write down the moment of the mass about the <i>x</i>-axis: </p><p>[[input:ans3]] [[validation:ans3]] [[feedback:prt3]]</p></li><li><p>Write down the expression representing the sum of the moments of all strips and the corresponding
integral obtained as \(\delta y \to 0\):&nbsp;<br></p>\(\displaystyle{\sum_{y=0}^{y=a}}\) [[input:ans6]][[validation:ans6]]<p>\(\displaystyle{\int_{0}^{a}}\) [[input:ans9]]\(dx\)[[validation:ans9]]</p><p>[[feedback:prt6]][[feedback:prt9]]</p></li><li>Evaluate the above integral by making the substitution \(u^2 = a^2 - y^2\) to obtain the total moment. Note that since \(x^2 + y^2 = a^2\) we have \(x =
\sqrt{ a^2 - y^2}\).<br><p> [[input:ans10]] [[validation:ans10]] [[feedback:prt10]]</p><p></p><p></p><p></p></li><li><p>The total area is half that of a circle of radius \(a\), that is \(\dfrac{\pi a^2}{2}\). The total mass is \(\dfrac{\pi a^2 \rho}{2}\) and its
moment is \(\dfrac{\pi a^2 \rho \overline{y}}{2}\).
</p><p> Deduce \(\bar{y}\):&nbsp;<span style="font-size: 0.9375rem;">[[input:ans11]] [[validation:ans11]] [[feedback:prt11]]</span></p></li>
</ol>]]></text>
<file name="15.2.2 Exercise 3.png" path="/" encoding="base64">[encoded data, 91508 characters]</file>
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      <text><![CDATA[<h4 class="HELM_solution">Solution</h4>

<ol class="HELM_parts">

  <li>\(2x \delta y,\ 2 x \rho \delta y\)</li>

  <li>\(2x \rho y \delta y\)</li>

  <li>\(\sum_{y=0}^{y=a}{2x \rho y \delta y},\  \int_{0}^{a} 2x \rho y \textrm{d}y\)</li>

  <li>\(\frac{2 \rho a^3}{3}\)</li>

  <li>\(\frac{\pi a^2 \rho \bar{y}}{2}  = \frac{2 \rho a^3}{3}\) from which \(\bar{y} = \frac{4a}{3\pi}\)</li></ol>]]></text>
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      <text>ta1:2*x*delta*y;
ta2:2*x*rho*delta*y;
ta3:2*x*rho*y*delta*y;

ta6:2*x*rho*y*delta*y;

ta9:2*x*rho*y;

ta10: (2*rho*a^3)/3;
ta11:(4*a)/(3*pi)</text>
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