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      <text>27.3.4.5 moment of inertia</text>
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      <text><![CDATA[<h4> Moment of inertia</h4>
<p>The moment of inertia \(I\) of a particle of mass \(M\) about an axis \(PQ\) is defined as \(\displaystyle I = \mbox{Mass}\times\mbox{Distance}^2\mbox{or} I=Md^2\) where \(d\) is the perpendicular distance from the particle to the axis.<br> 
To find the moment of inertia of a larger object, it is necessary to carry out a volume integration over all such particles. The distance of a particle at \((x,y,z)\) from the \(z\)-axis is given by \(\sqrt{x^2+y^2}\) so the moment of inertia of an object about the \(z\)-axis is given by 
\[\displaystyle I_z = \displaystyle\int_V \rho(x,y,z) (x^2+y^2) \;\mathrm{d}z.\]
Similarly, the moments of inertia about the \(x\)-axis and \(y\)-axis are given by 
\[\displaystyle I_x = \displaystyle\int_V \rho(x,y,z) (y^2+z^2) \;\mathrm{d}x \quad\text{ and }\quad I_y = \displaystyle\int_V \rho(x,y,z) (x^2+z^2) \;\mathrm{d}y\] 
In the case where the density is constant over the object, so \(\rho (x,y,z)=\rho\), these formulae reduce to \[\displaystyle I_x = \rho \displaystyle\int_V (y^2+z^2) \;\mathrm{d}x \quad\text{ , }\quad I_y = \rho \displaystyle\int_V (x^2+z^2) \;\mathrm{d}y\quad\text{ and }\quad I_z = \rho \displaystyle\int_V (x^2+y^2) \;\mathrm{d}z\]
When possible, the moment of inertia is expressed in terms of \(M\), the mass of the object.</p>]]></text>
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