Moment of inertia of an object (8-6)
Question
...and the square of the distance \(r_i\) from the \(i\)th piece to the rotation axis.
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Review
What is the quantity in parentheses in Equation 8-5, \(\sum_{i=1}^Nm_i r_i^2\)? Although it involves a sum over all the little pieces into which we’ve divided the turbine blade, it is \(\textit{not}\) simply the total mass \(M\) of the blade. That sum would be \(M = \sum_{i=1}^Nm_i\), without the factor of \(r_i^2\). Instead, the sum \(\sum_{i=1}^Nm_i r_i^2\) is a new quantity that tells us how the mass of the blade is \(\textit{distributed:}\) It depends on both the mass of each small piece \(m_i\) and how far away from the rotation axis that piece is (\(r_i\)). This quantity is called the \(\textbf{moment of inertia}\) (sometimes called \(\textit{rotational inertia}\)) of the turbine blade. We represent it by the symol \(I\):
