Moment of inertia of an object (8-6)

Question 1 of 4

Question

To find the moment of inertia $I$ of an object...

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Question

...we imagine dividing the object into $N$ small pieces, and calculate the sum...

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Question

...and the square of the distance $r_i$ from the $i$ th piece to the rotation axis.

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Question

...of the product of the mass $m_i$ of the $i$ th piece...

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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 $I$ :