Calculus Based Section

Calculus Based Section - Calculus Based Section: Rotational...

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Calculus Based Section: Rotational Inertia of Solid Bodies In our study of rotational dynamics we skipped over exactly how to calculate the rotational inertia of a solid body. The process for calculating this quantity is quite complicated, and requires quite a bit of calculus. Thus we devote a section to calculating this quantity. Consider a small section of a rod, a radius r from the axis of rotation, and with a mass δm , as shown below: Figure %: A small piece of mass on a rod being rotated about an axis. Because the volume of the section of the rod is sufficiently small, we can calculate the moment of inertia of this single piece: I = δmr 2 . To find the moment of inertia of the entire rod, we sum over all pieces of a similar size that compose the rod: I r k 2 δm k To get an exact answer for the moment of inertia, we take the limit as the δm gets smaller; as the rod is broken up into more and more pieces. Thus: I = r k 2 δm k = r 2 dm This integral equation is the basic equation for the moment of inertia of a solid body. Even with this equation, it is quite difficult to calculate the moment of inertia of a solid body. We
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This note was uploaded on 02/09/2012 for the course PHY PHY2053 taught by Professor Davidjudd during the Fall '10 term at Broward College.

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Calculus Based Section - Calculus Based Section: Rotational...

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