Moment of Inertia

The moment of inertia describes how mass is distributed around a point. How we calculate it depends on the type of system. We call a system discrete if it is composed of only point like particles. We call a system continuous if you cannot describe your system that way. Essentially, can you invoke the particle approximation for each constituent object in your system? If yes, your system is discrete If not, it is continuous. At the moment we consider discrete systems here.

Moment of Inertia in one dimension for a discrete system

We have a system of \(N\) objects that can each be modeled as a particle. Each object has mass \(m_i\) and position \(x_i\) where \(i\) is a index distinguishing one object from another and runs from 1 to \(N\). The moment of inertia about the origin is $$I = \sum_{i=1}^{N} m_ix_i^2$$

For the case where the moment of inertia is calcualted about some other point, replace \(x_i\) with the distance from the location of the particles to the axis of rotation.

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Moment of Inertia in 2D

There is nothing special about doing it in two dimensions, except you have to do it twice.

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Exercises

Solutions

Worked Examples

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