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Sunday, November 30, 2014

Moment of Inertia when there are Holes in the Object - Yikes!

Moments of inertia are essential to rotating objects. This quantity replaces mass in our equations that we are used to for linear motions, and it represents the distribution of mass about the axis of rotation. For a point mass, for instance, I = mr^2, where r is the distance from the axis to the mass. I have videos for sticks and disks, using the integral version for I.

Then there is the parallel-axis theorem, which is for finding the moment of inertia for an object that rotates about an axis that is not its center of mass. Imagine spinning a disk around a point on its edge! This theorem says I = I_cm + md^2, where I_cm is the usual moment of inertia for its axis at the center of mass, and d is the distance from the center of mass to the new axis of rotation.

Here, we take it one step further. What if the object has a hole drilled out, so some of the mass is missing, and the object still rotates? Huh?!?! Check it out. These are nice examples of rotations, moments of inertia, and the parallel-axis theorem for an unusual situation.

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