Can someone please explain what the parallel axis theorem is?

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The parallel axis theorem (PAT) allows for the calculation of an object's moment of inertia (MI) about any axis parallel to one through its center of mass, using the known MI at the center of mass. When the axis is shifted away from the center of mass, the moment of inertia increases, which is illustrated by the example of rotating a disc around a point on its circumference. The discussion also draws parallels between the PAT and concepts in statistics, such as standard deviation, highlighting that the second moment behaves similarly in both contexts. Additionally, it notes that the power of an AC waveform increases with a DC offset, further emphasizing the relationship between these mathematical principles. Understanding the PAT is crucial for applications in physics and engineering involving rotational dynamics.
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Can someone please explain what the parallel axis theorem is?
 
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The PAT tells you how the MI of an object changes when you use an axis that's not through the cm. Obviously, the mi will be higher (imagine trying to turn a disc around a point on its circumference - or even on the end of a long massless pole)
There is an analogous situation with standard deviation of a probability distribution. It is a 'second moment', in the same way. The standard deviation from the mean is less than the rms deviation from some other value. Also, the power of an AC waveform is greater when it has a DC offset component. Same sums for all three, basically.
 

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