Class Notes on Rotational Motion - Galileo and Einstein

11If let go from a horizontal position, the weight will completely dominate the situation, and accelerateddownwards very close to g. This means the far end of the light rod is forced downwards at close to 2g.Our example above is not that extreme, but the same idea.The Parallel Axis TheoremIf we know the moment of inertia of an object about a line through the center of mass, the moment ofinertia about any parallel line is easily found.We’ll prove this for a two-dimensional object (really a thin three-dimensional object):new axisrih m ir icenter of massThe moment of inertia of the two-dimensional object about any axisperpendicular to the object is simply related to that about theparallel axis through the center of mass.With the notation in the diagram,ICM2 miri, and recall mr i i0.iiNow I m r m ( r h)newaxisiCMCM2 2i i i ii I 2h m r m hIMh2.ii iii2This is the Parallel Axis Theorem:I I Mh2newaxisCM .where h is the perpendicular distance between the new axis and the parallel axis through the center ofgravity. It’s easy to extend this proof to a three-dimensional object: taking the parallel axes to be in thez-direction, replace r iwith (x i , y i ).