![]() ![]() Note: The lower limit is -h because the left side of the rod is -h units away from the axis of rotation. Substituting dI, (write the appropriate limits) Using the equation for dm, we substitute it into the first equation. Since the rod is uniform, the mass varies linearly with distance. The moment of inertia relative to the z-axis is then. Derivationedit We may assume, without loss of generality, that in a Cartesian coordinate systemthe perpendicular distance between the axes lies along the x-axis and that the center of mass lies at the origin. ![]() Hence, we have to force a dx into the equation for moment of inertia. Now, lets find an expression for dm. Parallel axes rule for area moment of inertia. Hey, there is a dm in the equation! Recall that we’re using x to sum. Now, we show our formula for the calculation for moment of inertia first: In other problem, the variable can be theta or r or … In this problem, we are summing from left of the axis to right of axis.
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