To calculate the moment of inertia of a cube about one of its edges, we can use the theorem of perpendicular axes. Here's how it is derived:
First, we consider the moment of inertia about the center of the cube. According to the theorem of perpendicular axes, the total moment of inertia
I can be found by adding the contribution from each axis. For a cube, with side length
a and mass
m, the moment of inertia about one of its center axes is:
I‌center ‌=‌Since the cube is symmetric, this value is the same for any of the three perpendicular axes intersecting at the center. The moment of inertia through the axis parallel to a face through the center can similarly be calculated.
Now, when calculating the moment of inertia about an edge, we can add the parallel axis contribution:
I=I‌center ‌+m(‌)2This accounts for the shift in the axis position. With simplification:
I‌=[‌+‌]+‌‌=‌ma2+‌‌=‌+‌‌=‌ma2Thus, the moment of inertia of a cube of mass
m and side
a about one of its edges is
‌ma2.