Question

About what axis would a uniform cube have its minimum moment of inertia

Answer 1

Compare the moment of inertia across different axes of the cube. Use the theorem of parallel axis of the moment of inertia to compare the moment of inertia across the axes with respect to the axis of the centre of mass. Hence we can easily come to a conclusion how the moment of inertia varies with respect to different axes and which one is the axis of minimum inertia.

Complete step by step answer:
The moment of inertia is the tendency of an object in circular motion to resist the change in its angular position on application of rotational force i.e. torque. The moment of inertia of any body is given by $I=M{{R}^{2}}kg{{m}^{2}}$ where M is the mass of the body and R is the distance from the distribution of mass to the axis about which the moment of inertia is to be calculated.
The theorem of parallel axis of moment of inertia, mathematically written as
$I={{I}_{CM}}+M{{d}^{2}}...(1)$ where I is the moment of inertia parallel through an axis which is parallel to the axis passing through the centre of mass, ${{I}_{CM}}$ is the moment of inertia along the axis passing through the centre of mass, M is the mass of the body and d is the distance between the axis whose moment of inertia is to be calculated and the parallel axis passing through centre of mass.
If we observe equation 1, then we can comment that Moment of inertia along any axis will always be more than the moment of inertia along the axis passing through the centre of mass. But now the question arises is it the axis through the diagonal or the axis in the middle along the length of the cube.
The moment of inertia in general is given by $M{{R}^{2}}$. If the axis is passed through the middle of the cube then R will be constant i.e. it will be half the length of the cube, but if the axis is passed through the diagonal, then the R will not be constant. This length will be sometimes greater by some factor x than R and sometimes less by x than R. Since this increase and decrease is the same we can conclude that the average distance from the axis to be R.
Hence the moment of inertia through the centre of mass will be minimum and the axes can be oriented in any direction.

Note:
The above condition is hypothetical i.e. The moment of inertia of a cube is minimum along the middle of the cube. There is no doubt that the moment of inertia will be minimum along the axis passing through the centre of mass. But if the mass distribution is not uniform then the orientation of the axis through the centre of mass also matters.