To find the moment of inertia of the disk with an off-center hole, we can use the parallel-axis theorem. The parallel-axis theorem states that the moment of inertia of a body about an axis parallel to and at a distance 'd' from an axis through its center of mass is given by:
I = Icm + Md^2
Where I is the moment of inertia about the parallel axis, Icm is the moment of inertia about the center of mass axis, M is the mass of the body, and d is the distance between the two axes.
In this case, we can consider the disk without the hole as a solid disk. The moment of inertia of this solid disk about its center can be calculated using the formula for the moment of inertia of a cylinder:
Icm = 1/2 * MR₀²
Where M is the mass of the disk and Râ‚€ is the radius of the disk.
Now, let's consider the moment of inertia of the disk with the hole. Since the hole is off-center, we need to use the parallel-axis theorem. The distance between the axis through the center of the disk and the axis through the center of the hole is h.
So, the moment of inertia of the disk with the hole about its center is:
I = Icm + Md^2
Substituting the values, we get:
I = 1/2 * MR₀² + Mh²
Therefore, the moment of inertia of the disk with the off-center hole when rotated about its center, C, is given by:
I = 1/2 * MR₀² + Mh²