The option that has the true statement about the moment of inertia of a circular disk, a ring, and a square is;
Option B; I_square > I_ring > I_disk
Formula for moment of Inertia of a circular disk about its' center of mass about an axis perpendicular to the plane of the paper is;
I_disk = ¹/₂Mr²
Formula for moment of Inertia of a ring about its' center of mass about an axis perpendicular to the plane of the paper is;
I_ring = Mr²
Formula for moment of Inertia of a hollow square about its' center of mass about an axis perpendicular to the plane of the paper is;
I_hollow square = ¹/₁₂(R⁴ - r⁴)
Where R is the outer width and r is the inner width.
We are told that the disk, ring, and square all have the same mass M and width 2r.
Thus;
I_disk = ¹/₂M(2r)²
I_disk = 2Mr²
I_ring = M(2r)²
I_ring = 4Mr²
I_hollow square = ¹/₁₂M((2r)⁴ - r⁴)
I_hollow square = ¹/₁₂M(15r⁴)
I_hollow square = ⁵/₄Mr⁴
Thus, we can see that The greatest is I_hollow square, second greatest is I_ring and third greatest is I_disk
In conclusion, the order from greatest to least is I_square > I_ring > I_disk
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