Determine the mass moment of inertia of a quarter of an annulus

[jaredogden]

Homework Statement

B.1

The quarter ring has mass m and was cut from a thin uniform plate. Knowing that r₁ = 1/2r₂ determine the mass moment of inertia of the quarter ring with respect to A. axis AA' B. The centroidal axis CC' that is perpendicular to the plane of the quarter ring.

CC' is located possibly where the center of mass is? (See attached file). EDIT: obviously it is the centroid.. the problem says centroidal axis, I'm dumb..

Homework Equations

I_AA' = 1/4mr²
I_CC' = 1/2mr²

Parallel Axis Theorem
I = I(bar) + md²

The Attempt at a Solution

I originally tried to just use the mass moments of inertia to calculate it. I then realized that the center of mass of the quarter annulus will not be at the origin O in this case so I probably will have to use the parallel axis theorem.

I am really lost on this and I originally calculated
I_AA' = (1/16)m(r₂² - (1/2)r₂²)
I_CC' = (1/8)m(r₂² - (1/2)r₂²)

 

[jaredogden]

I've calculated I(bar)_AA' = (1/4)[(1/4)(mr₂² - (1/2)mr₂²)] and simplified this to:
I(bar)_AA' = (1/32)mr₂2

Then adding that to md² where d = (1/2)r₁ → (1/4)r₂ from the parallel axis theorem to get
(1/32)mr₂² + m(1/4)r₂² = (9/32)mr₂²

The answer given by my professor is (5/16)mr₂²
Did I make a mistake somewhere or is the provided answer wrong?