Moment of inertia of a stiff wire bent into a wheel with spokes

[mcphyssics]

Homework Statement

A stiff uniform wire of mass M₀ and length L₀ is cut, bent, and the parts soldered together so that it forms a circular wheel having four identical spokes coming out from the center. None of the wire is wasted, and you can neglect the mass of the solder.

What is the moment of inertia of this wheel about an axle through its center perpendicular to the plane of the wheel? Express your answer in terms of the given quantities.

Homework Equations

I_rim = MR²
I_spoke = 1/3ML²
I_total = I_rim + 4*I_spoke

The Attempt at a Solution

The length of one spoke, which should also be the radius of the rim, should be L₀/(2\pi+4)

I am unsure of the mass of one spoke, as well as the mass of the rim. Would the mass of one spoke be the same ratio? (M_spoke = M₀/(2\pi+4)) ?
If so, then what would be the mass of the rim? I thought M_rim = M₀ - 4*M_spoke
therefore M_rim = (M₀*(1-4/(2\pi+4))
but I am not getting it right when I submit it.

 

[lewando]

What, exactly, are you submitting?

 

[mcphyssics]

This is what I have: I = (1-4/(2\pi+4))*M₀*(L₀/(2\pi+4))²+4*M₀/(2\pi+4)*(L₀/(2\pi+4))²

It seems very messy/cumbersome. Am I going about this wrong?