A straight wire has uniform density and total mass M. The wire is bent to form a closed loop, one section of which is a semi-circle of radius a, and the other section the diameter joining the two ends of the semicircle. The body is free to move about the midpoint O of its straight edge. It rotates at constant angular speed about an axis through O in the plane of the body and perpendicular to its straight edge.
Using T=Jij*ωi*ωj/2 or otherwise, show that T=(1/12)Ma^2ω^2((3π+4)/(π+2))
The Attempt at a Solution
I've worked out that ρ=M/(a(π+2))
I'm struggling with how to do the integrals, as its such an odd shape off to the examples we've done (cubes, etc). I tried to do the semi-circle bit and the diameter bit separately, but then I don't get aything like the answer I'm supposed to. Any pointers would be appreciated! :)