Tell me where I've gone wrong (here r(1,2) means position of point 1 with respect to point 2 etc):
ω x r(1,2)=v(1,2)
Differentiating both sides with respect to time:
ω x v(1,2) + \alpha x r(1,2) = a(1,2)
=>ω x (ω x r(1,2)) + \alpha x r(1,2) = a(1,2)
Now let us imagine a uniform rod in free...