we have same tension in both parts ( as it is the same string)
cancelling forces in horizontal drirection
T+Tcosθ=mw2(3L)
Tsinθ=mg
dividing
(1+cosθ)/sinθ =w2(3L/g)
further cosθ=3/5 sinθ=4/5
2=w2(3L/10)
w2=20/3L
we get w and hence all the value of speed tension etc.