We have shown that radius vector joining the bead to the centre of the wire makes an angle θ with the vertical downward direction.
If N is normal reaction, then as is clear from the figure,
mg=N=cos‌θ...(i)
mrω2=N‌sin‌θ..(ii)
or m(R‌sin‌θ)ω2=N‌sin‌θ or mRω2=N
from ( i),mg=mRω2‌cos‌θ
or cos‌θ−

g

Rω2

...(iii)
As |cos‌θ|≤1, therefore, bead will remain at its lowermost point for

g

Rω2

≤1, or ω≤√

g

R

When ω=√

2g

R

, from (iii)
cos‌θ=

g

R

(

R

2g

)=

1

2

∴θ=60°