$(1)$
\[
R\cdot \sin\theta=F\]
\[
\therefore
F-R\cdot \sin\theta=0
\]
$(2)$
\[R\cdot \cos\theta =mg\]
\[
\therefore
-mg+R\cdot \cos\theta=0
\]
$(3)$
\[
F=m r \omega^2 = m r \left(\frac{v}{r} \right)^2=m\cdot \frac{v^2}{r}\]
\[
\therefore
F=m\cdot \frac{v^2}{r}
\]
$(4)$
\[
\frac{R\cdot \sin\theta}{R\cdot \cos\theta}=\tan\theta=\frac{m\cdot \frac{v^2}{r}}{mg}=\frac{v^2}{rg}\]
\[
\therefore
\theta=\tan^{-1}\left(\frac{v^2}{rg} \right)
\]
$(5)$
\[
\theta=\tan^{-1}\left\{\frac{\left(30\times 10^3 \times \frac{1}{3600}\,\rm{m/s} \right)^2}{40\,\rm{m}\times9.81\,\rm{m/s^2}} \right\}
=10.03^\circ\]
\[
\therefore
\theta=10.0^\circ
\]