$(1)$
\[
U=\frac{1}{2}Iw^2=\frac{1}{2}I\left(\frac{\pi n}{30}\right)^2
\]
$(2)$
\[
\tau_{max}=\frac{T}{I_p}r
\]
$(3)$
\[
U=\frac{T^2}{2GI_p}l
\]
$(4)$
\[
U=\frac{T^2}{2GI_p}l=\frac{\tau^2I_pl}{2Gr^2}=\frac{\tau^2\pi r^4l}{4Gr^2}=\frac{\pi r^2\tau^2l}{4G}
\]
$(5)$
\[(1)=(4)より\\
\frac{\pi r^2\tau^2l}{4G}=\frac{1}{2}I\left(\frac{\pi n}{30}\right)^2\\
\tau^2=\frac{2G}{r^2l}\left(\frac{\pi n}{30}\right)^2\pi I
\]
\[
\therefore\tau=\frac{n}{30r}\sqrt{\frac{2\pi GI}{l}}
\]