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