The angle of twist in shaft $\left(1\right)$ can be determined by considering the angles of rotation at its two ends.

Moving in the $+\mathrm{x}$ axis direction from $\mathrm{A}$ to $\mathrm{B}$, the angle of twist is simply the rotation angle at the forward end $\mathrm{B}$ minus the rotation angle at the aft end $\mathrm{A}$:

${\mathrm{\varphi }}_1={\mathrm{\varphi }}_{\mathrm{B}}-{\mathrm{\varphi }}_{\mathrm{A}}$

Since $\mathrm{A}$ is a fixed support, the rotation angle at $\mathrm{A}$ is ${\mathrm{\varphi }}_{\mathrm{A}}=0$. Therefore:

${\mathrm{\varphi }}_1={\mathrm{\varphi }}_{\mathrm{B}}$