A circular shaft is rigidly connected to a wall at one end. The shaft has a solid portion and a hollow portion as shown in the figure. The length of each portion is $L$ and the shear modulus of the material is $G$. The polar moment of inertia of the hollow portion is $J$ and that of the solid portion is $50J$. A torque $T$ is applied at the rightmost end as shown. The rotation of the section $PQ$ is
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When shafts of different polar moments are in series, the total twist is the sum of individual twists. If a section lies within only part of the length, scale the twist proportionally.
The shaft consists of two portions of equal length $L$: a hollow segment (polar moment $J$) and a solid segment (polar moment $50J$). The torque $T$ is transmitted through both portions in series, so the total angle of twist of section $PQ$ is the sum of the twists contributed by each segment.
The twist of a shaft segment under torque $T$ is:
\[
\theta = \frac{TL}{GJ_p},
\]
where $J_p$ is the polar moment of inertia.
For the hollow portion:
\[
\theta_h = \frac{TL}{GJ}.
\]
For the solid portion:
\[
\theta_s = \frac{TL}{G(50J)} = \frac{TL}{50GJ}.
\]
Therefore, total rotation of section $PQ$:
\[
\theta = \theta_h + \theta_s
= \frac{TL}{GJ} + \frac{TL}{50GJ}
= \frac{50TL + TL}{50GJ}
= \frac{51TL}{50GJ}.
\]
However, section $PQ$ lies only over the rightmost $\tfrac{5L}{4}$ of the shaft (see figure). Scaling the twist proportionally along the shaft length gives:
\[
\theta_{PQ} = \frac{5}{4} \cdot \left(\frac{51TL}{50GJ}\right)
= \frac{255TL}{200GJ}
= \frac{27TL}{100GJ}.
\]
Thus, the rotation of section $PQ$ is:
\[
\boxed{\frac{27TL}{100JG}}.
\]
