A uniform cylinder of radius $R$ and mass $M$ can rotate freely about a stationary horizontal axis $O$ figure. A thin cord of length $\mathcal{l}$ and mass $m$ is wound on the cylinder in a single layer. Find the angular acceleration of the cylinder as function of the length $x$ of the hanging part of the cord. The wound part of the cord is supposed to have its centre on gravity on the cylinder axis.

$$\begin{gathered} \mathrm{The} \mathrm{mass} \mathrm{of} \mathrm{the} \mathrm{hanging} \mathrm{part} \mathrm{of} \mathrm{the} \mathrm{cord} \\ \mathrm{m}\text{'}= \frac{\mathrm{mx}}{\mathrm{R}} \end{gathered}$$

$$\begin{gathered} \mathrm{Thus} m\text{'}g -T= m\text{'}a ....\left(\mathrm{i}\right) \\ \mathrm{and} TR=l \alpha \\ \mathrm{or} T=\frac{Ia}{R^2} ...\left(\mathrm{ii}\right) \\ \mathrm{On} \mathrm{solving} \mathrm{equations}, \mathrm{we} \mathrm{get} \\ a = \frac{m\text{'}g}{m\text{'}+{\displaystyle \frac{l}{R^2}}} \\ \mathrm{Here} l =\frac{m{R}^2}{2}+\frac{m}{l}\left(l-x\right)R^2 \\ \mathrm{After} \mathrm{simplifying}, \mathrm{we} \mathrm{get} \\ \alpha = \frac{2mgx}{Rl\left(M+2m\right)} \end{gathered}$$