Cylinder and hanging mass

A uniform cylinder of mass M sits on a fixed plane inclined at an angle \theta. A string is tied to the cylinder’s rightmost point, and a mass m hangs from the string. Assume that the coefficient of friction between the cylinder and the plane is sufficiently large
to prevent slipping. What is m, in terms of M and \theta, if the setup is static?

Screenshot 2015-01-15 at 21.54.43

Problem Source : Introduction to Classical Mechanics

As always, we first identify the forces exerted on the cylinder :

Screenshot 2015-01-15 at 21.54.43(2)

In order to achieve a static state, we must have a balanced forces and balanced torques. It has stated that the coefficient of friction between the cylinder and the plane is sufficiently large to prevent slipping, so we can skip the calculation of balancing forces.

For balancing torques, we take the point where the cylinder is in touch with the plane as fulcrum. So we have

\begin{aligned}  \displaystyle Mg(R\sin \theta) &= mg(R-R\sin \theta) \\ \displaystyle m &= M\bigg ( \frac{\sin \theta}{1 - \sin \theta} \bigg )  \end{aligned}

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Cylinder and hanging mass

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