A spool of mass $m$, moment of inertia $I$ relative to its axis, and inner radius $r$ is placed on a

Question

A spool of mass $m$, moment of inertia $I$ relative to its axis, and inner radius $r$ is placed on a smooth inclined plane set at an angle $\alpha$ to the horizontal. A thread wound on the spool has its free end attached to a wall parallel to the incline. Find the acceleration $w$ of the spool's axis.

Professor PC Thomas Classes · Answer

Let tension be $T$. Force equation along incline: $mg \sin\alpha - T = mw$. Torque equation about CM: $Tr = I\alpha_{ang}$. Constraint for no slipping of the thread: $w = \alpha_{ang} r \Rightarrow \alpha_{ang} = w/r$. Substituting $T = Iw/r^2$ into the force equation gives $mg \sin\alpha - Iw/r^2 = mw \Rightarrow w = \frac{mg \sin\alpha}{m + I/r^2}$.