Maximum Angular Velocity Object falling disk

Question

We have a small problem. We designed a drum with a small rope around it, at the end of the rope a small weight is attached. We want to calculate the maximum angular velocity of the drum. The rope is 1m long. The weight attached are of a mass of 200 gram. The drum is hollow with an outer radius of R1 and inner radius of R2. And the drum has a mass of m kg. How can we calculate the maximum angular or rotational velocity of the drum?

Answer 1

A diagram would do wonders in this case :D. Since you omitted the fact that there is air resistance on the mass and roll friction on the drum, the only limiting factor is the length of the rope. Lets assume for simplicity that the mass (denoted with $M$) is located at the side of the drum initially: $$Mg = Ma = F_t$$ Where $g$ is the gravitational acceleration, $a$ the acceleration of the mass and $F_t$ is the tangential force on the disk. Furthermore: $$F_tR_1 = I\alpha$$ where $I$ is the moment of inertia of the disk (this can be computed using standard formulas found in your book or online) and $\alpha$ is the rotational acceleration. From here up to the moment the rope is fully unwound, these relations remain constant. This means that you have to compute when the mass has traveled $1m$ (through integrating for instance). Compute the rotational velocity using the resulting time from the angular acceleration.

EDIT: I am completely forgetting one fact, this only works when the rope is actually way longer and only travels 1m. If the endpoint of the rope has passed the startpoint of the mass (eg the side of the disk), the gravitional acceleration of the mass will not be transferred directly to the tangential force on the disk, but a component of it is transferred to the normal force $F_n$ on the disk (this points towards the center): $$F_t = \text{cos}(\theta)Ma$$ $$F_n = \text{sin}(\theta)Ma$$ where $\theta = 0$ at the startpoint, so at the bottom point of the disk, $\theta = 0.5\pi$ all the force from the mass is countered by the normal force on the disk. Since the tangential force will be negative for any angle afterwards, the maximum angular velocity will be reached here.