Calculate the acceleration of a Maxwell disc

Question

I've been working on a mechanics question recently and getting a bit stuck on it. The question is

if we have a Maxwell disc of radius $R$ and mass $M$ connected to a
  shaft of radius $r$ and mass $m$, and the system is allowed to fall
  freely under gravity, what is the acceleration of the system?

I tried to solve this problem using conservation of energy, assuming the system loses potential energy converted to translational + rotational kinetic energy as $-mgh=K_{rot}+K_{trans}$, but we are not given $h$ explicitly in the question, and so I can't use this exact method. I have thought about doing it with Newton's second law and force balancing, to obtain an equation such as: $$F_{net_y}=mg+Mg-2T=(m+M)a$$
where $a$ is the translational acceleration of the system, and $T$ is the tension in the strings of the apparatus, but I get stuck at this point.

Thanks.

Farcher · Answer

Use $(m+M)gh=K_{rm rot}+K_{rm trans}$ and the relationship between the final linear speed $v_{rm f}$ and the angular speed to find $h$ as a function of $v_{rm f}^2$.

As the acceleration is constant you can use the kinematic equation $v_{rm f}^2= v_{rm i}^2+2ah$ to find the acceleration.

Using forces you can use Newton's second law $F=ma$ as one equation with tension $T$ in it and you can also use the relationship between the angular acceleration $alpha$ about the centre of mass and the torque about the centre of mass $tau$ which again will be related to the tension $tau =I_{rm com}alpha$.
You can thus eliminate the tension and Using the relationship between linear acceleration and angular acceleration you can find the linear acceleration.

As an exercise it might be worth trying to use both methods?

Calculate the acceleration of a Maxwell disc

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