Does the mass of a pulley affect the speed of an object that's hung over it? Here is the question: a pulley has a mass M and an object of mass $m_1$ on a massless string is wrapped around it, does the mass of the pulley affect the acceleration of $m_1$? 
My reasoning is that it wouldn't, because the acceleration of the object only has to do with $a={F\over m}$, and there doesn't seem to be any place for the $M$ mass. If the mass $M$ does affect the acceleration, would the acceleration equal something like ${{(m_1+M)g}\over (m_1-M)}$?
From what I understand is that the pulley would affect the acceleration of the $m_1$, which leads me to believe that the mass of the pulley could have something to do with the acceleration (even though this isn't my first intuition). However, I'm not sure how to fit the $M$ into the equation for acceleration. How would I include $M$ into the equation for the acceleration of the mass $m_1$ (assuming that $M$ does need to be included)?
 A: The object is dropping and losing potential energy.
The object is also gaining kinetic energy as it falls faster and faster.
And, since the object is attached to the string wrapped around the pulley, the pulley is rotating faster and faster, in synch with the falling object.
If the pulley were massless, these two (the object's potential energy and kinetic energy) would be the only energy terms present, and conservation of energy could be used.
However, since the pulley is explicitly not massless, the increasing rotational velocity leads to an increasing rotational kinetic energy, another energy term that must be included.
But there is insufficient information to do this!  You would need the specifications of the pulley: radius and mass distribution to allow the calculation of the pulley's moment of inertia.
You can say that as the pulley mass increases, the acceleration of the object decreases.  How much is impossible to calculate...
A: The larger the mass of the pulley the less the acceleration of the object. If you know the the mass and moment of inertia of the pulley then you can calculate the acceleration. Note that for the most common pulley shapes (e.g. disc, hoop and disc, mostly hoop), the acceleration will be independent of the radius. Moment of inertia varies with $R^2$, while tangential force varies with $1/R^2$ so these cancel out. Have a look at the picture and a link to the slide containing that picture.
http://slideplayer.com/slide/8333507/
