General method for approaching block/rotating pulley/rotating object equations

[Same-same]

Not asking about a specific problem, but general methods. I'm having trouble with problems that usually involve tension, rotating objects, rotating pulleys, and one block free to fall due to gravity. I'm not asking for help with a specific problem, but with this approach in general. It seems like it should be simple, but for whatever reason I always freeze up whenever I try to solve one of these problems, so I'm trying to better understand rotation.

My current line of reasoning is to use force diagrams and then
Since $\tau$ = RF sin ($\theta$) =I$\alpha$,
F =((I$\alpha$)/(R sin($\theta)$))

Since the net force is generally provided by one or more objects allowed to fall
ƩF$_{y}$ (generally equal to m$_{falling object}$g- F$_{tension}$ )=m$_{total}$a + ((I $_{1}$ $\alpha$)/ (R$_{1}$ sin ($\theta)$))
+ ((I $_{2}$ $\alpha$))/ (R$_{2}$ sin ($\theta$) ))... and so on for any of the other I$\alpha$ and ma.
I will then generally plug in a/R for $\alpha$

At this point, I'm generally not sure what to do. Possible ideas are to do a sum of torques equation, or to attempt to solve for Tension in the x direction.

Another thing that confuses me is when to include translational velocity for a rotating object in my equations.

Thanks in advance.

 

[kuruman]

The general method is to
1. Draw free body diagrams (FBDs) for each identifiable component of the system. For example, if you have two masses hanging on either side of a pulley that is not assumed massless, you will need three separate FBDs, two for the hanging masses and one for the pulley.
2. Identify the forces in each diagram and label them appropriately. This means use different labels for different forces, however if $F_1$ in one FBD has a reaction counterpart in another, then they should be labeled the same and point in opposite directions in the diagram; this is the case most notably with tension. Also note that the tension on either side of a pulley with mass (and hence moment of inertia) has to be labeled differently, otherwise the pulley will not acquire angular acceleration.
3. Based on each of the FBDs write Newton's Second Law $F_{net,i}=m_i a_i$ or $\tau_{net,i}=I_i\alpha_i$ for each of the components of the system.
4. Establish relations between linear accelerations and between linear accelerations and angular accelerations. These will depend on how the total system is assembled.
5. The Newton's Second Law equations should be sufficient to solve for the unknowns.

You include translational velocity for a rotating object when the center of mass of the object is translating relative to your frame of reference. For example, if you are riding a unicycle, you do not include the translational velocity in the calculation of its wheel's kinetic energy. If you are standing by the side of the road and watch the unicycle go by, you do.