Two masses, M and m, are connected by a string passing over a pulley. Assume that M > m. The pulley is a solid disk of mass m_{p} and radius r. What is the acceleration of the two masses?
Step 1:
Take +y up for mass M and mass m.
Take into plane (clockwise) to be positive for the pulley.
Step 5:
For mass M:    For mass m:    For the pulley:  
ΣF_{y} = Ma_{1y}    ΣF_{y} = ma_{2y}    Στ = I α  
T_{1}  Mg = Ma_{1}    T_{2}  mg = ma_{2}    rT_{1}  rT_{2} = ½ m_{p}r^{2}α 
5 unknowns, 3 equations of motion, and two constraints:
a_{1} =  a_{2} ≡ a
α = a /r
The pulley equation then becomes: T_{1}  T_{2} =½ m_{p} a
Combining the three equations to eliminate the two tensions gives:
(Mg  Ma)  (mg + ma) =½ m_{p} a
a  = 

Note: if m_{p} = 0, then a = g (M  m) /
(M + m).