How to be verify fixed axis ?

Which forces needed for the rotational motion about a fixed axis ?

In the given figure linear acceleration of solid cylinder of mass $m_2$ is $a_2$. Then angular acceleration $\alpha_2$ is (given that there is no slipping).

What is combined translation and rotation motion ? 

Figure shows a small wheel fixed coaxially on a bigger one of double the radius. The system rotates about the common axis. The strings supporting $A$ and $B$ do not slip on the wheels. If $x$ and $y$ be the distances travelled by $A$ and $B$ in the same time interval, then

For the given figure find the acceleration of $1\, kg$ block if string is massless and mass of the pulley is $2\, kg$ and diameter of puller is $0.2\, m$ (in $m / s ^{2}$)