Question

Near the surface of the Earth, a wooden block with mass m = 2 kg is attached
to a string. The string is wrapped around a frictionless pulley with a radius
R = 0.5 m, and rotational inertia I = 2.5 kg·m2 as shown in the figure. The
pulley and the block are initially at rest. When the system is released and the
string begins to unwind, what is the tension in the string (in N)?

Answer 1

\[ma=mg-T\]
\[I\alpha=T\]
\[\alpha=a/R\]

Answer 2

we can consider the subsequent coordinate system:

on your mass are acting two forces, namely the tension T exerted by the string, and the weight m*g exerted by the eart.
Now the subsequent scalar equation holds:
\[\Large \left\{ \begin{gathered}
T - mg = m\frac{{{d^2}z}}{{d{t^2}}} \hfill \\
\frac{I}{R}\frac{{{d^2}z}}{{d{t^2}}} = - RT \hfill \\
\end{gathered} \right.\]
where R is the radius of your pulley.
Please, solve the second equation for d^2z/dt^2, and substitute it, in the first equation, so you will found the requested tension T