A string is wrapped around the edge of a uniform disc, and its free...

Let $M$ be the mass and $R$ be the radius of the uniform disc.

Let $a$ be the downward acceleration of the center of mass and $\alpha$ be the angular acceleration of the disc.

Since the string is attached to the ceiling and unwinds without slipping, the point of contact has zero acceleration, which gives the constraint equation:

$a = \alpha R$

The forces acting on the disc are its weight $Mg$ downwards and the tension $T$ upwards. The equation of translational motion is:

$Mg - T = Ma$

The torque about the center of mass is provided by the tension $T$. The equation of rotational motion is:

$T R = I \alpha$

For a uniform disc, the moment of inertia about its central axis is $I = \dfrac{1}{2} M R^2$. Substituting $I$ and $\alpha = \dfrac{a}{R}$ into the torque equation:

$T R =$\left( \dfrac{1}{2} M R^2 \right) \left(\dfrac{a}{R} \right)

$T =$\dfrac{1}{2}$M a$

Substituting the value of $T$ into the translational equation:

$Mg -$\dfrac{1}{2}$M a = Ma$

$Mg =$\dfrac{3}{2}$M a$

$a =$\dfrac{2g}{3}