A

Step 1: Identify the forces acting on the system. For the falling disc (mass M2), the gravitational force is acting downwards (F = M2*g) and the tension T in the string acts upwards.
Step 2: Apply Newton's Second Law (F = ma) to the falling disc:
$$M_2 g - T = M_2 a_2$$ where a_2 is the acceleration of mass M2 downward.
Step 3: For the rotating disc (mass M1), its rotational dynamics can be described by the torque due to the tension in the string:
$$T R_1 = I_1 \alpha$$ where I_1 is the moment of inertia of disc M1 and \alpha (angular acceleration) relates to the linear acceleration a_1 of the center of mass of M1 as follows: \( a_1 = R_1 \alpha \).
Step 4: The moment of inertia for a disc is given by \( I_1 = \frac{1}{2} M_1 R_1^2 \). Substituting I into the torque equation:
$$T R_1 = \frac{1}{2} M_1 R_1^2 \alpha$$
Step 5: Relating angular acceleration to linear acceleration, we have \( \alpha = \frac{a_1}{R_1} \) where \( a_1 = -a_2 \) (since they are connected by the thread). Combining these equations to eliminate T leads to:
$$M_2 g - (\frac{M_1 a_2}{2 R_1}) = M_2 a_2$$
Step 6: Rearranging gives a combined equation in a_2 to find its value:
$$a_2 = \frac{g M_2}{M_2 + \frac{M_1}{2}}$$
Therefore, the acceleration of the center of mass of the second disc is determined by the above relationship depending on the masses involved.