This problem involves a system with two blocks connected via a rope that runs over a pulley, which has a non-zero mass. The objective is to find the acceleration of both blocks and the pulley. To solve this, we need to analyze the forces acting on each block and the rotational dynamics of the pulley.

Key concepts involved:

1. Newton's Second Law: $F = ma$ for translational motion of the blocks.
2. Rotational Dynamics: $\tau = I\alpha$ for the pulley, where $I$ is the moment of inertia, and $\alpha$ is the angular acceleration.
3. Relationship Between Linear and Angular Acceleration: The linear acceleration $a$ of the blocks is related to the angular acceleration $\alpha$ of the pulley by the relation $a = r\alpha$, where $r$ is the radius of the pulley.

Step-by-step approach:

1. Free-Body Diagram (FBD) of Masses:

   • For mass $m_1$: The tension in the rope, $T_1$, acts horizontally.
   • For mass $m_2$: The tension in the rope, $T_2$, acts vertically, counteracted by the gravitational force $m_2g$.

2. Force Equations:

   • For $m_1$: $T_1 = m_1 a_1$
   • For $m_2$: $m_2g - T_2 = m_2 a_2$

   Since the rope is assumed to be inextensible, the accelerations of the two blocks will be related.

3. Pulley Dynamics: The pulley with mass $m_3$ has a moment of inertia $I = \frac{1}{2}m_3 r^2$ (assuming it's a solid disk). The torque $\tau$ is related to the tensions by: $\tau = (T_2 - T_1) r = I \alpha$ Substituting $I = \frac{1}{2}m_3 r^2$ and using $\alpha = \frac{a}{r}$, we get: $(T_2 - T_1) r = \frac{1}{2}m_3 r^2 \cdot \frac{a}{r}$ Simplifying: $T_2 - T_1 = \frac{1}{2} m_3 a$

4. Combining the Equations: From the force equations and pulley dynamics, we can solve for $a$, the acceleration of the system.

   • Substitute expressions for $T_1$ and $T_2$ from the force equations into the pulley equation.
   • Solve the system of equations to find the acceleration $a$ of both blocks and the angular acceleration $\alpha$ of the pulley.

Let me know if you would like me to proceed with the detailed calculations or if you have any specific questions!

Expanding on the solution, here are 5 related questions:

1. How would the solution change if the pulley were frictionless?
2. What is the effect of increasing the mass of the pulley on the system’s acceleration?
3. If the rope were replaced by one with a different mass, how would that affect the dynamics?
4. How does the solution compare if the pulley were massless, as suggested in the tip?
5. What would be the impact if the surface under $m_1$ had friction?

Tip:

Always ensure you account for rotational inertia when dealing with massive pulleys or rotating bodies in similar problems!