This problem involves three blocks connected by a rope and pulley system, with two blocks $m_1$ and $m_2$ on a horizontal, frictionless surface, and a third block $m_3$ hanging vertically. The surface is frictionless except for a coefficient of friction $\mu$ between blocks $m_1$ and $m_2$.

Problem Breakdown:

• We need to find the force that block 2 exerts on block 1.
• The system is moving together, so the acceleration of all blocks is the same.

Free Body Diagrams:

1. Block 1 (horizontal):

   • Tension $T_1$ pulls it to the right.
   • Force due to block 2's friction $f = \mu N_2$ pulls it to the left (where $N_2 = m_2 g$ is the normal force from block 2).

2. Block 2 (horizontal):

   • Tension $T_2$ pulls it to the right.
   • Force due to friction on block 1 $f = \mu N_2$ pulls it to the right (since this friction is an internal force in this two-block system).

3. Block 3 (vertical):

   • Gravity pulls it down with force $m_3 g$.
   • Tension $T_2$ pulls it upwards.

Forces and Equations:

• For block 1: $m_1 a = T_1 - \mu m_2 g$

• For block 2: $m_2 a = T_2 - T_1$

• For block 3: $m_3 a = m_3 g - T_2$

Since all blocks are moving together, we assume that the acceleration $a$ is the same for all three blocks.

Solution Strategy:

1. Use the equation for block 3 to solve for $T_2$ in terms of $a$.
2. Substitute this into the equation for block 2 and solve for $T_1$.
3. Finally, solve for $a$ by substituting into the equation for block 1.

Once we have the tensions, we can determine the force block 2 exerts on block 1, which is the frictional force $f = \mu m_2 g$.

Let me know if you would like the full step-by-step solution or if you have any specific questions about this process.

Related Questions:

1. How does the coefficient of friction $\mu$ affect the acceleration of the system?
2. What happens if the pulley has mass or friction?
3. How would the system change if block 1 had a different surface (not frictionless)?
4. How do the masses of the blocks influence the tensions in the ropes?
5. What would the free body diagram look like if the blocks were on an inclined plane?

Tip: Always check for equilibrium and consider the system's constraints, such as friction and rope tension, to set up your force equations correctly.