The problem states that there are two blocks with masses $M_1$ and $M_2$ hanging one under the other, and you are tasked with finding the tension $T_1$ in the upper rope. You are instructed to express the answer in terms of $M_1$, $M_2$, and $g$ (the acceleration due to gravity).

Approach:

1. Forces acting on the lower block $M_2$:

   • The tension in the rope directly supporting $M_2$ is $T_2$.
   • The force due to gravity acting on $M_2$ is $M_2g$.

   Using Newton's second law, the net force on $M_2$ (if the system is in equilibrium or moving at constant velocity) is: $T_2 = M_2g$

2. Forces acting on the upper block $M_1$:

   • The tension $T_1$ in the upper rope must support both the weight of $M_1$ and the tension $T_2$ (which is supporting $M_2$).
   • The force due to gravity acting on $M_1$ is $M_1g$.

   The total tension $T_1$ must balance these two forces: $T_1 = M_1g + T_2$ Substituting $T_2 = M_2g$ into the equation for $T_1$: $T_1 = M_1g + M_2g = (M_1 + M_2)g$

Final Answer:

The tension in the upper rope, $T_1$, is: $T_1 = (M_1 + M_2)g$

This expression shows that the tension depends on the combined weight of both masses.

Do you want more details on any step, or have any other questions?

Here are five related questions:

1. How would the answer change if the system is accelerating upward?
2. What if the blocks are moving downward with an acceleration $a$?
3. How would you solve this problem if $M_1$ and $M_2$ are connected by a pulley?
4. What would happen to the tension if the mass of $M_2$ is much larger than $M_1$?
5. How would the result be different on another planet with different gravity?

Tip: When solving tension problems, always identify all forces acting on each block and apply Newton's second law step by step.