To solve this problem, we need to consider both blocks together, since they are connected by a string and are subject to the same acceleration due to the force applied to $m_2$.

Step 1: Calculate the total mass

The two blocks, $m_1$ and $m_2$, are connected and move as one system. Therefore, the total mass is:

$M = m_1 + m_2 = 2 \, \text{kg} + 3 \, \text{kg} = 5 \, \text{kg}$

Step 2: Calculate the acceleration of the system

The force $F = 5 \, \text{N}$ is applied to the second block $m_2$. According to Newton's second law, the acceleration $a$ of the system can be found using:

$F = M \cdot a$

Solving for $a$:

$a = \frac{F}{M} = \frac{5 \, \text{N}}{5 \, \text{kg}} = 1 \, \text{m/s}^2$

Step 3: Calculate the tension in the string

Now that we know the acceleration of the entire system is $1 \, \text{m/s}^2$, we can calculate the tension $T$ in the string. The tension only affects block $m_1$, so we apply Newton’s second law to block $m_1$ alone:

$T = m_1 \cdot a$

Substitute the values:

$T = 2 \, \text{kg} \cdot 1 \, \text{m/s}^2 = 2 \, \text{N}$

Final Answer:

The tension in the string is $T = 2 \, \text{N}$.

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Would you like further clarification or details on any step?

Here are some related questions:

1. How would the tension change if friction was present?
2. What if the force was applied to $m_1$ instead of $m_2$?
3. How would the tension change if the masses were different?
4. What would happen if the string between the blocks was inextensible but had some elasticity?
5. How can you determine the net force acting on $m_1$ and $m_2$ separately?

Tip: Always remember that objects connected by a string experience the same acceleration, provided the string is inextensible.