As shown in the figure, a constant force F = m_2 g/2 is exerted on...

Let $a$ be the acceleration of the system, assuming $m_1$ moves towards the right and $m_2$ moves downwards.
Let $T$ be the tension in the string.

The equation of motion for mass $m_1$ in the horizontal direction is:
$T - F = m_1 a$

The equation of motion for mass $m_2$ in the vertical direction is:
$m_2 g - T = m_2 a$

Adding the two equations, we get:
$m_2 g - F = (m_1 + m_2) a$

Given that the applied force is $F = \dfrac{m_2 g}{2}$, substituting this value yields:
$m_2 g - \dfrac{m_2 g}{2} = (m_1 + m_2) a$

$\dfrac{m_2 g}{2} = (m_1 + m_2) a$

Solving for $a$, we get:
$a = \dfrac{m_2 g}{2(m_1 + m_2)}$

Since the calculated acceleration $a$ is positive, our initial assumption about the direction is correct. Therefore, the acceleration of $m_1$ is $\dfrac{m_2 g}{2(m_1 + m_2)}$ towards the right.