Answer
The largest magnitude of $F$ without the cord becoming slack is $~~F = 14.7~N$
Work Step by Step
The largest magnitude of $F$ (without the cord becoming slack) will cause $m_1$ to accelerate at the same rate as $m_2$ while the tension in the cord is zero.
We can consider $m_1$ to find an expression for $a$:
$F = m_1~a$
$a = \frac{F}{m_1}$
We can consider $m_2$ to find an expression for $a$:
$m_2~g~sin~\theta = m_2~a$
$a = g~sin~\theta$
We can equate the two expressions to find $F$:
$\frac{F}{m_1} = g~sin~\theta$
$F = m_1~g~sin~\theta$
$F = (3.0~kg)~(9.8~m/s^2)~sin~30^{\circ}$
$F = 14.7~N$
The largest magnitude of $F$ without the cord becoming slack is $~~F = 14.7~N$
