Physics - NLM Question with Solution | TestHub

The string goes slack when the acceleration of the upper block $m$ (pulled by $F_1$) is less than that of the lower block $2m$ (pulled by $F_2$) in the sense that the string would need to lengthen: $a_1<a_2$.

$\frac{F_1}{m}<\frac{F_2}{2m}\Rightarrow F_1<0.5F_2$. Among the options, $F_1=0.25F_2$ satisfies this.