Give me ID Card-2!

Answer

The length of the string is $2l \times 3 = 6l$ At their crossing point P, the length of the string connecting both the blocks will be equal

$\displaystyle\implies$ Length of the sting connecting the one of the block from one of the pulleys

$\displaystyle AP = \frac {6}{4}l = \frac {3}{2}l$

But the length between the pulleys remains same $( = 2l)$

Let $θ =\widehat{ABP}$

$∴\displaystyle \cos \theta = \frac{2}{3}$

$\Rightarrow\displaystyle \sin \theta = \dfrac{ \sqrt{3^2 - 2^2}}{3} = \dfrac{\sqrt{5}}{3}$

$\displaystyle OP = BP \cdot \sin \theta = \frac {3}{2}l \times \frac {\sqrt {5}}{ 3} = \frac {\sqrt {5}}{2}l$

Now, for the two blocks to cross each other,
$\displaystyle \left |\bigtriangleup U(m)\right | > \left|\bigtriangleup U(M)\right|$

$\displaystyle \implies mg \frac {\sqrt {5}} {2} l > Mg\left(\sqrt {3}- \frac {\sqrt {5}} {2}\right)l$

since $l>0$ and $g > 0$ ,

$\displaystyle m\frac {\sqrt {5}} {2} > M\left( \sqrt {3}- \frac {\sqrt {5}} {2}\right)$

$\displaystyle \implies \frac {m}{M} > \frac {2\sqrt{3} - \sqrt{5}}{\sqrt {5}}$

$\Huge\therefore\displaystyle \boxed {\displaystyle \frac {m}{M} > 0.54919338}$

so the final generalised form you get will be $\frac { m }{ M } \ge 2\sqrt { \frac { \eta +1 }{ \eta +3 } } -1$