Answer
$s=3.38m$
$F=76N$
Work Step by Step
We can find the required force and $s$ as follows:
From the given figure, $AB=\sqrt{16+s^2}$
and $AC=\sqrt{4+s^2}$
$sin\theta=\frac{s}{AB}$
$cos\theta=\frac{4}{AB}$
similarly, $sin\phi=\frac{s}{AC}$
and $cos\phi=\frac{2}{AC}$
Now, the sum of the forces in the y direction is given as
$\Sigma F_y=0$
$\implies T_1cos\theta-T_2cos\phi=0$
$\implies 4\frac{4}{16+s^2}=4\frac{2}{\sqrt{4+s^2}}$
This simplifies to:
$s=3.38m$
Now, the sum of the forces in the x direction is given as
$\Sigma F_x=0$
$\implies F-T_1sin\theta-T_2sin\phi=0$
$\implies F-4\frac{s}{\sqrt{16+s^2}}-6\frac{s}{\sqrt{4+s^2}}=0$
This simplifies to:
$F=76N$
