Length of rod given forces and lengths to calculate torque

Question

In the question and solution given below, I haven't understood why is the solution asking us to take torques about the center of mass.
In my approach I equated the net torque about the lower end B of the rod to zero

MY APPROACH:
let distance between line of action of F₂ and B = k cm
then length of the rod = k+20 cm
taking torques about B, we have
F₂k - F₁(k+20) = 0 (since rod is only translating)
Putting F₂ as 5N , and F₁ as 3N, we have k=30cm
thus my answer comes out as 50cm



*ps : we cant take F₁ as 7N. I already tried that. torque being zero is impossible in that case

Answer 1

You can only sum torques about any point and expect them all to be the same if and only if the body is in translational and rotational equilibrium.

Let $A$ and $B$ be two point on some body. Let $\vec \tau $ denote the torque, and $\Sigma \vec F$ be the net force. Then, the torque about A is related to the torque about point B by the following equation $$\vec \tau_B = \vec \tau _A + \vec r_{AB} \times \Sigma \vec F$$

This body will rotate about its center of mass, so to break down the motion in rotational and translational motion, you should consider the translation of the center of mass and rotation about the center of mass.

See here: Why if the torque equals zero measured from one point in space it equals zero measured from any other point?.