It is because the moment is being obtained with respect to point O, which if you take a rubber band and put it at one end at O and at the other end at FS1 and apply movement in the direction of FS1, the moment will be clockwise, but at F , the moment is anticlockwise, remember that for the equilibrium equation of the moment at Point O, the anticlockwise moments were taken into account as positive, the analogy is simple if you see it as a clock hand, without wishing to offend but It is something very basic in static balance, I recommend strengthening that knowledge because it is essential for mechanical vibrations.
Why Fs1 has -ve moment while the direction is the same as F at point A? at 5:10
It is because the moment is being obtained with respect to point O, which if you take a rubber band and put it at one end at O and at the other end at FS1 and apply movement in the direction of FS1, the moment will be clockwise, but at F , the moment is anticlockwise, remember that for the equilibrium equation of the moment at Point O, the anticlockwise moments were taken into account as positive, the analogy is simple if you see it as a clock hand, without wishing to offend but It is something very basic in static balance, I recommend strengthening that knowledge because it is essential for mechanical vibrations.
why there is x/2 in the potential energy method