Moments and Couples

For part a, resultant force of a couple iz zero, so F = i - 3j , so magnitude = root 10, and direction = -71.56 degrees. Not at all sure where to start with part b , guidance appreciated thanks

Take moments about O say. What is the moment due to the forces at A and B? So what is the opposing moment to ensure equilibrium?

So :
A : Perpendicular distance from O = Root 13. Magnitude of force at A = Root 5. Direction of force at A = 26.565 . Moment at A = Root 13 x Root 5 x Cos26.565.

Moment at A = 2 x 13^0.5 .

Do I just do the same for B ? And then the magnitude of the couple required to keep it equilibrium is the sum ? What do I then do about the direction ?

You seem to have chosen a most difficult route to obtaining the answer.

Yes, you'd need to do the same for B.

If you assume clockwise is positive, then add/subtract those two moments as appropriate to get the resulting moment. Then the required couple will be the negative of that result. Direction will be clockwise if positive, and anticlockwise if negative.

HOWEVER, it is far, far simpler, to:

Draw a diagram marking points and forces.
Again, assign one direction as postiive - clockwise or anticlockwise.

Split each force into its two components. The perpendicular distance of each component is just the absolute value of one of the coordinates of the point where it's acting - diagram helps here.
The whole working can be done in two lines!

Two lines, wow. I am definitely overcomplicating it then haha. I am not entirely sure I understand your simpler method though, would you please be able to elaborate further. Thanks a lot

Oh I see. stupid me haha . So we do 2 x 3 + 1 x 2 + -3 x 1 + 2 x 2 to get the total moment clockwise. So the magnitude and direction of the couple required to keep it in equilibrium is 9N anticlockwise ?

Close. Which direction is the j force acting at A? Clockwise or anticlockwise?

And the units for moments would be Nm if it were Newtons and metres, but certainly not just "N".

anti-clockwise. So 7Nm anticlockwise.

Ha, my mistake. 5Nm. I just automatically subtracted 2 not 4.

Agreed - 5Nm anticlockwise is the required couple.

And not a square root or trig function in sight!