Is this the right way to calculate the force on a screw? My brother wants to mount his T.V. to the wall and wants to know if his drywall anchors will be enough. We haven't done any free body diagram analysis in a long time so we just want to know if what we did is right.
This is the diagram:

So it is a metal plaque with 4 screw holes in the corners. the plaque is 22cm tall and the holes are 1cm inside from the border so they are 20cm apart. Then, a metal arm comes out of the plaque approx. 60cm. The whole thing is about 3Kg and assume the center of mass to be at the middle, so 30cm from the wall, to keep things simple.
We started by calculating the torque of the T.V. and arm:
$$150N * 0.6m +30N * 0.3m = 99Nm$$
We then assumed that this torque would be applied in such a way that it will turn from the bottom of the base (like if you were opening a door and the hinge would be at the bottom of the base, horizontally to the wall). As such, the screws would have to pull the plaque towards the wall and create an opposite torque. So our equation to have torque equilibrium was:
$$F_a*0.21m + F_b*.01m = 99Nm$$
Where $F_a$ is the force the top screws would need to apply and $F_b$ the force the bottom screws would need to apply.
Now, if we converted this 99Nm to a single force acting along the arm, we could convert that to:
$$99Nm = F_c * 0.11m$$
$$F_c = 99Nm / 0.11m = 900N$$
Then we do a forces equilibrium equation:
$$F_a + F_b = F_c = 900N$$
We then solved the equation system and arrived to $F_a = 450N$ and $F_b = 450N$. Which was odd for us to see they were the same force. That means the torque in A is $94.5Nm$ and the torque in B is $4.5Nm$. We divide both by 2 since there are actually 4 screws, not 2 and arrive to $47.25Nm$ and $2.25Nm$
Since A is a bigger torque, we wanted to look for a drywall anchor that can support that torque, but I have the feeling we did something wrong and I can't see it.
 A: There are a couple things.
Most of the time when you mount something on the wall, it is right next to the wall like a picture. The force on the anchors is more or less straight down parallel to the wall. There isn't much torque. Anchors have a weight rating on the package like 20 lb or 50 lb. The rating is for this kind of situation.
In the situation you have, there is a significant torque. The force on A is trying to pull it out of the wall. I don't know if the rating is the same for this kind of force.
For what it is worth, your calculation of $99$ Nm is right. I would assume all of that torque is trying to pull the top anchor loose. The force is $F = \tau/r = \frac{99 Nm}{0.21 m} = 471 N$.
That is the force on the anchor. The force on the screw is probably much larger. After you mount the anchor in the wall, you put the TV bracket against the wall, and screw a screw into the anchor. You tighten the screw. The anchor and head of the screw squeeze the bracket from opposite sides. How much force depends on how hard you tighten the screw. As long as you don't strip the threads on the inside of the anchor, you should be OK. Squeezing the bracket and screw into the anchor doesn't change how much weight is hanging on the anchor, not the torque pulling the anchor out of the wall. A bigger anchor has screw threads on the outside that attach to the drywall. Those are the ones you need to worry about.
A: The problem is  in your equation of bending moment balance:
Fa *0.21 + Fb *.01 = 99 Nm is incorrect!!! Remember that the force is distributed as per a heel and toe concept.
The correct equation is:
Fa *0.21 + Fa *(.01/0.21)*0.01 = 99 Nm ....If we solve for Fa...then we have:
Fa = 384 N  = 86.3 lbs
Fb = 18.3 N
The issue here is that we want to be sure that the gypsum wallboard can hold those forces.  You did not mention the thickness of the wall...If we assume 1/2 inch...then if you use  1/8 toggle bolts in this  1/2 inch wall...they have a 200 lbs pull out force...or about 400 lbs for the top two bolts. Basically a safety factor of 5 !!!!
Have fun!!!
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