How Much Force is Needed to Lift a Wheel Over an Obstacle?

[Curieuse]

1. Problem statement
In figure, what magnitude of (constant) force F applied horizontally at the axle of the wheel is necessary to raise the wheel over an obstacle of height h=3.00cm ? The wheel's radius is r=6.00cm , and it's mass is m=0.800 kg.

Homework Equations

Balance of forces: F_net=0 (xy plane)
Balance of torques: τ_net=0 (z axis)

3. The Attempt at a Solution
By balance of torques about point of contact with obstacle,
F(r-h)=mgr
I chose that point so as to not account for forces of contact there!
The answer i got
F=(mgr)/(r-h)
F= 15.7 N
The answer at the back of the text is 13.6N.
What did i do wrong?

Also i have this lingering doubt, will the normal forces at the point of contact with the horizontal floor , be included?
I worked another problem where the solution i found ignored it and when i tried putting it in, it simply nullifies the weight , which lead to unreasonable answers... But still why is it not to be included? I mean it's also a force no.. Of course it disappears as soon as the wheel lifts off due to that horizontal F .. Thanks in advance..

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[Orodruin]

What did i do wrong?

What is the moment arm of the gravitational force wrt the point of contact?

 

[Curieuse]

The moment arm was (r^2 - (r-h)^2)^(1/2)
I mistook it to be r as i was doing the eye approximations :p
Thanks a lot @Orodruin
I'll be more careful!