Apparently, the question isn' t as boring as it looks.
"Due to P-factor, the right-hand engine typically develops its resultant thrust vector at a greater lateral distance from the aircraft's C.G. than the left-hand engine."
P-factor: also known as asymmetric blade effect and asymmetric disc effect, is an aerodynamic phenomenon experienced by a moving propeller with a high angle of attack that produces an asymmetrical ceter of thrust
According to the above statement P-factor is experienced only by a moving prop with HIGH ANGLE OF ATTACK but the propeller does not have a high angle of attack during a take off run.
During the take-off run, the fixed-pitch propeller has the lowest TAS, thus also the highest angle of attack.
Propeller angle of attack is the angle between the propeller' s chord and the relative airflow, the latter being a combination of the propeller' s movement in the air and the relative air from the movement of the airplane.
Next, the P-factor is caused by a positive angle between the propeller disc' s axis and the relative airflow of the airplane.
The down-going blade will have a higher angle of attack than the up-going blade because of how the airplane' s relative airflow bounces against them.
Also, the relative airflow vector will be larger, adding to the equation.
The real culprits behind yaw in this stage for a nosewheel-equipped airplane are the corkscrew-shaped propwash striking the empennage, torque produced by the engine itself (Newton's Third Law), and, as has been mentioned, the crosswind direction. P-Factor and gyroscopic precession are of primary concern for taildraggers in this stage.
I don' t agree. The corkscrew effect is irrelevant on twin engined airplanes.
Also, the torque produced by the Newton' s 3rd law itself only affects single-engine airplanes. On a twin engine, they more or less compensate themselves because the left engine will try to twist the left wing counter-clockwise from the propeller/engine' s axis, thus try to push the fuselage up, the right engine will also try to twist the wing counterclockwise but this time the fuselage being on the other side, it will try to pull the fuselage down.
The resulting roll torque from this is insignificant to create any additional weight on any of the main wheels, meaning that the wheel drag will be approximately equal and that no yaw will result.
The problem of the critical engine on the ground for twins is solely a P-factor problem, itself caused by the postive tilt of the propeller and/or the angle of the propeller disk during rotation.
It is not that significant (or non-existing if there is no tilt by design) when the nose gear is still on the ground but it is significant during rotation and until the airplane sticks off.
When does it make a big difference?
100kts.....V1.. BANG BANG BONG (left engine torquemeter drops)..CONTINUE (apply right rudder to make-up for the thrust assymetry)... reading XXX kts nearing V2 of XXX kts..rotate... (additional correction for P-factor with rudder = more drag!!)....(due to impaired performance and additional drag, more time between rotation and unsticking)...minimum V2 is XXX kts.
After the rotation, the airplane with only one engine, and all the added drag from the rudder and the windmilling engine could take several seconds to unstick and this is where the P-factor is going to hurt your performance most.
This portion of the take-off run is why the V1 must be higher than Vmcg (minimum control speed ground using aerodynamic controls only, mainly the rudder). The critical engine scenario affects Vmcg calculation and as a result also the V1 calculation.
The critical engine awareness is an important factor for a safe post-V1 continue call. It defines your error margin and the level of precision you need to operate at. You should also know that you can expect to need a little more rudder input during rotation.
In any case, this is basic theory and you should always operate your aircraft in accordance with the procedures as described in your airplane manual.