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We know that the effect of P-factor on puller propellers is the yawing moment in the opposite direction. I mean if we had a load (thrust or lift) on the right side of propeller, then the airplane will yaw to the left.
Now my question is: what about pusher propellers? Is it the same scenario or is the direction of load and resulting yaw the same? I mean in pusher propellers the right load causes right yaw?
What about pitch motion?
Initially I thought your question was poorly worded but now I think you actually do mean what you are asking but in a sort of nonsensical way because I think you have misunderstood something.
You might be misunderstanding what P-factor actually is. It is a yawing action produced by a propeller when it is tilted so the airflow through it is not perpendicular to the disc. The most extreme example of this is on a helicopter where when it flies forward, the blades on one side move almost directly into the wind while the blades on the other side move almost directly with the wind. A so-called advancing and retreating blade which causes asymmetrical lift production. We don't call this P-factor on helicopters though.
On an airplane, the tilt is caused by the fact that airplanes need to fly at a non-zero AOA to produce left which tilts the propeller so that it the wind isn't flowing straight through the discs of the propeller vertically. The blade on the advancing side experiences both a higher airspeed and AOA and while the blade on the retreating side experiences less and so there is assymetrical thrust produced in the propeller which tries to yaw. If an airplane is flying with zero AOA, then the air is flowing straight through the propeller disc perpendicularly and produces no yaw due to P-factor.
That means this phrase here is wrong and poorly worded
"We know that the effect of P-factor on puller propellers is the yawing moment in the opposite direction."
Opposite direction of what? Because as far as the propeller is concerned there are only two directions: forward/reverse or CW/CCW. What you mean is that the plane yaws to the left for right handed propellers with positive AOA (or various other combinations).
So from that description it should be evident there isn't a difference in P-factor between tractors and pushers.
However, P-factor isn't the only propeller effect that causes yaw. Propeller swirl pushes asymmetrically on on vertical stabilizers which are not symmetrical (i.e. above and below) which also produces a yaw. But this isn't part of P-factor. This yawing effect exists regardless the AOA.
ADDENUM: Due to gyroscopic precession, this effect causes a helicopter to pitch upward rather than roll (resulting movement is 90 degrees after the location the force where the force is applied around the disc). However, P-factor is often said to cause yaw on airplanes where the upward pitching is never mentioned. The physics don't change so I am not sure why this is.
The only think I can think of is that both pitching due to advancing/retreating blades as well yawing due to swirling slipstream pushing on an symmetrical vertical stabilizer are both always present. But on an airplane the yawing is much more noticeable whereas on a helicopter the pitching is much more noticeable (probably because on a helicopter you are already fighting the yaw due to rotor counter torque and on an airplane you are have very stable pitch control).
