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From William Kershner's “The Advanced Pilot’s Flight Manual”- page 2-17, P-Factor:
If the airplane is yawed, the P-factor effect is encountered. A left yaw would mean a slight nose down tendency and a right yaw a slight nose up tendency (you can reason this out).
Unfortunately, I couldn’t reason this out :( Anyone care to explain? Thanks!
As a starting point, you need to thoroughly understand why P-factor tends to make a left yaw torque (in the case of a clockwise-rotating prop, as viewed from behind) when an aircraft flies in a nose-high "angle-of-attack". Note that by the aircraft's "angle-of-attack", we mean the aircraft's pitch "attitude" in relation to the flight path, not in relation to the ground. Do you see how this increases the lift or thrust generated by the down-going half of the prop disk, and decreases the lift or thrust generated by the up-going half of the prop disk? Thus creating extra thrust from the right side of the prop disk and making the aircraft tend to yaw to the left, in the case of a clockwise-rotating prop (as viewed from behind)? If not, you might benefit from reading the section entitled "P-factor" from John Denker's "See How It Flies" on-line tutorial website.
Once you've digested that, you'll observe that for a mental "shortcut", we can model the effect of P-factor as follows: think of the spinning prop as a solid disk. Pick the point on the disk that is "tilted backwards" in relation to the actual flight path. This would be the top of the disk in the case of an aircraft flying at a high angle-of-attack. Now rotate your chosen point 90 degrees in the direction that the prop is rotating-- which for this explanation we'll assume to be clockwise, as viewed from behind. So now you are looking at a point on the right edge of the prop disk, as viewed from behind. P-factor can be modeled as an extra forward "thrust" or "push" applied to the prop disk at this point. This will tend to make the aircraft yaw to the left. Note that this has nothing to do with gyroscopic precession-- it's a purely aerodynamic effect. This mental shortcut isn't meant to explain why P-factor occurs, just to quickly determine in which direction it acts.
Once you are comfortable with this "shortcut" for figuring out which part of the prop disk will be "loaded up" by the P-factor effect, consider that sideslip tilts the prop disk in relation to the direction of the flight path, just as flying at a high angle-of-attack does. In a sideslip with the nose pointing to the left of the flight path (ball to the right, yaw string to the left), which part of the prop disk is "tilted backwards" in relation to the direction of the flight path? The left-hand side. So if the prop is rotating clockwise as viewed from behind, the P-factor effect can be modeled as an extra forward "push" on the top of the prop disk, and a decreased forward "push" on the bottom of the prop disk. This will create a nose-down pitch torque.
Kershner could have been more clear in his language here, particularly in relation to his use of the word "yaw". It's critical to note that we are actually talking about the sideslip angle here, not an actual yaw rotation rate.1 An actual non-zero yaw rotation rate creates a pitch torque in the opposite direction, through gyroscopic precession -- yawing to the left will pitch the nose up. Just as flying at a high angle-of-attack creates a left yaw torque through P-factor, while pitching the nose down (as when a tail-wheel aircraft lifts the tail during the take-off roll2) creates a left yaw torque through gyroscopic precession. One effect is related to the aircraft's attitude in the relation to the direction of the flight path, and the other effect is related to the direction and rate of change of the aircraft's attitude in space.
Footnotes--
Clearly, the word "yaw" is often used in several different ways in aviation. Most correctly, it refers to an actual rotation about the aircraft's vertical axis, but it's not uncommon to use it to mean essentially the aircraft's "yaw attitude" in relation to the "relative wind" or actual direction of the flight path though the surrounding airmass-- which is exactly the same thing as the "sideslip angle". (The term "yaw string" is a classic example of this usage-- and the "yaw string" is sometimes also called the "slip string".) When Kershner says "if the aircraft is yawed", you should read "if the aircraft is yawed into a sideslip or skid and then held there", or "if the aircraft is allowed to fly in slipping or skidding attitude".
Harvey S Plourde's book "The Compleat Taildragger Pilot" contains a great graph showing the individual and combined effects of all the different factors that contribute a left-yawing tendency during the takeoff roll in a tailwheel aircraft, from the start of the takeoff roll all the way to the moment of liftoff. Different factors are dominating at different points in the takeoff roll.
Think of an aircraft at a 90 degree left bank, knife edge flight. How would normal p-factor affect the aircraft? For clockwise rotating propellers of course, it would crate a left yaw and causes the nose to drop towards the earth. The same thing happens when you introduce a left yaw to a wings-level aircraft. P-factor moves the nose down towards the earth, so the aircraft pitches down. In both cases the airflow is coming from the same side of the propeller if you can just ignore the aircraft itself. Since the effect is the same, then resulting action should be the same as well. That’s why left yaw causes slight nose down, and right yaw causes slight nose up.
It’s less prominent because pilots fly just a fraction of flight time with side slip, if they do at all, versus when compared with angle of attack. More over, the pitch change effect due to side slip is not discernible in the noise, e.g. turbulence or trim changes.
It is like an helicopter. If it hovering in one place the blades will meet the airflow at the same angle. The forward speed of the helicopter make them to change their AOA(increase) for half of the circle of the rotating blades (the retreated blades)to compensate (bite more air for that) asymmetric vertical pulling . But the aircraft propeller don't have swash plate to help with this so P- factor bites you. Means that the half of the disc will help you to have a roll rate better for leftside due to the Increased AOA to the one side of the propeller , but when you try to do it to the right side must use more resources (force on the stick) and rudder That why it is take off trim for rudder @6 degree( angle-of-attack is more than in cruising. Hope you're understand the analogy. It is about the amount angle-of-attack of the propeller for only half of the rotation when we put the nose up atitude (P factor). it is not identical. About the nose down attitude that came from the aircraft wings. The yaw change the angle of incidence for airflow meeting them(specifically if they are straight wings, not sweept)
There are two factors that will make the difference in (single) propeller aircraft. One is airflow from the propeller that will be more speedy on the root of the left wing, than the airflow over the right side (slower), and two it is about the left side of the rudder that have same effect (airflow coming on the left side has an effect on the aircraft),That it is way aircraft engine is slightly offset by the factory in longitudinal axis.
