# In the gyroscopic precession left turning tendency, why does a pitch down cause a forward force at the top of the propeller?

I'm a student pilot, and all the textbooks and online resources I'm reading show a pitch down as a forward force acting on the top of the propeller (causing the resulting forward force on the right side of the prop, thus a left yaw). Where does this force come from? Isn't a pitch down a downward force on the prop hub, since the rotation is occurring about the airplane's center of gravity (behind the prop)?

The prop is a big gyro. It's the gyroscopic precession force created at the propeller when it's subjected to an input changing its axis of rotation, where, as with any gyro, the inertial resistance to the change in axis acts at 90 degrees rotationally to the input.

The airplane pitches down, rotating the axis (extending forward) of the propeller down, acting through the hub. Being a gyro that is being forced to change its axis vertically, it results in precession feedback acting through the hub at 90 degrees (as if there was a thrusting force at 3 o'clock), yawing the airplane to the left.

It doesn't matter where the prop's axis is in relation to the rotational axis of the airplane in pitch, just that its axis changes as the airplane pitches.

The best way to experience it is to take off in a tail dragger with a large prop, like a Cessna 180, and force the tail up early in the takeoff run before the rudder is having any effect. The swing to the left from propeller precession is pretty strong.

• Oh I see, so the "force" at the top of the prop is actually just a representation of the rotational acceleration about the longitudinal axis of the prop. Your second paragraph is exactly what I was looking for. Many thanks! Commented Apr 15, 2019 at 3:14

Grab a bicycle wheel, get it spinning, then try to change its plane of rotation.

Rotating objects (such as a frisbee) tend to stay in the same plane of rotation, indeed, this is what makes them valuable as orientation references when visual cues are lost.

A spinning prop is no different in its behavior. When you pitch down, you are changing the plane. One way to see it is to consider the prop as a spinning disk. When you tilt it, the parts of the disk 90 degrees from where you tilt experience the greatest change in direction. This is why pitching at 6 and 12 o'clock produces yaw at 3 and 9 o'clock.

It is the rotational energy of the disk changing direction that causes the yaw.

There is not actually any force applied at the top of the propeller. This is just a shorthand way of describing the effect of the rotational inertia of the propeller. The propeller is initially spinning clockwise around the axis pointed straight through the spinner. Nothing is spinning in the other two perpendicular axes.

Mathematically, we can represent angular momentum with a conventional direction where angular momentum in the direction $$\rm\hat x$$ corresponds to rotation in a clockwise direction as viewed if you were to face in the direction $$\rm\hat x$$.

The propeller has an angular momentum of $$L = I\omega \rm\hat x$$, where $$I$$ is the "moment of inertia" of the propeller (essentially how much inertia it has), $$\omega$$ is the rotational speed of the propeller in RPM, and $$\rm\hat x$$ represents the axis that goes forward through the spinner.

Now introduce a pitching moment, is lowered by a small angle $${\rm d}\theta$$ in a short time $${\rm d}t$$. After this short time, the new angular momentum of the propeller will be $$L = I\omega({\rm\hat x} - {\rm\hat z\,d}\theta)$$, where $$\rm\hat z$$ is the upward direction.

But angular momentum cannot be created or destroyed, so this has to come from somewhere. Where it comes from is the rest of the plane: after the same time $${\rm d}t$$, then, it has gained angular momentum of $$dL_{\rm plane}=I\omega{\rm\hat z\,d}\theta$$, so that the total angular momentum $$L+L_{\rm plane}$$ is unchanged. This manifests as a torque on the plane directed along the $$\rm\hat z$$ axis equal to the time derivative of the plane's angular momentum:

$$\tau=\frac{{\rm d}L_{\rm plane}}{{\rm d}t} = I\omega{\rm\hat z}\frac{{\rm d}\theta}{{\rm d}t}$$

This torque creates a clockwise motion as viewed from below the plane (i.e. looking up, along the $$\rm\hat z$$ axis), or in other words a yaw to the left.

In short, by pitching the propeller down you are "creating" counter-clockwise angular as viewed from below the airplane, and the plane has to rotate clockwise in order to balance it out and conserve angular momentum.