why or how does gyroscopic precession work?
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Conservation of angular momentum. Gravity pulling the gyroscope down will try and change the angular momentum, but since it has to be conserved, it has to go somewhere it ends up being made perpendicular to both the force and the axis of rotation. Feeling this force is really the best way to describe it. If you are holding a spinning bicycle wheel, if you try to rotate the whole thing perpendicular to the axis of rotation, you feel another force fighting you perpendicular to both your rotation and the wheel’s rotation.
When the gyroscope is spinning, it has angular momentum, just like regular momentum, but for spinning objects. L = p × r = I * ω. Angular momentum is linear momentum cross product with the distance to the origin, or the moment of inertia times angular velocity. Moment of inertia is basically angular mass, whereas in linear motion, mass is directly proportional to inertia, moment of inertia depends on how far that mass is from the origin. Ie something close to the center of rotation doesn’t need to move as much to have the same angular speed. Understanding these terms isn’t entirely necessary for what we are going to talk about.
If we have a gyroscope spinning, and it’s axis of rotation is at some angle φ from the vertical, we will see gyroscopic precession. Gravity will try and pull the gyroscope down, and surface the gyroscope is on will push it up. These forces balance, so we don’t get linear acceleration (the gyroscope doesn’t fall down) but the torques (angular force) they create are not balanced, so we get angular acceleration (the gyroscope begins to be rotated). That torque will be τ = F × r = ΔL/Δt (that second half is just the same relationship between force and linear momentum F = Δp/Δt)
This is where things get hard to explain without pictures. ΔL = Δt(F×r) now the way cross products work is when you multiply 2 vectors, you get a 3rd vector perpendicular to both initial vectors. Again, understanding it isn’t critical, but knowing the direction is all we are gonna care about for now. Gravity is acting down, and r is just the distance from the center of mass to where the gyroscope touches the ground, so it’s at an angle of φ. Our resulting change of angular momentum is going to be perpendicular to both of those, and since it has to be perpendicular to the downward direction, it must be horizontal. And since it has to be perpendicular to our axis of rotation, it can’t speed up or slowdown our gyroscope. The result is the entire gyroscope being pushed around seemingly by nothing.
[Video to hopefully show what I said](https://youtu.be/ty9QSiVC2g0)
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