# Gyroscopic Precession

This whole text is hereby placed in the public domain. Apart from the image from How Stuff Works, which remains their copyright, and is included under fair use rights as an example for educational purposes.

## Wrong

Almost every description of how gyroscopes work is wrong in some way. This is sometimes because they attempt to simplify things to make them understandable. Others explain it in terms of properties which are not really relevant. This will hopefully be one of the former.

I was inspired to put this together after seeing Destin, from Smarter Every Day's amazing explanations of how a helicopter works. He openly admitted that the gyroscopic effect was a bit spooky and that he found it hard to explain. He explained it in terms of "phase" and the lift acting later than when it was applied, which I think confuses matters way more than it explains them.

## Right!

Since writing this, I've discovered a youtube video on gyroscopic precession by Veritasium, who basically nails it. He keeps it beatifully simple, and having the animated and annotated visuals is a winner. I'm glad to see that he chose exactly the same hand-waving simplifications as me, and to focus on exactly the same properties too. If I'd have known about that video, I would not have felt the need to create this page. Good work, dude! Go watch his other vids too.

## Fundamentals

The only thing you need to know about is angular momentum and turning forces (a.k.a. moments, or torque).

If you're familiar with linear momentum - the tendency of a moving body to keep moving - angular momentum is little different; it is the tendency of a spinning body to keep spinning. It's definitely simplifying things to say that the faster a body is spinning, the more angular momentum it has, and that the heavier a body is, the more angular momentum it has, because the arrangement of the mass about the axis on which it is spinning which is fundamentally important. See how an ice-skater can change the rate of his spin by pulling his arms and legs (i.e. mass) inwards to speed up, and extending his arms and legs to slow down. All the time, he's preserving his angular momentum. However, if you have a rigid body, such as a gyroscope or top, the simplification becomes valid.

Secondly, angular momentum is a vector. This means it has a direction and a magnitude. You may think of it as an arrow, that's often how mathematicians draw them, they have a direction, and their length represents their magnitude. Angular momentum is represented by a vector whose direction is that of the axis of rotation.

Thirdly, in the same way that forces acting on an object moving in a straight line will change its momentum, turning forces acting upon spinning objects will change their angular momentum. And like angular momentum, they are represented by vectors. And like angular momentum, their direction is that of the axis about which the turn is being performed. This makes moments and angular momentum remarkably similar objects. So similar that, with some hand waving - namely assuming the moment is applied for a short constant period of time - we can add them. I expand on this simplification below.

Finally, when you add vectors, you get another vector. Just lay the two arrows head-to-tail and the sum is the vector between the tail of the first one and the head of the final one.

## Example: Spinning Up A Park Roundabout

If a park roundabout is already spinning, then its angular momentum vector is vertical. If you then try to speed it up by tugging bits of it as they go past, then the moment you are applying is also rotating about the same axis, and so the vector is also vertical, in the same direction. The sum (again hand-waving, I've not mentioned time) of these two vectors is just a longer vector in the same direction, and indeed you've not changed the axis of rotation, but you have made the rotation faster. The vectors explain the change you've just seen.

Similarly, if you'd have been trying to slow the roundabout down, your moment would have had the opposite direction, and the (hand wavy) sum would be a shorter vector than the original, but with the same direction. Again, you didn't change the axis of rotation, and you made it slower, and your vectors have explained what you've just seen.

## Example: Precession Of A Hanging Gyroscope

No point in not diving straight in to the "hard" example - you have all the tools at hand that are necessary to intuitively see what is happening and why. That's a typical hanging gyroscope, taken from How Stuff Works: "How Gyroscopes Work", which like all others explains things in the wrong terms (it's obsessed with the linear, and sneakily avoids trying to apply its explanation to the parts of the system where it doesn't work), Its angular momentum is a great big vector pointing mostly left-right, along the wheel's axis. We can overlook the up/down component, all that matters is that this axis, this vector, is flat in the webpage.

The turning force applied by gravity is trying to rotate the wheel counter-clockwise as we look at it. The axis of this rotation is directly into the page. As the centre of gravity of the system is quite close to the fulcrum, the vector will be quite short.

So the (hand-wavy) sum of the angular momentum with the effect of the turning force is a vector that's still mostly left-to-right, but has a small component into the webpage.

Hold a pencil out in front of you to represent the original axis. Then push its tip slightly into the screen, holding the other end stationary. You've just added the effect of the gravitational force. The new vector is the new axis of rotation. And that's precessed the gyroscope - the vectors have explained what happens.

If you are unsure of what you've just done, consider the wheel when it's spinning in the plane of the webpage - its angular momentum vector will be pointing straight out of the webpage towards you. Point the pencil that way. Now consider gravity's turning force. It's trying to tip either the nearest or furthest bit of the wheel downwards, its axis, and thus moment vector, is left-right, in the plane of the webpage. So add a little bit of left to the position of the tip of your pencil. The wheel no longer spins in the plane of the webpage. You've just precessed the wheel again! The angular momentum and turning force vectors have explained everything.

## How Wrong Is This?

The horrible "sum" handwaving, and the lack of mention of time. In simple terms, the effect of a constant force is proportional to how long the force is applied. This applies to both linear forces and to turning forces. So force times time is the actual change to momentum. I've assumed the force is applied over a short unspecified period of time, and therefore the time component of that is just a constant that can be hand-waved away. As the system is precessing, the force of gravity is constantly changing where it acts, so the "constant force" assumption doesn't hold true. The actual "summation" is "integration", which doesn't treat time as discrete chunks, and can take into account the constantly changing nature of the setup. However, if you are looking at the system for a sufficiently short instant of time, what you will see corresponds to what is described above.

Also, at no point did I mention which of the two possible ways a vector can point in a direction - forwards or backwards. As long as you are consistent with your choice, it doesn't actually matter which way you choose. This choice is called "handedness". Clench your hand and stick your thumb up. Imagine a rotation around your thumb that flows out of your fingertips - the direction of your thumb is the vector you would use to describe the rotation. It depends on whether you used your left or right hand. Everyone uses right hands conventionally, but the maths doesn't change whichever you chose. Another hastily constructed page by Phil Carmody
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