Can someone please explain Gyroscopic precession?

Question

So I watched this video

It is a fairly popular one by veritasium, and one that confused me. I was wondering if someone can cover the physics behind this in a intutitive and easy to understand manner. I understand torque and angular momentum come into play here, but what I don't get is the vector pointing away from the wheel(the right hand rule). I spent a lot of time thinking but it just doesn't make sense.

Edit: Ok so I realized that this may be confusing. I particularly want to know why the angular momentum vector is perpendicular to the plane of rotation.

Answer 1

The intuition about vector is an arrow pointing in a given orientation. It is clear for the position vector $\mathbf r$, pointing from the wheel center to outside, or for linear momentum $\mathbf p$ pointing in the direction of the movement.

In the case of angular momentum $\mathbf L$ it is not intuitive. There is nothing really pointing in a direction perpendicular to the plane of rotation.

But the quantities: $yp_z - zp_y$, $zp_x - xp_z$ and $xp_y - yp_x$ are conserved for an object with no forces applied. Moreover, if we make a coordinate rotation to another orientation , that conserved quantities change as vector components. So, $(L_x, L_y, L_z)$ is a vector.

The advantage of representing it pointing as described in the video is that the rate of change of that vector with an applied torque can be visualized as another arrow, which represent exactly that torque.

Answer 2

This answer isn't yet about gyroscopes.
What this answer does do: I support that the concept of vector angular momentum doesn't help with understanding gyroscopes.

About how angular momentum is commonly represented:

Our space has three dimensions of space. In a space with three spatial dimensions each plane has a single line that is perpendicular to that plane. That is, with three spatial dimensions specifying the perpendicular line (in vector form) is sufficient to specify a particular plane (in this case a plane of rotation). By contrast: in a space with four spatial dimensions a plane has two lines that are perpendicular to each other, and perpendicular to that plane.

Rotation is in a plane.

Let's say that you want to represent rotation in such a way that your method of representation works for any number of spatial dimensions of the space. That narrows things down to representing rotation in terms of the actual plane of rotation.

So:
Representing angular momentum in terms of the line that is perpendicular to the plane of rotation isn't the only choice, and it isn't the best choice. Then again, it's not a particulary bad choice either; changing the representation isn't worthwhile.

Answer 3

"what I don't get is the vector pointing away from the wheel(the right hand rule). I spent a lot of time thinking but it just doesn't make sense."

It's fundamentally arbitrary. But it works.

Mathematicians had a way to describe things spinning in two dimensions, with complex numbers. If you have a wheel with a marker on it, you can describe the location of the marker with a complex number to describe its position compared to the center of the wheel. Then you can describe it moved to some other place on the circle by multiplying by another complex number that has length 1. You can rotate to the left or the right.

But in 3 dimensions it's more complicated, and anyway there are questions of mass and force. If you hit a spinning baseball, some of the force goes into changing its linear velocity and some into changing the spin. Lots of stuff to figure out. The first step is deciding how to measure spin.

You can measure the amount of spin by how fast the angles change. Then you want to measure the direction of spin in 3D. If it's a wheel that spins around an axle, you might want to measure the direction of the axle. And then there are two directions it can spin, right and left, just like the complex plane. Since it's completely arbitrary which of them you choose to write which way, they chose the Right Hand Rule. They could have used the Left Hand Rule just as easily, but they didn't. If you use the Left Hand Rule you'll confuse people.

There are lots of other ways to do it. For example you could have a vector to represent a radius around a center, and a second vector to represent the velocity of a point at that distance. The way it gets done generally works.

Except that in some circumstances you can calculate things in the wrong order and have part of the angular motion cancel out wrong and get "gimbal lock". This will not happen in 3D if you use 4D math instead. You can have the space part of the number show an axis, and if it is perpendicular to the orbit you want, it will act just like the complex-number case. But if you multiply the axis by something that isn't perpendicular, it will trace out an elliptical orbit and the time part will show how far ahead or behind in time it will be at that fraction of the cycle. Or you can do a more complicated operation to rotate or revolve a 3D structure as a unit.

This was first worked out as quaternions, but it has been repeated as part of geometric algebra, and it was repeated again as spinor theory.

Answer 4

The product of force with distance is work done or energy transfer. It's maximum when force and distance changed are parallel and minimum when they are perpendicular. This product of force and perpendicular distance has no work done but produces a kind of force at the other end of distance which could be fixed but allow particle to move around it. That makes distance fixed or called radius for rotational motion. The force at centre is different for rotational motion because it depends upon force applied and distance from centre or fixed point. As it is given by cross product of distance from centre and force applied instead of work done which is dot product and scalar, this force at centre is torque and vector quantity. The direction of torque is perpendicular to the plane of force and radius. As after force is applied, particle has momentum in direction of applied force, so in rotational motion after torque is applied, particle has angular momentum in direction of torque applied.

Instead of showing expression for angular momentum as $\vec l=\vec r\times \vec p$ and saying that direction of angular momentum is perpendicular to the plane having radius and momentum. It is more logical to explain torque for layman.

Now come to gyroscopic motion in the video. A wheel is rotating with speed, it has angular momentum in horizontal direction. But it weight is pulling downward its centre due to gravity, which is prevented by support. When support is removed, a torque is produced perpendicular to the page if gravity is downward and angular momentum of spinning wheel is horizontal into the plane of page. This torque cause change in angular momentum into the direction of torque due to gravity. This causes spinning wheel to rotate about its point of contact with hanging cord. The direction of rotation is vertically upward against gravity because torque at every point of rotation produces toque at centre upward that counter gravity. The speed of rotation depends upon weight of a wheel and inversely to mass of spinning wheel and its speed of spinning.