Accelerometers, spin, and signal-to-noise

18:30the the sensor is not going to register a lot of movement. And these things are accelerometers at the end of the day. And an accelerometer measures linear acceleration. It discerns spin based on linear acceleration and how >> like oh >> at one point there's a acceleration in the X direction, then it's in the Y direction, then it's in the X direction, then it's in the Y direction. And this cycling is going to get you the spin rate. Well, if you're closer to the edge of the ball, then as the ball spins, you have higher linear acceleration, and so your signal-to-noise effectively is like larger. >> Right. >> Right? >> Right. It makes sense. >> Um and and so that makes sense why it would be over there. Um but that means that you have to balance it. And actually, um it I thought back to my

19:11classical mechanics days, and it's very easy to balance something and make it look like a sphere because all you have to do is if you have one thing over here and you know you want the center axis to be like that origin point, then you just make a similar weight on the other side, but that can't be enough because now you have a preferred axis. So, you need two other on this end and on this end. And then what ends up happening is um for those who are in undergraduate physics and have taken an advanced course in classical mechanics, what you're doing is making the inertia tensor diagonal. And then like there's a matrix and everything else is a zero except the stuff in the diagonal, and all of the diagonal elements are the same because you have the same mass here, here, and

19:51here. And so, you can just factor out everything and it becomes the identity matrix where you have zeros everywhere and just 1 1 1, and it looks exactly like a sphere. And this is one of the coolest problems I remember homework problems in my classical mechanics was like proving that like a sphere spins the same way as a cube in for all intents and purposes. Which is kind of crazy to think about like a sphere literally looks the same from every side, right? In the sense that like for example, I can take a sphere, like let's say a uniform sphere, and I can attach a a string to it to the to the to the ceiling, and then when I when I like twist it,

20:32it's going to oscillate like this, and the frequency of oscillation like this doesn't matter where I attach on the sphere. Because if I rotate the sphere this way and then try to try to do it, the torsion pendulum is going to be exactly the same, okay? The frequency of oscillation. For a cube, naively you would think oh if I if I if I attach the string on the on the flat part, or if I attached it to the corner, the frequency of oscillation should be different. But no, because the inertia tensor is identical in the sphere and the cube for even one orientation, no matter how you rotate it, those things like don't change the inertia tensors.

21:14It's called a unitary transformation. I just remember this from like um uh it was like a undergrad problem uh at Princeton. It was the the class was called Death Mech. Um and and this was