Conservation of angular momentum and the gyroscopic force.
In the pictured path of the wheel, the wheel starts out rolling forward from our perspective. The rolling rotation has an angular momentum represented by a vector pointing out from the left face. By western convention, if you give a thumbs up with your right hand and orient your fingers to point in the direction of rotation, your thumb points in the direction of the angular momentum vector. That vector is what must be conserved. The thumb vector is useful to think about and keep track of, but it’s the curling fingers (rotation) that’s actually physically relevant.
When pushed over, the angular momentum vector tilts upwards. To keep angular momentum conserved, gyroscopic forces will add rotation with an angular momentum pointing downwards. To use the right-hand rule again, point your right thumb in the direction of the vector (down) and your fingers curl in the direction that the gyroscopic force will turn the wheel (to the right, as pictured in your path drawing).
Silly notes on the right hand rule:
Of course, the vector pointing out along either pole/axel/axis could be used as the angular momentum vector. Early mathematicians chose the right hand rule to define which was the positive vs negative vector. You can try with your left hand to show to yourself that that convention choice still gives the same result. You may have learned the “gang sign” method of the right hand rule, but it’s terrible and I recommend forgetting it in favour of the rotation/curl based definition I use above that is much more physically intuitive and quick to assess. The gang sign method is needlessly tied to the mechanical steps of calculating a cross product which is less intuitive than thinking about the underlying rotation that the cross product represents.
I did focus attention on the fingers, which follow the direction of rotation. The thumb is just a good way to keep track of which way the fingers are going. It’s the rotation that has inertia. Your hand is just being used to keep track of the direction of rotation.
“Physics cannot explain why, only how” - Richard Feynman.
Why is angular momentum conserved is a major philosophical question beyond the scope of this discussion, unless you’d like to bring us down that tangent? We could talk about Noether’s Theorem, and the nature of rotational symmetry. I just don’t think that starting from “the laws of physics contain an azimuthal symmetry from which a law of conservation of angular momentum can be derived” is the right vibe here, and it only pushes the unanswered “why” into the symmetry of our equations instead. OP wanted the how, not the philosophical why.
Conservation of angular momentum and the gyroscopic force.
In the pictured path of the wheel, the wheel starts out rolling forward from our perspective. The rolling rotation has an angular momentum represented by a vector pointing out from the left face. By western convention, if you give a thumbs up with your right hand and orient your fingers to point in the direction of rotation, your thumb points in the direction of the angular momentum vector. That vector is what must be conserved. The thumb vector is useful to think about and keep track of, but it’s the curling fingers (rotation) that’s actually physically relevant.
When pushed over, the angular momentum vector tilts upwards. To keep angular momentum conserved, gyroscopic forces will add rotation with an angular momentum pointing downwards. To use the right-hand rule again, point your right thumb in the direction of the vector (down) and your fingers curl in the direction that the gyroscopic force will turn the wheel (to the right, as pictured in your path drawing).
Silly notes on the right hand rule:
Of course, the vector pointing out along either pole/axel/axis could be used as the angular momentum vector. Early mathematicians chose the right hand rule to define which was the positive vs negative vector. You can try with your left hand to show to yourself that that convention choice still gives the same result. You may have learned the “gang sign” method of the right hand rule, but it’s terrible and I recommend forgetting it in favour of the rotation/curl based definition I use above that is much more physically intuitive and quick to assess. The gang sign method is needlessly tied to the mechanical steps of calculating a cross product which is less intuitive than thinking about the underlying rotation that the cross product represents.
I’m sorry but this answer is not wrong, it’s just not an answer.
You are not explaining the reason why, you are explaining how to solve it using some useful tricks.
Physics doesn’t act according to your thumb and surely your thumb does not encapsulate the essence of inertia.
It’s akin to answering “What is 2 x 3?” and answering proving a table where to look up the answer for each combination of number to multiply.
I did focus attention on the fingers, which follow the direction of rotation. The thumb is just a good way to keep track of which way the fingers are going. It’s the rotation that has inertia. Your hand is just being used to keep track of the direction of rotation.
“Physics cannot explain why, only how” - Richard Feynman.
Why is angular momentum conserved is a major philosophical question beyond the scope of this discussion, unless you’d like to bring us down that tangent? We could talk about Noether’s Theorem, and the nature of rotational symmetry. I just don’t think that starting from “the laws of physics contain an azimuthal symmetry from which a law of conservation of angular momentum can be derived” is the right vibe here, and it only pushes the unanswered “why” into the symmetry of our equations instead. OP wanted the how, not the philosophical why.
Putting it this way blew my mind a little bit, for some reason. Great answer.