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I would approach this problem by "thinking" that the gear $A$ is like on a treadmill. So the gear $O$ is like moving "backward" while the gear $A$ is trying to move forward. You can easily compute the velocity of the two contact points and see what happens.


In the reference frame of the car, the axle is stationary, but the ground is moving below at speed $v_C$. If the car doesn't skid, then the surface of the tyre must move at the same speed, but with a velocity that is directed backwards in the bottom and forwards in the top. At half the distance between the tyre surface and the hub, the speed is $v_C/2$. ...


Electron spin is not the result of a rotation of the electron around itself. In this case, of course (2) also doesn't hold. In fact, one can show that there is a double implication as follows: 1) if $\vec v$ is defined as in (1) one gets $$ \frac {\vec r \times \vec v}{r^2} = \vec {\omega} - \vec r \frac {(\vec r \cdot \vec {\omega})}{r^2}. \tag{I}$$ So, ...

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