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I want to know whether the condition of rolling has been proven or if it is just the result of observations only?
rolling without slipping
{ $v = \omega r$ where $v$ is transactional velocity of object, $\omega$ is angular velocity and $r$ is radius of object }
“Rolling without slipping” isn’t something that can be proven. Rather, it’s a model, a way of thinking about a situation, that can often be a useful approximation. You use it when it’s accurate enough to help. Don’t use it when the situation is different and it doesn’t help.
Car tire on F1 race car leaving the pits in a cloud of tire smoke: not ok
For a navigation system that wants to decide exactly how far your car has gone: maybe yes, maybe no. Depends on how accurate you need to be.
Lots of physics models are like this. There’s judgement in using them, they sometimes aren’t what you need for the situation you have, but they’re taught because they’re often useful.
If you're asking for a proof that "rolling without slipping" means $v=r\omega$, here's one:
When we say an object is "rolling without slipping," that means that the distance covered by the object in a certain time $\Delta t$ is exactly the same as the arc-length that a point on the wheel travels while rotating. (To see why this is true, consider the opposite: if a wheel is slipping, then one of two things are happening: it either travels some distance without turning, or turns without traveling any distance. In the former case, the arc-length that a point on the wheel travels is less than the distance covered; in the latter case, it is greater. Requiring that it doesn't slip therefore means that neither of the above are true, guaranteeing equality. You can also see this if you imagine "rolling up" the ground onto a wheel as it rolls, or equivalently, if you imagine the wheel as a spool of string that unravels as it rolls.)
In a time interval $\Delta t$, a wheel rotating with constant angular velocity $\omega$ will rotate a total arc-length of $s=r\theta=r\omega\Delta t$. Similarly, in a time interval $\Delta t$, an object traveling at constant velocity $v$ will cover a distance $d=v\Delta t$. Setting $s=d$ as per our argument in the first paragraph, we see that
$$v\Delta t=r\omega\Delta t$$
or equivalently
$$v=r\omega$$
Since we have imposed no other assumptions besides "rolling without slipping" in the above, we have therefore proven that saying an object is rolling without slipping is equivalent to setting $v=r\omega$.
