Problem with Angular velocities

You're wrong because you're assuming some relations between W0 and Wc that can't be true.
V=rw, as you've already shown you know. So if W0 is 1/6 of Wc, you then KNOW that W0 is measured at a radius R from the centre of the biggest gear... but I'm not really sure why you would need both of them, and frankly, I'm struggling to understand the information you're posting in regards to the questi
on.

Wo and Wc are different bodies, Wo is the angular velocity of a large metal bar that unite the Gears with radius R and 2R and Wc is the angular velocity of the surface (radius 6R)where the 2 wheels are rotating. To explain what I am thinking let's make it more simple because my english is pretty bad... Let's assume the Bar and the gears A and B are static, if C moves the gears would rotate too right? And if we assume C static and the bar mobile the gears would rotate too, In those cases the wheels would have the same linear velocity of the Surface C and the Bar respectively, making the only difference the radius and the spinning direction.

The Problem is that here we mix Both cases since the Surface is moving at the same direction of the Bar (against clock), one entire cycle of the Gears would take a lot more than if the Surface C were static. If C moved Clock wise and the bar Stayed the same, the cycle of the gears would take less time.

That's why I assume that the Real linear velocity of the wheels is the same as the relative linear velocity between the Bar and the Surface C, Since they are connected by the gears.

Which is something you can say in physics, because you define directions and therefore which way a value would be to be positive and which would be negative, because most values are gauge invariant and can be defined arbitrarily. Defining upwards as a positive value is the most common definition, so your teacher at the very least not wrong.

It was a projectile problem, we assumed the start point to be the center and up to be positive and down negative. She said that the the gravity was additive to the initial speed and because of that the projectile was gaining more and more speed to the point it turned out to be infinite when we tried to find the middle point of the trajectory.
In other words when the particle was going up she added the speed generated by the gravity when we assumed that going up was positive and since the gravity pulled the projectile down It should be negative but she only added it and when the results weren't the same as a stupid guide they used, she only said "I don't know why it's wrong and class is over" when we said that gravity should be negative not positive because of the reason above stated.

[MENTION=846]Corn[/MENTION]; I know that linear Velocities are the same, that's why I used them to make the equations to get the angular speed