Minimun angular velocity for a wheel to maintain contact with a plate

[samee]

I guess point A is the origin of my stationary axis, with Y going up along AB, then X going along AC. My moving axes would be at the center of the moving disk at point C.

So, the w of the small disk is induced by the Ω of the main shaft along AB. If the radius of the small wheel is r, then w=-R/rΩi+Ωj

because; V(D)=RΩ from the main shaft. V(D)=rw2 from the disk, but V(D)=V(D), so rw2=RΩ and w2=RΩ/r

okay!

This is all well and good, but shouldn't the small wheel touch the plate no matter what it's angular velocity? The answers are (a) Ω=√(g/a), and (b) Ω=√(2g/a) so I guess there is an answer that I'm just not understanding...

 

[KataKoniK]

Hello there,

Thank you for your post. It seems like you have a good understanding of the concept of rotating axes and the relationship between angular velocity and linear velocity in a rotating system. However, I would like to clarify a few points for you.

Firstly, point A is indeed the origin of your stationary axis, but we typically label the axes as X and Y, rather than AB and AC. This is just a convention and does not affect the calculations or understanding of the system.

Secondly, your moving axes would indeed be at the center of the moving disk at point C, as you correctly stated. This is because the center of the disk is the point of rotation and therefore has an angular velocity of Ω.

Moving on to the equations, you are correct in stating that the angular velocity of the small disk is induced by the angular velocity of the main shaft along AB. However, the equation w = -R/rΩi + Ωj is not entirely accurate. The correct expression for the angular velocity of the small disk, w, would be w = -RΩ/r i + Ω j. This is because the angular velocity is a vector quantity and needs to include both magnitude and direction. The negative sign in front of RΩ/r is to account for the opposite direction of rotation between the main shaft and the small disk.

As for your question about the small wheel touching the plate, it is true that the small wheel will always touch the plate, regardless of its angular velocity. This is because the small wheel is directly attached to the main shaft and therefore has the same angular velocity as the main shaft. The equations you mentioned, (a) Ω = √(g/a) and (b) Ω = √(2g/a), are not related to the small wheel touching the plate. These equations are actually the equations for the angular velocity of a pendulum, where g is the acceleration due to gravity and a is the length of the pendulum arm.

I hope this helps clarify any confusion. If you have any further questions or would like to discuss this topic further, please don't hesitate to reach out. Keep up the good work!