- #1
Ailo
- 17
- 0
Hi! I would appreciate your thoughts on something.
Let's say you have a ring with radius R rotating with angular velocity [tex]\omega[/tex] about a vertical axis. A little bead is threaded onto the ring, and the friction between the bead and the ring is negligible. The bead follows the ring's rotation, and will for a given [tex]\omega[/tex] place itself on a position on the ring which makes an angle [tex]\theta[/tex] with the vertical.
By applying Newton's laws, one obtains the following equations for a given omega (N is the magnitude of the normal force from the ring on the bead):
[tex] N cos(\theta) = mg [/tex]
and
[tex] N sin(\theta) = m \omega^2 (R sin(\theta)) [/tex].
This gives a formula for the cosine of theta;
[tex] cos(\theta)=\frac{g}{\omega^2 r}[/tex].
So the problem is: what is the physical intepretation of what happens when omega gets so small that the right side of the equation exceeds 1?
My thoughts are that, since N both has to balance the force of gravity and simultaneously create a centripetal acceleration, N obviously has to be larger than mg. When omega sinks below a certain value, that just isn't the case anymore. So if you were to have it at an angle theta at a high angular velocity and then gradually lower omega until you hit that lower bound, it will just slide to the bottom.
Am I right?
Let's say you have a ring with radius R rotating with angular velocity [tex]\omega[/tex] about a vertical axis. A little bead is threaded onto the ring, and the friction between the bead and the ring is negligible. The bead follows the ring's rotation, and will for a given [tex]\omega[/tex] place itself on a position on the ring which makes an angle [tex]\theta[/tex] with the vertical.
By applying Newton's laws, one obtains the following equations for a given omega (N is the magnitude of the normal force from the ring on the bead):
[tex] N cos(\theta) = mg [/tex]
and
[tex] N sin(\theta) = m \omega^2 (R sin(\theta)) [/tex].
This gives a formula for the cosine of theta;
[tex] cos(\theta)=\frac{g}{\omega^2 r}[/tex].
So the problem is: what is the physical intepretation of what happens when omega gets so small that the right side of the equation exceeds 1?
My thoughts are that, since N both has to balance the force of gravity and simultaneously create a centripetal acceleration, N obviously has to be larger than mg. When omega sinks below a certain value, that just isn't the case anymore. So if you were to have it at an angle theta at a high angular velocity and then gradually lower omega until you hit that lower bound, it will just slide to the bottom.
Am I right?