- #1

Päällikkö

Homework Helper

- 519

- 11

The classical(?) problem:

Suppose you have a ring, where practically all the mass is concentrated on the edges (like the tire of a bicycle). There is, however, a massless rod placed at the central axis.

The ring's given some [itex]\omega _i[/itex] amount of angular speed (about the "logical" axis).

The system is then placed by the end of the rod to lean horizontally on another rod that is placed vertically. The system will now rotate about the second rod too at an angluar speed [itex]\Omega[/itex].

In case you found my description confusing (I sure did), here's a diagram I drew:

http://img473.imageshack.us/my.php?image=img0157us.jpg

Anyways, my question is, how much initial angular speed [itex]\omega _i[/itex] must be given in order for the system to execute the described motion?

If you still think my desciption is confusing, please ask for more precise info .

This is what I tried:

----

I assumed that energy is conserved (

[tex]\frac{1}{2} I_\omega \omega _i ^2 = \frac{1}{2} I_\omega \omega _f ^2 + \frac{1}{2} I_\Omega \Omega _f ^2 [/tex]

[itex]\omega _f[/itex] and [itex]\Omega _f[/itex] are connected by the equation solved in 1). So plugging that in gives me:

[tex]I_\omega ^3 \omega _i ^2 \Omega ^2 = I_\omega (rmg)^2 + I_\Omega I_\omega ^2 \Omega ^4[/tex]

If I solve for [itex]\Omega[/itex], I can have real answers only if discriminant [itex]\geq[/itex] 0:

[tex]\left( -I_\omega ^3 \omega _i ^2 \right) ^2 - 4I_\Omega I_\omega ^3 (rmg)^2 \geq 0[/tex]

[tex]\Rightarrow \omega _i \geq \left( 4 \frac{I_\Omega}{I_\omega}(rmg)^2 \right) ^{\frac{1}{4}}[/tex]

Is it all wrong?

(If) so, could someone give me a hint to the right direction?

EDIT: Corrected a typo.

Suppose you have a ring, where practically all the mass is concentrated on the edges (like the tire of a bicycle). There is, however, a massless rod placed at the central axis.

The ring's given some [itex]\omega _i[/itex] amount of angular speed (about the "logical" axis).

The system is then placed by the end of the rod to lean horizontally on another rod that is placed vertically. The system will now rotate about the second rod too at an angluar speed [itex]\Omega[/itex].

In case you found my description confusing (I sure did), here's a diagram I drew:

http://img473.imageshack.us/my.php?image=img0157us.jpg

Anyways, my question is, how much initial angular speed [itex]\omega _i[/itex] must be given in order for the system to execute the described motion?

If you still think my desciption is confusing, please ask for more precise info .

This is what I tried:

**1)**I assumed that as the rod's placed it is given some initial [itex]\Omega _i[/itex]. For [itex]\Omega _i[/itex] to stay constant, [itex]\Omega _i = \frac{rmg}{I_\omega \omega}[/itex] (this I derived and it should be right). [itex]I_\omega[/itex] is the moment of inertia of the ring about the axis that was first set to motion.----

**2)**So, the [itex]\Omega[/itex] solved above will be the final [itex]\Omega _f[/itex] for the case where [itex]\Omega _i[/itex] = 0.I assumed that energy is conserved (

*is it?*), so:[tex]\frac{1}{2} I_\omega \omega _i ^2 = \frac{1}{2} I_\omega \omega _f ^2 + \frac{1}{2} I_\Omega \Omega _f ^2 [/tex]

[itex]\omega _f[/itex] and [itex]\Omega _f[/itex] are connected by the equation solved in 1). So plugging that in gives me:

[tex]I_\omega ^3 \omega _i ^2 \Omega ^2 = I_\omega (rmg)^2 + I_\Omega I_\omega ^2 \Omega ^4[/tex]

If I solve for [itex]\Omega[/itex], I can have real answers only if discriminant [itex]\geq[/itex] 0:

[tex]\left( -I_\omega ^3 \omega _i ^2 \right) ^2 - 4I_\Omega I_\omega ^3 (rmg)^2 \geq 0[/tex]

[tex]\Rightarrow \omega _i \geq \left( 4 \frac{I_\Omega}{I_\omega}(rmg)^2 \right) ^{\frac{1}{4}}[/tex]

Is it all wrong?

(If) so, could someone give me a hint to the right direction?

EDIT: Corrected a typo.

Last edited: