# Another Lagrangian question: Bead sliding along a horizontalotating ring

1. Jan 12, 2009

### wdednam

Another Lagrangian problem: Bead sliding along a horizontal rotating ring

1. The problem statement, all variables and given/known data

A horizontal ring of mass M and radius a rotates freely about a vertical axis passing through a point on its circumference. If a bead of mass m slides along the ring without friction, what is the Lagragian function of the bead-ring system? Also write down the Kinetic energy of the bead if the ring rotates at a constant angular velocity $$\omega$$ about the vertical axis.

2. Relevant equations

$$T = \displaystyle{\frac{1}{2}}I\dot{\theta}^2 + \displaystyle{\frac{1}{2}}m\bold{v}^2$$

$$L = T - V$$

$$I = I_{CM} + MR^2 = 2Ma^2$$ in this case

$$\bold{v}^2 = \bold{v_{radial}}^2 + \bold{v_{tangential}}^2$$

3. The attempt at a solution

Okay, since the motion is horizontal, the gravitational potential energy of the system is constant and the Lagragian function of the system will simply be the kinetic energy of the system: L = T. Now, I can write down an expression for the kinetic energy of the ring (it is simply $$Ma^2\dot{\theta}^2$$) and the radial kinetic energy of the bead ($$\displaystyle{\frac{1}{2}}m\bold{v}^2 = \displaystyle{\frac{1}{2}}m\dot{r}^2$$ right?), but I don't know what the tangential component of the bead's velocity will look like. Is it simply the algebraic sum of the bead's and ring's tangential velocities? I'd appreciate any help, thanks.

Last edited: Jan 12, 2009
2. Jan 12, 2009

### wdednam

Hi again,

I found the solution. If you are interested in seeing it, send me a message and I'll give you the link. (I don't think it is within forum rules to post it here)

Thanks to everyone who started thinking about this problem and was going to get back to me later.

Cheers.