Lagrangian Example: Sliding Bead on a Rotating Hoop
I don’t know who came up with this problem, but it’s actually pretty fun. Here’s the deal:
- Start with a hoop with a radius of 0.05 meters. It rotates about an a vertical axis (in the y-direction) that passes through the plane of the hoop (see the sketch above).
- The hoop rotates with a constant angular velocity of 10 rad/s.
- On the hoop, there’s a bead with zero friction such that it can slide along the hoop. Let’s say it has a mass of 0.05 kg (but of course the mass doesn’t actually matter).
So, let’s find an equation of motion for this bead. Just to be clear, we are pretty much forced to use Lagrangian mechanics (here is my introduction to that). I mean, let’s consider what would happen if I tried to use Newtonian mechanics. For this, I would have to somehow find the force the hoop exerts on the bead. That’s pretty tough. Oh, I can’t even use conservation of energy to find this force since there’s some type of external driving force that rotates the hoop. So, that’s a no-go for Newtonian.
With Lagrangian mechanics, the first thing we need to do is to determine the degrees of freedom. Even though the bead can move in all 3 dimensions, the bead is constrained to move along the hoop. That’s just one degree of freedom.
