# Ferris wheel dynamics (swinging bucket)

1. Mar 19, 2015

### eric_phsyics

What mass is required to keep the seat of a ferris wheel from swinging no more than 30 degrees measured from the vertical axis?

So lets say we have a big ferris wheel with radius R and its spinning at 10 RPM. At the end of the ferris wheel, is a hinge, and a hanging bucket (like the picture). How do we figure out the swinging motion as the wheel rotates?

On the bucket hinge (where it attaches to the wheel) we have a torque created by gravity, and a centrifugal force from the momentum of the bucket. I can't seem to figure out the geometry though.

2. Mar 19, 2015

### Matternot

I haven't thought about it too long, but this is my first thought:

Consider the energy of the bucket (linear kinetic, rotational kinetic and gpe) If the ferris wheel is rotating at a constant speed, the ferris wheel must be imposing a force upon the bucket. You would have to consider all work is done on the bucket by the ferris wheel. From this, you should be able to calculate how much energy the bucket has and how much rotational energy it has.

Modelling the angle exactly as a function of time sounds fairly close to the double pendulum problem. Maybe take the equation from that and set the derivative of theta 1 to your 10 RPM and second derivative to 0.

Hope this helps.

3. Mar 19, 2015

### eric_phsyics

Thanks that was useful. After a quick search, I found this link:

http://www.astro.uwo.ca/~houde/courses/PDF files/physics350/Lagrange.pdf

Equation 4.37 is very close to what I need. I also need to account for friction at the hinge. How do I incorporate a constant frictional force?

Also, this is a differential equation, correct? So to solve my problem, will I just need to take this equation into maple and get a solution for theta?

4. Mar 19, 2015

### Matternot

Sure! This looks good. Here he uses principle of least action, but differentiating the energy should give the same result. (It's debatable which method is easier). This equation would assume the bucket acts like a point pendulum. It might be a bit tricky to convert to a rigid pendulum if e.g. they want you to assume it's in the shape of a rectangle. In the legrangian in that paper, replace the potential with that taken from the centre of mass and replace the kinetic with the linear kinetic (in the a omega t terms) and the rotational (in the b thetadot terms) using moment of inertia.

5. Mar 19, 2015

### Matternot

I don't know much about hinges, but you might want to model it as a spinning disc. The frictional force would require knowledge of the radius of the hinge. Using the equation you already have, you could just include a torque into the equation (being very careful about signs) You could have to put it in the energy equation. i.e. d/dt(T+V)=Energy lost to friction per dt to avoid sign errors. Modelling a disc spinning as your hinge, calculate the energy lost in that per dt (it will be dependant on theta dot). Using Lagrangian mechanics, you might have to use the Rayleigh dissipation function, but my knowledge is a bit limited if you use that.

6. Mar 19, 2015

### BvU

Did you notice your equation 4.37 doesn't contain a mass any more ?

7. Mar 19, 2015

### Matternot

Why should it? both the potential and the K.E. is proportional to the mass. They cancel in the Euler Lagrange equation.

8. Mar 19, 2015

### BvU

When all you have is an equation of motion with no mass in it, there's no solving for the original question. And if all you know is this 10 RPM you don't have enough to go on even if you can solve the equation of motion.

But I may be wrong; if so, please put me right !

9. Mar 19, 2015

### Matternot

Aaah I see your confusion. Equation 4.37 does not consider a fringe with constant friction. The potential and kinetic energies (and therefore the Lagrangian) are proportional to the mass of the bucket, and so the equation of motion is independent of the mass. Energy lost to friction will not be proportional to the mass, and so from this, the mass becomes relevant and the question becomes solvable.

10. Mar 19, 2015

### Matternot

Hence I would actually expect an upper bound on the mass as opposed to a lower bound on the mass... Which at first might seem counter-intuitive

11. Mar 20, 2015

### BvU

Is this a genuine exercise ? If so, what information do you have to work from ? The 10 rpm is all I have seen thus far.
Or are you bringing in all kinds of things as you go ? The word friction doesn't appear in your problem statement. Any other things that are going to come out of the blue sky in the near future ?

This isn't meant to be grudgy or anything: the subject is interesting. So indulge potential helpers with a bit of completeness in context, if you can !

12. Mar 20, 2015

### Matternot

It's not my question...

The friction only really arrives in the second post:
Though I agree it's usually helpful to have all information.

A google search suggests it's not a question from an online question sheet