Olympiad Mechanics problem (torque)

[zwierz]

there are a lot of different interesting motions in this problem

 

[Dr.D]

IIRC, J.P. Den Hartog discussed this problem in detail in his Mechanical Vibrations text from many years ago.

 

[Dr.D]

An end of a light wire rod is bent into a hoop of radius r.

This quote, and the associated figure indicate that there is support for a bending moment transfer between the pendulum shaft and the ring. Saying this differently, the pendulum is not pivoted on the ring, but is rather fixed on the ring. This means that there is only one DOF, not two.

 

[zwierz]

IIRC, J.P. Den Hartog discussed this problem in detail in his Mechanical Vibrations text from many years ago.

That is strange. This problem is too simple to discuss it in a monograph in detail.

 

[Dr.D]

That is strange. This problem is too simple to discuss it in a monograph in detail.

By George! You are correct. He did not discuss it in detail, but only offered it as a homework problem in such a way that the person working the problem would develop a rather comprehensive discussion of the several possible cases. I was recalling my own detailed write up of this problem.

 

[fedecolo]

He did not discuss it in detail, but only offered it as a homework problem in such a way that the person working the problem would develop a rather comprehensive discussion of the several possible cases.

So what are the several possible cases?

 

[Dr.D]

Several different friction models.

 

[zwierz]

It does not make sense to consider friction models whenever non slipping condition has already been imposed.

The phase portraits of described above system essentially differ from each other for the following four cases $\epsilon r/g>1,\quad \epsilon r/g=1,\quad 0<\epsilon r/g<1,\quad \epsilon=0$

The center of the ring can rotate about the shaft; it can oscillate near its stable equilibrium; it can even perform asymptotic motions

 

[Dr.D]

I'll leave you to discuss it with Den Hartog.

 

[RoloJosh16]

I understand the fact that on an incline the slipping begins at a angle $\alpha$ s.t tan $\alpha = \mu$ , but how can you assocciate that fact to this problem?

 

[haruspex]

I understand the fact that on an incline the slipping begins at a angle $\alpha$ s.t tan $\alpha = \mu$ , but how can you assocciate that fact to this problem?

Are you referring to the original problem in post #1 or to the very different problem zwierz introduced (in violation of forum guidelines!) in post #29?

 

[RoloJosh16]

Are you referring to the original problem in post #1 or to the very different problem zwierz introduced (in violation of forum guidelines!) in post #29?

To the original problem.

 

[RoloJosh16]

Are you referring to the original problem in post #1 or to the very different problem zwierz introduced (in violation of forum guidelines!) in post #29?

Do you understand what I am asking?

 

[haruspex]

To the original problem.

In the original problem the shaft is turning at constant speed and the system is in steady state, so we are only concerned with kinetic friction.

Edit: but, interestingly, if we define α as tan(α)=μ_k then the answer can be written as (R+L)sin(θ)=Rsin(α ).

 

[TSny]

I understand the fact that on an incline the slipping begins at a angle $\alpha$ s.t tan $\alpha = \mu$ , but how can you assocciate that fact to this problem?

The original problem is similar to having a conveyer belt inclined at angle $\alpha$ with the belt moving upward at a constant speed. Suppose a box is placed on the moving belt and the box happens to stay at rest relative to the earth. Then you can ask for the relation between $\alpha$ and the coefficient of kinetic friction $\mu_k$.

 

[RoloJosh16]

The original problem is similar to having a conveyer belt inclined at angle $\alpha$ with the belt moving upward at a constant speed. Suppose a box is placed on the moving belt and the box happens to stay at rest relative to the earth. Then you can ask for the relation between $\alpha$ and the coefficient of kinetic friction $\mu_k$.

Ahhh, thank you. I was thinking about that relation but didn' t think about a moving surface. The box in that case would be the tiny part of the hoop in contact with the shaft?

 

[TSny]

Ahhh, thank you. I was thinking about that relation but didn' t think about a moving surface. The box in that case would be the tiny part of the hoop in contact with the shaft?

Yes, that's right.