A thin disk rolls without slipping

[Tomtam]

Homework Statement

A very thin disk of mass m and radius R rolls without slipping along a horizontal plane. The disk is constrained to remain vertical. Let ψ be the angle between the plane of the disk and the x-axis of a fixed frame and θ be the angle measuring spinning of the disk about its center. By neglecting the gravitational force, show that the center of mass of the disk moves in a circle, if dψ/dt(t=0) ≠ 0 ; and along a straight line, if dψ/dt(t=0) = 0.

Homework Equations

I_θ and I_ψ are moment of inertia.

The Attempt at a Solution

I've tried to solve this problem, but I couldn't find any clues from what I've got.
what I've got are

constrain equations:

f1 = dx/dt - R(dθ/dt)cosψ = 0
f2 = dy/dt - R(dθ/dt)sinψ = 0

Lagrangian :

L = ½m(dx/dt)²+½m(dy/dt)²+½I_θ(dθ/dt)²+½I_ψ(dψ/dt)²

Lagrange's equations:

m(d²x/dt²) = -(dλ₁/dt)
m(d²y/dt²) = -(dλ₂/dt)
I_ψ(d²ψ/dt²) = λ₁R(dθ/dt)sinψ-λ₂R(dθ/dt)cosψ
I_θ(d²θ/dt²) = -λ₁R(dψ/dt)sinψ+(dλ₁/dt)Rcosψ+λ₂R(dψ/dt)cosψ+(dλ₂/dt)Rsinψ

I don't know what I should do next.

 

[TSny]

constrain equations:

f1 = dx/dt - R(dθ/dt)cosψ = 0
f2 = dy/dt - R(dθ/dt)sinψ = 0

Lagrangian :

L = ½m(dx/dt)²+½m(dy/dt)²+½I_θ(dθ/dt)²+½I_ψ(dψ/dt)²

Looks good to me so far.

Lagrange's equations:

m(d²x/dt²) = -(dλ₁/dt)
m(d²y/dt²) = -(dλ₂/dt)
I_ψ(d²ψ/dt²) = λ₁R(dθ/dt)sinψ-λ₂R(dθ/dt)cosψ
I_θ(d²θ/dt²) = -λ₁R(dψ/dt)sinψ+(dλ₁/dt)Rcosψ+λ₂R(dψ/dt)cosψ+(dλ₂/dt)Rsinψ

None of these equations looks correct to me. How did you get time derivatives of the λ's in the first, second, and fourth equations? The coefficients of the λ's in the last two equations are incorrect.

Review the mathematical formalism of setting up Lagrange's equations with constraints. A very brief summary is here
http://homepages.wmich.edu/~kamman/Me659LagrangesEquationsConstraints.pdf
See equation (2) in the link.

 

[Tomtam]

I followed methods from Classical Mechanics 3rd , Goldstein poole & Safko . page 47. Thank for your suggestion.

 

[TSny]

I followed methods from Classical Mechanics 3rd , Goldstein poole & Safko . page 47. Thank for your suggestion.

OK. This is interesting. I'm an old-timer who used the first edition of Goldstein where the formalism is developed differently in the section on Lagrange's equations with constraints. In the first edition it is assumed that the constraints can be put into the form of equation (1) of the link that I gave in post #2. Then Goldstein derives essentially equation (3) of the link. (You can set the nonconservative forces $\left(F_{q_k} \right)_{nc}$ equal to zero in your problem.) Note that this equation does not involve time derivatives of the λ's. Unfortunately (3) is not derived in the third edition.

In the 3rd edition, it appears that they are considering more general constraint equations that cannot necessarily be put into the form of equation (1) of the link. They then derive more complicated equations of motion that involve time derivatives of the λ's.

I'm a little surprised that the 3rd edition does not contain the simpler equation (3) of the link for constraints of the form (1). In both editions of Goldstein, there is an example of a hoop rolling down an inclined plane. This example is word-for-word the same in both editions. Can you follow that example based on the formalism in the 3rd edition? It appears to me that in both editions the solution uses the formalism of the 1st edition, where use is made of equation (3) of the link. So, I would think a student would have a hard time following this example in the 3rd edition. Maybe I'm overlooking something obvious whereby equation (3) is easily obtained from the equations of motion as presented in the 3rd edition of Goldstein. But I don't see it off-hand.

[EDIT: For the hoop example, the constraint equation $r d\theta = dx$ can be integrated and arranged as $r \theta - x +C = 0$, where $C$ is a constant of integration. Then you can easily use the formalism of the 3rd edition with this integrated equation as the constraint. But, the solution in the 3rd edition does not mention this. It sticks with the differential form $r d\theta = dx$ and uses the equivalent of equation (3) of the link. In your problem, the differential form of the constraints would be $R d\theta \cos \psi = dx$ and $R d\theta \sin \psi = dy$, which are not integrable. But the formalism of the 1st edition (or the link) can be used.]

Anyway, your problem seems to me to be much easier if you use the formalism of the 1st edition (i.e., equations (1) and (3)) of the link.

 

[TSny]

In the 3rd edition there is a footnote referencing a paper by J. Ray in the American Journal of Physics published in 1966.

Apparently this paper was the genesis of the switch in treatment from the 1st edition. There is an example in this paper that is identical to an example in the 3rd edition of Goldstein.

What's interesting, is that a few months after this paper was published, J. Ray published an erratum in the same journal in which he states that the first paper has errors which invalidate the treatment (and therefore the treatment in Goldstein's 3rd edition). The formalism in Goldstein's 1st edition remains valid.

It is odd that the 3rd edition of Goldstein (2001) would present Ray's treatment of 1966 when Ray showed in 1966 that the treatment is not valid.