# Lagrangian "constant" string length constraint for unwinding string in pulley system [closed]

The following is a problem from Patrick Hamill's A Student's Guide to Lagrangians and Hamiltonians.

A string passes over a massless pulley. Each end is wound around a vertical hoop, as shown in the figure below. The hoops tend to descend, unwinding the string, but if one hoop is much more massive than the other, it can cause the lighter hoop to rise. The hoops have masses $$m_1$$ and $$m_2$$ and radii $$R_1$$ and $$R_2$$. Show that the tension in the string is $$\tau = \dfrac{gm_1m_2}{m_1+m_2}$$.

Q: What is the approach for defining the constant string length constraint, when the string is unwinding as is the case here?

• @user256872 well added length will add to the wheels angular & translational energy in the exact same way a wheel would along some distance on the ground. Should be an exact ratio of translational to angular momentum/energyj that I'm too lazy to calculate Jun 17, 2021 at 6:07
• This problem is constrained to one degree of freedom I disagree. If we restrain the pulley, each of the hoops would descend independently, hence we have 2 dof. Pulley would only ensure that the rope tension is the same throughout. Jun 17, 2021 at 8:34
• @A.V.S. you have three generalized coordinates, see my answer
– Eli
Jun 18, 2021 at 11:56
• @ACuriousMind: there's a lot to learn in this question. I didn't find the formulation of this constraint as trivial as it may sound. There were several points to be made regarding the physics of the problem. In my post my intention was to explore and highlight those aspects and my use of any eqns was to the same effect. . The interesting points that I am talking about are 1. this is the only case where both, by themselves perfectly valid, perfect roll and constant length constraints intersect. Inspite of that, there is no perfect rolling happening here. 2. The final tension is agnostic to ... Jun 19, 2021 at 17:58
• @lineage I didn't delete your answer because I thought it was trivial or anything like that. I deleted it because it is our policy (linked in the comment I left when deleting it) to delete detailed answers to homework-like questions, which we consider off-topic here. The community has closed this as such a question, and so I deleted your answer. Jun 19, 2021 at 18:23