# How to use Lagrangian Mechanics, not Newtonian Mechanics, to find tension in a system of two hoops attached to a massless string over a pulley? [closed]

This question asks me to find the tension force in the rope connecting two hoops, masses $$m_{1}$$ and $$m_{2}$$ radii $$R_{1}$$ and $$R_{2}$$ attached to the massless rope and hung one each side of a massless pulley.

Here is a figure:

This is my sketch of the problem based on the description in the exercise, taken from Chapter 1 of A Students Guide to Lagrangians and Hamiltonians by Patrick Hamill. In the book the arrangement of the hoops, pulley and rope are shown and I have added the labels in my figure based on my current understanding.

My attempt at a solution:

No constraints are given but I am assuming $$C = x_{1} + x_{2}$$ where $$C$$ is the length of the rope. Also assuming the motion is in 1D along the string.

$$T = \frac{1}{2}m_{1}\dot{x_{1}}^2 + \frac{1}{2}I_{1}\omega_{1}^{2} + \frac{1}{2}m_{2}\dot{x_{1}}^2 + \frac{1}{2}I_{2}\omega_{2}^2 \\ V = -m_{1}gx_{1}-m_{2}g(C-x_{1})$$

and using $$L = T - V$$, substituting $$I = MR^2$$ and $$\dot{x}=R\omega$$ I get $$L = m_{1}\dot{x_{1}}^2 + m_{2}\dot{x_{1}}^2 + m_{1}gx_{1}+ m_{2}g(C-x_{1})$$

and after applying Lagrange's equation $$2m_{1}\ddot{x_{1}} + 2m_{2}\ddot{x_{1}} + m_{1}g - m_{2}g = 0$$

giving $$\ddot{x_{1}} = \frac{(m_{2}- m_{1})g}{2m_{1}+2m_{2}}$$

I tried to use $$ma = T - mg$$ to obtain the given solution which is $$T = \frac{m_{1}m_{2}g}{m_{1} + m_{2}}$$ but got this: $$\ddot{x_{1}} = \frac{(m_{2}- m_{1})g}{2m_{1}+2m_{2}} = \frac{T}{m_{1}}-g \\ \frac{T}{m_{1}} =\frac{m_{2}g- m{1}g+ 2m_{1}g+2m_{2}g}{2m_{1}+2m_{2}} = \frac{3m_{2}g+ m_{1}g}{2m_{1}+2m_{2}}$$

• Hello there, and welcome to the Physics Stack Exchange! Homework and "check my work" questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. Please read this post on asking homework questions and this post for "check my work" questions. – Vincent Thacker Jun 16 at 15:02
• Hi, all the work above is my own, my summary above is the result of 3 days of work. Please explain how I can show more effort? I have also searched the web over this period and my textbooks. There are no solutions I can find to this problem, and on Stackexchange only the Newtonian method is present. Please also explain how this question is not of use to the community, as there are no solutions to this problem anywhere I can find on the Web, except the Newtonian method, which does not help me to learn to apply the Lagrangian. I am asking this question as a last resort.. Please help. Thanks. – Dr Caroline O'Brien Jun 16 at 15:30
• Please also explain why this Lagrangian question is allowed and has been answered but not mine. I am keen to learn. physics.stackexchange.com/questions/433777/… – Dr Caroline O'Brien Jun 16 at 15:38
• Can you post a figure to show how the loops connected? It is hard to image how the linear motion related to the loops' rotation. – ytlu Jun 16 at 17:46
• Yes, of course - I have added it to the post above. Many thanks. – Dr Caroline O'Brien Jun 16 at 18:13

Oh.. everything you did is right except the fact that the you didn't take the initial conditions.

Take, $$\dot{x_1}, \dot{x_2}=0$$,

i.e.,

$$T = \frac{1}{2}I_{1}\omega_{1}^{2} + \frac{1}{2}I_{2}\omega_{2}^2$$

Although I haven't verified this, but a quick troubleshooting tells me you have unnecessary factors of 2 coming up.