# Falling disk with string

1. Aug 15, 2016

### Vrbic

1. The problem statement, all variables and given/known data
Let's have a disk and massless rope tangled in it. One end of rope is tied to the ceiling and the disk is falling freely down. System has one degree of freedom. As a coordinate we can choose angle $\phi$ which says an angle of rotation from the start position. Find from the Lagrange equations of the first kind equation of motion and binding force.

2. Relevant equations
Lagrange equtions of the first kind:
$$m \ddot x = F_x+ \lambda \frac {\partial f} {\partial x} \\ m \ddot y = F_y+ \lambda \frac {\partial f} {\partial y}$$, where f is binding function

3. The attempt at a solution
I suppose this term $\lambda \frac {\partial f} {\partial y}$ is binding force, but I don't have no idea how to find it.
Next here we use coordinate $\phi$, I should transforme y -> $\phi$. I suppose diameter of disk $r$ so $y=r\phi$. Do you agree?
I probably know how to solve it other way: $-ma = -mg +F1, M=F1*R=I*\alpha$(angular acceleration) $=mR^2/2*a/R=amR/2$ from this I know anything what I need: $-ma=-mg + am/2 => a=2g/3$ Is it right?
But I don't know how to solve it through Lagrange equation. Can you advice me?

2. Aug 15, 2016

### drvrm

the disc is falling as well as rotating so initially two degrees of freedom of rotational motion as well as translational motion appears.
now one can talk about constraints. and what will be constraining equations which can render one degree of freedom dependent on the other.,
if ds is displacement then it must be equal to radius times the change in angle
ds - R. d(phi) =0
so, lagranges equati on has to be written first in generalized coordinates and one can use method of lagranges multipliers ; one can look up Goldstein's book on classical mechanics....

3. Aug 16, 2016

### Vrbic

It's really wide book :)
I've read some paragraphs and something is clear. But I don't understand why is sometimes convenient to have a constraint written in derivatives (velocities)? And how I find out, if it is my case? I know a procedure of solving minimalization problem for function of more variables by Lagrange multipliers, but generally I don't get relation between minimalization problem and this problem.
When I look up at Lagrange equations of first kind which I wrote, I guess that $f$ function should be $f=y-R\phi$. And one equation will be for $y$ and second for $\phi$ do you agree?