Tension on the rope (classical mechanics problem)

[allshaks]

Homework Statement

The situation is that of Goldstein's problem 1.21 (or 1.19 in some editions):
"Two mass points of mass m1 and m2 are connected by a string passing through
a hole in a smooth table so that m1 rests on the table and m2 hangs suspended.
Assume m2 moves only in a vertical line."
But the question is the following: what is the tension?

Homework Equations

The Euler-Lagrange equations of the system are:
$$ \frac{d}{dt}(m_1 d^2 \dot{\theta})=0$$
$$ (m_1+m_2)\ddot{\theta}=-m_2 g+m_1 d \dot{\theta}^2$$
The constraint equation is:
$$r_1-y_2=l$$
Where $l$ is the length of the rope.

The Attempt at a Solution

I started by using the fact that the only force applied on $m_1$ is the tension, and that this tension must be in the direction of the string; that is, always radial (using the hole as the frame of reference). This means:
$$F_{r_1}=\frac{a_{r_1}}{m_1}= \frac{\ddot{r_1}-r_1 \dot{\theta}^2}{m_1}=-T$$
Where $T$ is negative because I define it as being positive when it is pulling up mass 2 (the other mass) such that:
$$T=m_2 \ddot{r_1}+m_2 g$$
Because $\ddot{y_2}=\ddot{y_1}$
I've been playing with all of these equations for a while, but I couldn't find the solution. I'm not sure if there is a way of obtaining an expresion $T$ that involves Lagrangian mechanics, or if there are any other techniques to do it.
I'm following a problem set from a Classical Mechanics course I'm taking at college, and there is another question in the problem set: "In order to calculate the constraint forces on a system, what are the methods that could be employed?". I'm not sure how to answer that question, and I think that maybe some info on that (maybe some textbook that covers the subject?) may help me solve the problem. Thank you very much.

 

[Nidum]

There is very little maths needed to solve this problem .

Draw free body diagrams for the two masses and just look at them .