Rotating mass pulled in at constant rate

Question

I had a an idea about a situation. Assume a mass swinging around an axis at a constant angular velocity at a radius $R_0$ (circular motion), attached to the axis of rotation by some weightless string. Also assume no friction or air resistance. Then, a winch pulls the string in at a constant rate, effectively decreasing the length of the string at a constant rate $S$. What path would the mass take if the winch started pulling when the mass was at $R_0$ and $\theta_0$, so any given starting angle and radius.

My attempt was to use Lagrangian analysis:

$$\mathcal{L}=\frac{1}{2} MR^2 \dot \theta^2$$

And then from here I would use $R$ as my variable of interest to solve for theta as a function of $R$, and then write $R$ as a function of time because $R$ is decreasing linearly.

I have two questions:

1. Would this process work?
2. Is there a classical way to solve this problem without Lagrangian analysis (i.e. just with Newtonian mechanics)?

Answer 1

Your radial coordinate should be changing, so

$$\vec r = (r_0-v_r t)\begin{pmatrix}\cos\theta\\\sin\theta\end{pmatrix}$$

Derive it with respect to $t$ to get the velocity $\vec v=\dot{\vec r}$ and plug it into $L=\frac{1}{2}m\vec v^2$.