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:

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)

Thanks!

(Also I'm new to this website still so if my question is bad or doesn't follow the rules just let me know with patience and kindness please. I will try to fix it if so! Thanks!)

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)?