# Lagrangian problem

## Homework Statement

Mass m is connected to a string and is being whirled in a circle in a horizontal plane of a table. The string passes through a hole in the center of the circle and is being pulled with a constant velocity V starting at t=0 so the radius decreases. Initially the mass is at a distance r0 from the hole and is revolving with angular speed $$\omega_0$$. Use Lagranges equations and Lagrange multipliers to find then tension in the string as a function of time.

## Homework Equations

I have attempted to write down the Lagrangian with the lagrange multipliers but i am not sure it is correct.
what i have is that the speed of the mass is given by:
$$v=r\omega = r\dot{\theta}$$ and i have that r = r0 - Vt so $$v=(r_0-Vt)\dot{\theta}$$
Using this in 1/2mv^2 for the Lagrangian along with the constraint r-r0+Vt=0 my Lagrangian with lagrange multiplier was this:
$$\frac{m}{2}(r_0^2-2r_0Vt + V^2t^2)\dot{\theta}^2 + \lambda(r-r_0+Vt)=L$$
and when doing the derivative $$\frac{\partial{L}}{\partial{\dot{r}}}$$ i have this is the partial with respect to V. is this correct? then i would solve for lambda and this would be my force of tension?

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