I would like to know if you can help me determine the restraining force for an elastic pendulum. The problem is the following
A particle of mass $m$ is suspended by a massless spring of length $L$. It hangs, without initial motion, in a gravitational field of strength $g$. It is struck by an impulsive horizontal blow, which introduces an angular velocity $\omega$. If $\omega$ is sufficiently small, it is obvious that the mass moves as a simple pendulum. If $\omega$ is sufficiently large, the mass will rotate about the support. Use a Lagrange multiplier to determine the conditions under which the string becomes slack at some point in the motion
Since it asks me to solve by Lagrange multipliers I need the constraint force. My attempts were the following:
Since the mass is suspended from a spring, when it hits the spring, an angle theta will be formed and the spring will deform a length $z$. So in polar coordinates I could write $r=L+z$, however I'm not sure about this.
Another constraint I can use would be $\dot\theta=\omega$ since there will be angular velocity at the moment the pendulum is struck.
However, since you ask me for conditions for $L$, I believe that there should be no constraint force $Q_\theta$ and therefore, there should only be constraint force $Q_r$, this means then that my multiplier should only depend on $r$ and It makes me think my first attempt is the right one but I'm not sure. I hope you can help me, I will appreciate it. I only want the constraint equation, I can solve the rest.
Best Answer
The elastic pendulum have two generalized coordinate $~x,y~$
from here the constraint equation
$$\tan(\theta)=\frac xy\quad,\omega\,t=\arctan(x/y)$$
and with $~x=r\,\sin(\theta),y=r\,\cos(\theta)$
you obtain two degrees of freedom $~r~,\theta~$ and one constraint equation $~\omega\,t-\theta=0~$
with $$U=-k\frac 12\,{r}^{2}+k\,L\,r-m\,g\,r\cos \left( \theta \right)$$
$$T= \frac 12\,m \left( \left( \sin \left( \theta \right) {\dot r}+r\cos \left( \theta \right) \dot\theta \right) ^{2}+ \left( \cos \left( \theta \right) {\dot r}-r\sin \left( \theta \right) \dot\theta \right) ^ {2} \right) +\lambda\, \left( \omega\,t-\theta \right) $$
where L is the pendulum initial length , and $~\lambda~$ the Lagrange multiplier
you can obtain the equations of motions and the constraint force