A mass at the top of a vertical hoop. (Lagrangian Mechanics)

[FLms]

Homework Statement

A heavy particle is placed at the top of a vertical hoop. Calculate the reaction of the hoop on the particle by means of the Lagrange's undetermined multipliers and Lagrange's Equations. Find the height at which the particle falls of.

Homework Equations

$$\frac{d}{dt} \frac{\partial L} {\partial\dot{q}} - \frac{\partial L}{\partial q} = 0$$

The Attempt at a Solution

$$L = \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\theta}^2) - mgr cos(\theta)$$

Euler-Lagrange Equation with respect to $r$ becomes:

$$m \ddot{r} - m r \dot{\theta}^2 + m g cos(\theta) = \lambda$$

And, with respect to $\theta$:

$$mr^2 \ddot{\theta} + 2 m r \dot{r} \dot{\theta} - mgr sin(\theta) = 0$$

As the radius is constant, it's derivative is zero. So, we have:

$$- m r \dot{\theta}^2 + m g cos(\theta) = \lambda$$
$$m r^2 \ddot{\theta} - m g sin(\theta) = 0$$

Now, I don't really know what's the next step.
What should I do?

Any help appreciated.

 

[Fightfish]

$\lambda$ is actually the reaction force of the surface of the hoop on the particle.

We can include the reaction force as a potential in the formulation of the Lagrangian:
$$L = \frac{1}{2} m (r^2 + r^2 \dot{\theta}^2) − m g r cos(\theta) - V (r)$$
Comparison with your E-L equations will then give $- \frac{dV(r)}{dr} = \lambda$
Hence, the reaction force, taking the boundary condition r = R, is
$$F_{R} = - \frac{dV(r)}{dr}|_{r = R} = \lambda |_{r = R} = mg cos(\theta) - mR\dot{\theta}^2$$

As for when the particle falls off, that means that the reaction force must be zero since it is no longer in contact with the hoop.

 

[FLms]

$\lambda$ is actually the reaction force of the surface of the hoop on the particle.

We can include the reaction force as a potential in the formulation of the Lagrangian:
$$L = \frac{1}{2} m (r^2 + r^2 \dot{\theta}^2) − m g r cos(\theta) - V (r)$$
Comparison with your E-L equations will then give $- \frac{dV(r)}{dr} = \lambda$
Hence, the reaction force, taking the boundary condition r = R, is
$$F_{R} = - \frac{dV(r)}{dr}|_{r = R} = \lambda |_{r = R} = mg cos(\theta) - mR\dot{\theta}^2$$

As for when the particle falls off, that means that the reaction force must be zero since it is no longer in contact with the hoop.

Ok, that was helpful.

I guess I could use Newtonian Mechanics to find out that $v^2 = 2 g R (1 - cos(\theta))$, and then substituting in the equation, which gives me:

$$\lambda = mg (3 cos(\theta) - 2)$$

But, is there any other way to find $v^2$ or $\ddot{\theta}^2$? I.e., by using only the Lagrangian Mechanics.

PS: Advanced Physics?! Really? That was kind of a surprise to me...