- #1

- 24

- 3

## Homework Statement

Please see attached image :)

## Homework Equations

Euler-Lagrange Equation

[itex] \frac{\partial{L}}{\partial{q}} - \frac{d}{dt}\frac{\partial{L}}{\partial{\dot{q}}} = 0[/itex]

[itex] L = T - V[/itex]

## The Attempt at a Solution

a. The potential energy V is the potential energy from the spring and the gravitational potential energy. The kinetic energy is the energy in the radial direction and in the theta direction.

[tex] L = \frac{1}{2}m(\dot{r}^2+\dot{\theta}^2(r_0+r)^2) + mg(r_0+r)cos\theta - \frac{1}{2}kr^2[/tex]

b. Use one Euler-Lagrange equation for r and one for theta and I get: [tex] m\dot{\theta}^2 + mgcos\theta - kr = \frac{d}{dt}(m\dot{r}) [/tex] [tex] -mg(r_0+r)sin\theta = \frac{d}{dt}(m\dot{\theta}(r_0+r)^2)[/tex]

Which are the F=ma equation for the radial direction and the Torque = (d/dt) Angular momentum equation.

c. Confused here, if angular position and velocity are to be fixed, I assume they mean [itex] \theta [/itex] and [itex] \dot{\theta}[/itex], do I consider them both to be zero, or just a constant number? And if the position is fixed doesn't that mean that the velocity should be zero? Is this a differential equation I have to solve?

d. and e. Need c to continue.

Any help is appreciated, thanks !