# Spring Pendulum - Lagrangian Mechanics

• bigguccisosa
In summary, the conversation discusses the use of the Euler-Lagrange equation to solve for the motion of a spring-mass system with potential and kinetic energies. The equations for the radial and angular directions are derived, and the concept of fixing the angular position and velocity is explained as an approximation for the short timescale of the motion. The error in the first term of the radial equation is noted and corrected.

## Homework Equations

Euler-Lagrange Equation
$\frac{\partial{L}}{\partial{q}} - \frac{d}{dt}\frac{\partial{L}}{\partial{\dot{q}}} = 0$
$L = T - V$

## 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.
$$L = \frac{1}{2}m(\dot{r}^2+\dot{\theta}^2(r_0+r)^2) + mg(r_0+r)cos\theta - \frac{1}{2}kr^2$$

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

#### Attachments

• homework1.PNG
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"Fixed" means "constant." The position and velocity aren't actually fixed (so ##\dot{\theta} \ne 0##), but because the timescale over which the radial motion occurs is so short, you can analyze the motion as if ##\theta## and ##\dot{\theta}## were constant. It's an approximation.

By the way, the first term in the radial differential equation is incorrect.

Thanks I must have missed a (r_0 + r) factor in the first term when I was putting it into tex, I'll give it a shot with theta and theta dot as constants.