# Accelerating Elastic Pendulum (Lagrangian)

1. Feb 24, 2008

### darkfall13

1. The problem statement, all variables and given/known data

A pendulum of mass $$m$$ is suspended by a massless spring with equilibrium length $$L$$ and spring constant $$k$$. The point of support moves vertically with constant acceleration. Write the Lagrangian of the system and the equation of motion.

2. Relevant equations

$$L = T - U$$

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

3. The attempt at a solution

We can draw a vertical and call the angle of the pendulum to this $$\theta$$. I know for an elastic pendulum we have (where $$l$$ is the variable length) $$T = \frac{m}{2} \left( \dot{l^2} + l^2 \dot{\theta^2}\right)$$ and $$U = \frac{1}{2}k\left(l-L\right)^2 + mgy$$ but I'm not sure how to involve the acceleration term. A hint in the right direction would be greatly appreciated :D