# Small oscillations of a spring-pendulum

1. Sep 10, 2013

### thecommexokid

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

A spring of rest length L (no tension) is connected to a support at one of the ends and has a mass m attached to the other. Write the Lagrange equations. Discuss the movement for small departures from equilibrium.

2. Relevant equations

I did the first part, and I found the equations of motion to be
$$\ddot r = r\dot\phi^2 + g \cos\phi - \frac km (r - L)$$
and
$$\ddot \phi = -\frac gr \sin\phi - \frac 2r \dot r \dot\phi,$$
where r measures the distance of the mass from the pendulum pivot and ɸ measures the angle the pendulum makes with the vertical. I am told that these are correct.

3. The attempt at a solution

Now I want to think about the case of small oscillations, so I will discard all terms which are second-order or higher in small quantities. Therefore I can lose the $\dot\phi^2$ term, the $\dot r\dot\phi$ term, and all powers of ɸ greater than 1 in the expansion of sin ɸ and cos ɸ.

This gives me the right answer...but I don't actually understand what I just said. How do I prove that $\dot r$ and $\dot\phi$ are, in fact, small quantities?