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Small oscillations of a spring-pendulum

  1. Sep 10, 2013 #1
    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?
     
  2. jcsd
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