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What is the equations of motion for a pendulum and spring

  1. Feb 3, 2009 #1
    1. The problem statement, all variables and given/known data

    This is for a math based physics class. I need the equation ofa pendulum and of a spring.

    2. Relevant equations

    Spring: x=Asin(wt-phi)+B
    Pendulum: theta=theta0cos(wt+phi) or theta=theta0cos(sqrt(g/l)sin(theta))

    3. The attempt at a solution

    I don't know which of these are correct. Please let me know the right equation for each; if it's a differential equation, please let me know what it is solved if possible. Thank you.
  2. jcsd
  3. Feb 3, 2009 #2


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    For small displacements of the pendulum when [tex] sin \theta \simeq \theta [/tex] and ignoring friction in both cases, the equations governing the motion of the mass-spring and the pendulum are those of simple harmonic motion.

    The D.E. is [tex] \frac {d^2 x}{ dt^2} + \omega^2 x = 0 [/tex]

    Which has a solution with two arbitrary constants (necessary because the D.E. is second order). There are various equivalent forms for the solution. One of which is

    [tex] x = A sin(\omega t + \phi) [/tex]

    [tex] \omega [/tex] is called the angular frequency and its exact form in terms of the physical parameters in your problem can be determined when you set up the differential
    equation. A is the amplitude and [tex] \phi [/tex] the phase. These two parameters are determined from initial conditions. All of this is covered in standard physics textbook.
    Last edited: Feb 3, 2009
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