Equation of motion of a mass-spring system

  1. hi, all. I am trying to derive the equation of motion of a mass spring system without using the
    energy method but I am wrong somewhere and I cant find it, can you help me find where I am
    wrong. Equation of motion of a simple mass spring system is indeed mx''+kx=0 but here I am
    thinking that when we pull the mass, motion arises from the spring force which is trying to bring back the mass and it is -kx due to our choice of negative direction but when the force is negative,
    i.e -kx, the acceleration x'' must also be negative because they are in the same direction and sense. Here their sense both are negative. So equation should be -mx''=-kx(sum of forces equal mass product acceleration) and thus -mx''+kx=0 Can you explain me where I am wrong?
  2. jcsd
  3. DrClaude

    Staff: Mentor

    Exactly. Therefore, writing
    \ddot{x} = \frac{-k x}{m}
    ensures that the acceleration ##\ddot{x}## is negative when the displacement ##x## is postive. If you add a minus sign in front of ##m \ddot{x}##, you get a positive acceleration for a positive displacement.
  4. DrClaude

    Staff: Mentor

    I should also add that the base formula is ##F=ma##. Once you have figured out what ##F## is, the equation must be applied directly, without modifying the ##ma## part.
  5. It is very clear, thanks a lot.
    Last edited: Feb 5, 2014
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