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Period of a mass oscillating on a spring

  1. Feb 15, 2009 #1

    jgens

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    Gold Member

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

    The period of a mass m oscillating on a hookian spring of negligible mass.

    2. Relevant equations

    F = -kx

    3. The attempt at a solution

    I'm using x in place of ∆x for convienance.

    F = -kx

    dv/dt = -kx/m

    ∫vdv = -k/m∫xdx

    v^2 = C - k(x^2)/m: Placing the restraint that if v = 0, x = xi

    v^2 = k(xi^2)/m - k(x^2)/m

    v = sqrt(k(xi^2)/m - k(x^2)/m)

    t = sqrt(m/k)∫dx/sqrt(xi^2 - x^2)

    t = arcsin(x/xi)sqrt(m/k). Therefore, if we allow ∆x = 0

    t = arcsin(0)sqrt(m/k). Hence, if we want the time for one period:

    t = 2pisqrt(m/k)

    Is this correct? Thanks!
     
  2. jcsd
  3. Feb 15, 2009 #2

    rock.freak667

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    Homework Helper

    Yes that is correct.
    But if you didn't want to integrate you could have just put F=ma=-kx
    such that a= -(k/m)x
    which is in the form a=-(w^2)x meaning that w=sqrt(k/m)

    and the period T=2pi/w
     
  4. Feb 15, 2009 #3

    jgens

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    Gold Member

    Thanks! That's a much simpler derivation.
     
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