Propagation of error

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

    this is regarding propagation of error for a lab i did:

    we measured the amplitude of a damped harmonic oscillation over a time period, taking amplitude measurements every 1 second for 14 seconds. when graphed (by excel), the plot has the form of y = Ae^(-gt), where A is the amplitude, t is time and 1/g = the damping time.

    how would the uncertainty of g be calculated, if the uncertainties of A and t are known for each measurement?


    2. Relevant equations

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    3. The attempt at a solution

    i have no idea how to do this.
     
  2. jcsd
  3. mgb_phys

    mgb_phys 8,952
    Science Advisor
    Homework Helper

    The effect of an uncertainity in A is simple, if you double A what effect does this have on y ? Similairly if A changed by 10% what effect would this have on y?

    T is a bit more complicated but you can always do this experimentally if you can't do the maths.
    Calculate y for some value of t, now change t by a small amount and see how y changes.
    Do this for a few values and you will see if the change in y is proportional to change in t or some other function.
     
  4. This is what I did. Tell me if I’m wrong:

    Let:
    y(A,t) = Ae^(-gt)
    uncertainty of A = dt
    uncertainty of t = dt
    uncertainty of y = dt

    then:
    dy = {[dy1)^2 + [dy2]^2}^(1/2)

    such that:
    dy1 = y(A + dA, t) – y(A, t)
    dy2 = y(A, t + dt) – y(A,t)

    dy1 = dA e^(-gt)

    dy2 = Ae^(-g(t+dt)) – Ae^(-gt)
    = Ae^(-gt-gdt) – Ae^(-gt)
    dy2 = Ae^(-gt) [e^(-gdt) – 1]

    dy = e^(-gt) * {(dA)^2 + A^2 (e^(-gdt) – 1)^2}^(1/2)

    so:
    dy = e^(-gt) * {(dA)^2 + A^2 (e^(-2gdt) – 2e^(-gdt) + 1)}^(1/2)
     
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