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Propagation of error 
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#1
Sep2307, 03:34 PM

P: 60

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  3. The attempt at a solution i have no idea how to do this. 


#2
Sep2307, 05:28 PM

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P: 8,953

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. 


#3
Sep2307, 06:30 PM

P: 60

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^(gtgdt) – 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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