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.
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.
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)