Calculating Uncertainty of g in Propagation of Error

[smithg86]

Homework Statement

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?

Homework Equations

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The Attempt at a Solution

i have no idea how to do this.

 

[mgb_phys]

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.

 

[smithg86]

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)