Question

Error Propagation What is error propagation? A question in error propagation is that when we take a product of measurements we do what with the uncertainties? Should our uncertainties get bigger or smaller as they propagate through the formulas? Take a square and measure one side. What happens to the uncertainties when you calculate Area? Can this be beneficial when our product contains measurements of different units? The rule is to find the relative uncertainty in a product of measurements simply add the relative uncertainties. But why the 2 in front of the ? Also, notice I have suppressed the 100%. And the result is not the uncertainty in g, but the relative uncertainty in g. How do we find the uncertainty in g? Oh (comes from resolution of vernier caliper. 0.00005[m]) OL (comes from 2pt measurement with resolution uncertainty.) OD (comes from 2pt measurement with parallax and motion.) ot (comes from (cheap) Standard Deviation of a trial.) For the time uncertainty either use Standard Deviation from one trial's t1,t2,13; or use (t_max-t min)/2 as the uncertainty for one particular trial.

Answer 1

Error propagation is the effect of errors in a measured value on the values of a function depending on that value. For example, if for a function, f(x,y,z) there are errors in the measurement of x, y and z, given by then there will be an error in the function f(x,y,z) as well. This phenomenon is known as error propagation.

ܠ ܕܠ ,Tܠ

While doing a product of measurements these uncertainties are added.

For example,

Suppose, where, x, y and z has the uncertainties same as
ܠ ܕܠ ,Tܠ
respectively. Then, the uncertainty in is given by,

Tܠ * ܀ * v + ܐܠ * 3 * L +2ܠ ' T * = ܠ

As it is clear from the above equation, the uncertainties get bigger as they propagate through the formulas.

.

Let's suppose the length of one arm of a square is with uncertainty $\Delta a$ .

So, are is,

And the uncertainty of area is,

$\Delta S = 2*a*\Delta a$

Even if the product contains measurements of different units it won't be any different from the case where the product contains same units of measurement.

For example,

where, = distance, = velocity, = time, then the uncertainty of will be,

$\Delta x = v*\Delta t + t*\Delta v$

where, $\Delta t$ = uncertainty of time and $\Delta v$ = uncertainty of velocity.