Error analysis and math

Problem:

An experimentalist makes independent measurements of the length and height of a rectangular feature.  The values and their standard deviations are 10.62 ± 0.46 microns and 12.46 ± 0.52 microns.  Calculate the perimeter P and the area A of the feature including the standard deviation of each.

Solution:

Problem:

Propagation of Errors:  Snell's law relates the angle of refraction θ2 of a light ray travelling in a medium of index of refraction n2 to the angle of incidence θ1 of a ray travelling in a medium of index n1, as shown in the figure. 

image

From the knowledge that n1 = 1.000 and from independent experimental measurements of
θ1 = (22.03 ± 0.8)°, and θ2 = (10.45 ± 0.8)°, find
(a) n2 ;
(b) the percentage errors in sinθ1 and in sinθ2;
(c) the uncertainty in n2.

Solution:

Problem:

A force F is applied on a square plate of side L.  If the percentage error (standard deviation) in determination of L is 3% and that in F is 4%, what is the error in pressure?

Solution:


Problem:

Suppose C is the capture rate of dark matter in an astrophysical body.  Let CA be the dark matter annihilation rate per effective volume.  Then an approximate Boltzmann equation governing the number N of dark matter particles in the astrophysical body is
dN/dt = C – CAN2.
If N(0) = 0, find N(t).
Hint:  ∫du/(1 – u2) = tanh-1u

Solution: