Date: Mar 7, 2013 2:48 PM
Author: Red Star
Subject: Theory of errors: center of gravity
Let's suppose we have two point masses:
m1 = 4 kg, m2 = 3.5
Their associated standard deviation are:
dm1 = 0.5 kg, dm2 = 0.2 kg
Their distance from the origin of the reference frame are (1D problem):
x1 = 0.7 m, x2 = 0.8 m
Their associated standard deviation is:
dx1 = 0.01 m, dx2 = 0.01 m
I have to find the center of gravity (CoG) of this system and its uncertainty. The nominal value can be found with the well-known relation:
X = (m1*x1 + m2*x2)/(m1+m2) = 0.7467 m
In order to find the associated uncertainty, I used the relationship for non-linear combinations (cf. http://en.wikipedia.org/wiki/Propagation_of_uncertainty), evaluating the four partial derivatives of the previous relationship.
Hence dX = 0.0079 m
In my opinion is not an intuitive result: in fact the uncertainty of the center of gravity is less than the uncertainty on the position of each mass.
Are you able to justify this result?