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., 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?