The goal for today is to locate the center of mass of an aircraft (or similar object). (This is more commonly called the center of gravity, but center of mass is the more scientific term.)
After measuring the weight on each wheel, we need to analyze the data, as follows:
We start with the following definitions:
and since there are no other forces, we immediately see that
|W = Wf + Wm (2)|
In general, the defining property of the center of mass is a vector equation. However, in this simple situation, we will (a) exploit the left/right symmetry of the aircraft, (b) assume that the forces are all vertical, and (c) assume that we only care about the horizontal component of the location of the center of mass. Subject to those assumptions, the defining property of the center of mass can be simplified to the following:
|Xc W = Xf Wf + Xm Wm (3)|
Physics tells us that we can choose any reference point we like, and measure positions relative to that reference point. In this context, the reference point is called the datum.
To say the same thing in algebraic terms, we can multiply both sides of equation 2 by any constant D and then subtract that from equation 3. That gives us:
|(Xc−D) W = (Xf−D) Wf + (Xm−D) Wm (5)|
which is true for any datum D.
It is sometimes convenient to choose the main-gear location as the datum. This is not the conventional choice that engineers would make when designing an aircraft, but it is convenient in the maintenance shop when determining the center-of-mass location by weighing the aircraft.
Choosing D=Xm causes the last term in equation 5 to drop out, leaving us with:
As a cross-check on the work, you can easily verify that when the entire weight is on the front wheel, Wf = W and Xc = Xf. On the other side of the same coin, you can easily verify that when all of the weight is on the main wheels, Wf = 0 and Xc = Xm.
More generally, if you want to use some arbitrarily-chosen datum D, the fully explicit for the center of mass relative to D is:
as you can easily derive from equation 3 by using equation 2 to eliminate Wm.
The FAA has a pamphlet on this topic, reference 1.