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Scrap document in a Word template, apparently part of a draft on mechanics frames, with Phil's initials and the date 1.11.15. It defines quantities A and B from the gravitational pulls of m1 and m2, writes the force components Fy and Fz, and computes F squared. It sets up the vector rcog from mass m to the center of gravity point and notes the resulting expression is very complicated. Many equations were lost in extraction.
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This is the Title PhL 1.11.15
Note that page numbering is turned on in this template and view is 125%,
located in phil/roaming/microsoft/templates size about 219K.
= (Gm)[ h {m1/r31+ m2/r32} + rcosθ {m1/r31- m2/r32} ]
Define
A ≡ (Gm){m1/r31 + m2/r32}
B ≡ (Gm){m1/r31 - m2/r32}
Then
Fy = rsinθA
Fz = hA + rcosθB
To find the center of gravity point, we will need the magnitude of F
F2 = Fy2 + Fz2 = (rsinθA)2 + (hA + rcosθB)2
= (r2sin2θ + h2) A2 + r2cos2θ B2 + 2hrcosθ AB
F =
Then from *** we have
rcog = =
We know that
= =
and this vector points generally in the up direction in Fig (D.2.1). We want rcog to be a vector from mass m to the center of gravity, so then
rcog = rcog (-) = rcog (-) // vector from mass m to the cog point
and in theory we have all the pieces to compute rcog as a function of r,θ,h,m1,m2. As the reader can see, even for this very simple problem of a system of just two masses, the resulting expression is very complicated.