new section 13.3
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Draft notes in Phil's 'new frames doc' folder, marked as already installed and not to be viewed again. It plans to rebuild the section from Section 12.1 using Swap Rules. It collects position, velocity and acceleration transformations between frames S and S', angular momentum and torque relations, and fictitious forces (centrifugal, Coriolis, Euler) and torques, with cross-references to earlier equation numbers. Text is partly garbled where symbols dropped.
AI-written summary; may contain errors. This description is approximate.
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New Section 13.3
This is installed, do not view again.
I should just start with Section 12.1 and edit it using the Swap Rules! I will make red after I edit each equation with the Swap Rules.
(12.1)
Definitions and Equations
r', v', a' position, natural velocity and natural acceleration in Frame S'
r, v, a position, natural velocity and natural acceleration in Frame S
ω angular velocity of Frame S relative to Frame S'
b vector directed from origin of Frame S' to origin of Frame S
r = b + r' (6.1)
v = v' + ω x r' + S (6.6a)
v = v' + ω x r + S' (6.6c)
a = a' + x r' + 2 ω x v' + ω x (ω x r') + S (7.6a)
S S' Euler Coriolis centripetal frame
a = a' + x r + 2 ω x v' + ω x (ω x r) + 2ω x S' + S' (7.6b)
L(c) = L'(c') + (r' - c + b ) (ω x r' + S) + (c'- c + b) x v' (11.2.14)
(c) = '(c') + (c'- c + b) x a' – S x [v' + ω x r' + S] + 'S' x v'
+ (r' - c + b) x [ x r' + 2 ω x v' + ω x (ω x r') + S] (11.2.15)
Fictitious Forces (Section 8)
For using fictitious forces we have (Frame S is inertial)
F = ma // true Newton's Law in inertial Frame S (8.1.2)
F'eff = ma' // fake Newton's Law in rotating Frame S' (8.1.4)
F'eff = F + F'fict . (8.1.7)
For the general case, the fictitious forces can be expressed as
F'fict = – mS – mω x (ω x r') – 2m ω x v' – m x r' . (8.1.8)
frame centrifugal Coriolis Euler
For Special Case # 1 problems (ω axis passes through Frame S origin), we have
F'fict = – mω x (ω x r) – 2m ω x v' – m x r . Special Case #1 (8.4.1)
centrifugal Coriolis Euler
Fictitious Torques (Section 11)
N'(c')fict = – (c'- c + b) x ma' + S x [mv' + mω x r' + mS] – 'S' x mv'
– (r' - c + b) x [ m x r' + 2mω x v' + mω x (ω x r') + mS] (11.3.8)
where the second line is just (r' - c + b) x F'fict from (8.6).