Section 12_1 new
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Summary section from a longer document on moving reference frames, apparently written by Phil. It collects velocity, acceleration, angular momentum and torque relations between frames S and S' from earlier sections, along with fictitious forces and torques (centrifugal, Coriolis, Euler). It then simplifies them for Special Case 4, where the origins coincide (b = 0, r = r', c = c' = 0) and S is inertial.
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12.1 Summary of the Forward Problem equations (non-swap notation)
We now summarize the results of Sections 6, 7, 8 and 11. The first set of equations below is valid regardless of whether either of these frames is inertial (they could both be non-inertial). The second set of equations involving fictitious forces and torques assumes that Frame S is inertial (and Frame S' is not).
(12.1.1)
Definitions and Equations
r, v, a position, natural velocity and natural acceleration in Frame S (12.1.2)
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) (a)
v = v' + ω x r' + S (6.6a) (b)
v = v' + ω x r + S' (6.6c) (c)
a = a' + x r' + 2 ω x v' + ω x (ω x r') + S (7.6a) (d)
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) (e)
L(c) = L'(c') + m(r'-c') x [ (ω x r') + S] (11.2.14) (f)
(c) = '(c') + m(r'-c') x [ x r' + 2 ω x v' + ω x (ω x r') + S]
– m(' + ω x c' + S) x ( v' + ω x r' + S) + m ' x v' (11.2.15) (g)
Fictitious Forces and Torques (Section 8 and 11)
F = ma N(c) = (c) // true law
F'eff = ma' N'eff(c') = '(c') // fake law
F'eff = F + F'fict N'eff(c') = N(c) + N'fict(c')
F'fict = ma' - ma = ' - N'fict(c') = '(c') – (c) . (12.1.3)
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) (12.1.4)
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 (8.4.2) (12.1.5)
centrifugal Coriolis Euler
The fictitious torques may be written,
N'(c')fict = (r'-c') x F'fict + m(' + ω x c' + S) x ( v' + ω x r' + S) – m' x v' (11.3.8)
or
N'(c')fict = - (r'-c') x [ mS + mω x (ω x r') + 2m ω x v' + m x r'] (12.1.6)
+ m(' + ω x c' + S) x ( v' + ω x r' + S) – m' x v' (11.3.10)
Special Case #4 (Forward Problem, non-swap notation)
If the origins of Frame S and Frame S' coincide, then b = 0. In this case one of course has r = r' and also c = c' for the torque and angular momentum reference points. It is usual in this case to select c = c' = 0. We then write L ≡ L(0), L' ≡ L'(0), N ≡ N(0) and N' ≡ N'(0). The new drawing is this,
(12.1.7)
We have placed the common origin in the plane of paper and are viewing things from a direction which causes the instantaneous ω vector to point toward the viewer. Frame S' is rotating relative to Frame S at rate ω. The vector r in general is not in the plane of paper. Here are the simplified equations obtained from those above with r = r', b = S = S' = S = c = c' = 0 :
r, v, a position, natural velocity and natural acceleration in Frame S (12.1.8)
r, v', a' position, natural velocity and natural acceleration in Frame S'
ω angular velocity of Frame S' relative to Frame S
r = r' (a)
v = v' + ω x r' ALL SPECIAL CASE #4 (b) = (c)
a = a' + x r' + 2 ω x v' + ω x (ω x r') (d) = (e)
S S' Euler Coriolis centripetal
L = L' + mr' x (ω x r') (f)
= ' + mr' x [ x r' + 2 ω x v' + ω x (ω x r') ] (g)
For the following items, Frame S is inertial and Frame S' is rotating
F = ma N = // true law
F'eff = ma' N'eff = ' // fake law
F'eff = F + F'fict N'eff = N + N'fict
F'fict = ma' - ma = ' - N'fict = ' – (12.1.9)
F'fict = – mω x (ω x r') – 2m ω x v' – m x r' (12.1.10)
centrifugal Coriolis Euler
N'fict = r' x F'fict = – r' x [ mω x (ω x r') + 2m ω x v' + m x r'] (12.1.11)