Section 13_4 new
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A section of a document on mechanics in moving and rotating reference frames. It restates the Section 13.3 results with Frames S and S' swapped, listing relations for position, velocity, acceleration, angular momentum and torque. It then specializes to Special Case #4, where the origins coincide (b = 0), giving simplified equations with Coriolis, centrifugal and angular-acceleration terms.
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13.4 Summary of the Inverse Problem Equations (swap notation)
The results of this section are those of Section 13.3 but with S ↔ S' and v'↔v for all vectors except b and ω. Again, this is just a change of labeling. We repeat Fig (12.2.1) which shows the swap notation case :
(13.4.1)
r, v, a position, natural velocity and natural acceleration in Frame S (13.4.2)
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' (a)
v = v' – ω x r' – S (b)
v = v' – ω x r – S' (c)
a = a' – x r' – 2 ω x v' + ω x (ω x r') – S (d)
a = a' – x r – 2 ω x v' + ω x (ω x r) + 2 ω x S' – S' (e)
L(c) = L'(c') + m(r'-c') x [ -(ω x r') - S] (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' (g)
Special Case #4 (Inverse Problem, 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,
(13.4.3)
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 (13.4.4)
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' (b) = (c)
a = a' – x r' – 2 ω x v' + ω x (ω x r') (d) = (e)
L = L' – m r' x (ω x r') (f)
= ' + mr' x [- x r' - 2 ω x v' + ω x (ω x r')] (g)