SEction 9 rewrite
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A rewrite of Section 9 from Phil's document on non-inertial frames, marked "installed, do not edit here" as an obsolete draft. It restates his rotating-frame equations (position, velocity, force and effective force with centrifugal, Coriolis and Euler terms) in a "swap" notation, then matches them to equations in Marion and Thornton & Marion. It ends with a translation table of symbols between the two notations.
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installed, do not edit here
9. Notation comparison with Marion (1970) and Thornton & Marion (2003)
In this we Section compare our notation for rotating-frame kinematics and non-inertial-frame physics to that of Marion (1970) and Thornton and Marion (T&M 2003).
Their notation is close to our "swap" notation, so before making any comparisons, we restate various of our equations in swap notation. A swap notation equation has an S subscript on the equation number and is obtained from the corresponding non-swap equation by prime ↔ noprime ( b and ω do not change) :
r' = b + r (6.1)s
v' = v + ω x r + S' (6.6a)s
F' = ma' = mS' + ma + mω x (ω x r) + 2m ω x v + m x r (8.1.3)s
ma = Feff = F' - mS' – mω x (ω x r) – 2m ω x v – m x r (8.1.5)s
In the above swap notation equations, Frame S' is fixed (f) and Frame S is the rotating frame (r) :
We now compare these equations with those of the Marion series authors:
r' = b + r (6.1)s
r' = R + r // Marion p 341 (11.1)
// T&M p 388 (10.1)
v' = v + ω x r + S' (6.6a)s
vf = vr + ω x r + V // Marion p 344 (11.12)
// T&M p 391 (10.17)
F' = ma' = mS' + ma + mω x (ω x r) + 2m ω x v + m x r (8.1.3)s
F = maf = mf + mar + mω x (ω x r) + 2m ω x vr + m x r // Marion p 344 (11.17)
// T&M p 392 (10.23)
ma = Feff = F' - mS' – mω x (ω x r) – 2m ω x v – m x r (8.1.5)s
mar = Feff = F - mf – mω x (ω x r) – 2m ω x vr – m x r // Marion p 344 (11.19)
// T&M p 392 (10.25)
Based on these comparisons, we make the following translation table:
our swap Marion
notation authors
S r name of the rotating frame (r = rotating)
S' f name of the fixed frame (f = fixed)
∂S (d/dt)rotating time derivative in the rotating frame
∂S' (d/dt)fixed time derivative in the fixed frame
r r position in rotating frame
v vr velocity in rotating frame
a ar acceleration in rotating frame
r' r' position in fixed frame
v' vf velocity in fixed frame
a' af acceleration in fixed frame
F' F force in fixed frame = true force in rotating frame
Feff Feff total effective force in the rotating frame
Ffict total fictitious force in the rotating frame
b R location of the rotating frame origin (measured in the fixed frame)
S' f,V velocity of the rotating frame origin (measured in the fixed frame)
S' f acceleration of the rotating frame origin (measured in the fixed frame)