old earth problems section
DOCX · 19.5 KB
Open DOCX file
An old draft section of notes on mechanics in non-inertial frames, apparently by Phil, from a scrap-files folder. It starts from the effective force with centrifugal, Coriolis and Euler terms and applies it near the Earth's surface. It neglects the Euler term, defines g_local = g - w x (w x r), and estimates that g and g_local differ by up to about 1/3 of 1%, leaving only the Coriolis force for moving objects.
AI-written summary; may contain errors.
Extracted text (machine-read; may contain errors)
old earth problems section
(d) Earth problems
The arrangement of frames was shown in the Figure in Section 4 (g). Recall that
F'eff = F – mS – mω x (ω x r') – 2m ω x v' – m x r' (8.4a)
frame centrifugal Coriolis Euler
For problems involving motions of objects near the surface of the Earth, one has
F = mg + possible other forces (8.12)
where g = -g . Possible other forces might include air friction, wind, the action of magnetic fields on charged particles, etc. For the earth ~ 0.5 msec/day ~ 10-8 sec-2 for seasonal variations and much less for short term variations. Even for r' ~ 1 km, we can neglect the Euler term x r' ~ 10-5 ~ 10-6 g, so we are left with
F'eff = (mg + possible other forces) – mS – mω x (ω x r') – 2m ω x v' . (8.13)
Carrying out a static experiment (v' = 0) to measure the g vector at some location on the Earth (no other forces in this experiment), one finds
F'eff = mg – mS – mω x (ω x r') = m[g – S – ω x (ω x r')] = mglocal
where
glocal ≡ g – mS – ω x (ω x r'). (8.14a)
In this special case #1 problem we have from (7.12), neglecting the x b term,
S = ω x (ω x b) . (7.12)
Then since (6.1) r = r' + b, we can write glocal in the more intuitive form
glocal ≡ g – ω x (ω x r) . (8.14b)
Note that glocal is not proportional to . In practice, the locally flat Earth is perpendicular to glocal and that is the "g" one would be using in any local calculations. Then for an object in motion near the surface of the Earth, one has
F'eff = mglocal – 2m ω x v' + possible other forces
and so only the Coriolis force of our earlier discussion is of concern. No terms are neglected in this equation, there is no approximation other than = 0. The frame and centrifugal effects are both included in mglocal .
We can ask how much g and glocal differ. For the Earth ω = 7.27 x 10-5 sec-1, so if |r'| is say 1 km,
ω2r' = (7.27)2 10-10 103 ~ 5 x 10-6 m/sec2 ~ 10-6 g // ω x (ω x r') term very small
On the other hand, S = ω x (ω x b) ~ ω2b. Since b ≈ 6.37 x 106 m we have
ω2b = (7.27)2 10-10 6.37 x 106 = 344 x 10-4 ~ 3 x 10-2 m/sec2 ~ 3 x 10-3 g ≈ ω2r
The conclusion is that g and glocal differ by up to 1/3 of 1% with direction as in (8.14b).
In some problems this S ~ ω2b ~ 3 x 10-2 may in fact be negligible, justifying the list of approximate equations given below (8.6). But when using mglocal, the S term vanishes altogether from the problem along with the centripetal term.