Ross Finlayson wrote:
When you think about Newton's laws, or when I do,
or when one does, or as one may, think about Newton's
laws, as we think about Newton's laws, one might make
for a "deconstructive account" of them.
rest/rest motion/motion equal/opposite
<- isn't that three laws?
Once again your writing is so confused that one cannot discern what you mean.
F = ma
This is a formulation of Newton's Second Law of Motion.
Originally (in 1686/1687) Newton wrote this law in Latin words that as an equation he would have later written (or has written?)
F = p|c,
(F = dot p, where by a dot above a symbol he defined the derivative of the corresponding quantity with respect to time, a notation that we are still using today), where he had previously defined the equivalent of
p = m v.
He called this, what we call "momentum" today, "the quantity of motion", and defined that it "arises from Velocity [v] and the quantity of Matter [mass
m] together" ("Philosophi|a Naturalis Principia Mathematica": "Definitiones").
[The notion of vectors did not exist yet -- they were introduced only in
the 19th century by Hamilton --, so Newton would not have written this
as a vector equation.]
<https://cudl.lib.cam.ac.uk/view/PR-ADV-B-00039-00001/25>
(Cambridge University, Cambridge Digital Library. High resolution digitised version of Newton's own copy of the first edition, interleaved with blank pages for his annotations and corrections. Cited in: <https://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica#Latin_versions>)
One can see that this is equivalent to
F = m a,
considering that the mass is assumed to be constant-|, and that acceleration is the derivative of velocity with respect to time:
F = p|c = m v|c = m a.
I have heard that the law was first formulated as "F = m a" by Leonhard
Euler ca. 100 years later, but so far I have not found corroborating evidence.
___
-| This assumption must be challenged in special cases. For example, the
mass of a rocket decreases due to pushing out fuel for propulsion. As
a result, the *rocket equation* features a time derivative of mass that
arises from the product rule of derivation: F = p|c = m v|c + m|c v = m a + m|c v.
<https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation#Most_popular_derivation>
<- that's also F(t) = m d^2 x /dt^2, differential d
Correct.
F = g Mm / d^2
That is a *wrong* notation of Newton's Law of Universal Gravitation.
AISB, one must use "G" for the gravitational *constant* here; "g" is the symbol for the gravitational *acceleration* instead, whose magnitude is given by
g(d) = G M/d^2,
which is therefore NOT constant in general.
However, this form of the magnitude of the gravitational acceleration only holds for spherically-symmetric objects with mass M when d means the
distance from the center of mass, i. e. the _radial_ distance.
In general, the *field of* gravitational acceleration grau (the gravitational field) is given by Gauss' Law for gravitation:
rec ria grau = -4-C G -U,
where -U is the mass density field.
AISB (multiple times) for a *uniform* *spherically-symmetric* mass density, using Gauss' Theorem, one finds Newton's Law of Universal Gravitation:
re#_V dV (rec ria grau ) = re>_A dArau ria grau
= -4-C r-# g(r) = -4-C G re#_V dV -U = -4-C G M
<==> g(r) = G M/r-#
===> grau(rrau) = -(G M/r-#) rrau/r
<==> Frau_g = m grau(rrau) = -(G M m/r-#) rrau/r,
where V is a spherical volume that fully contains the region where the mass density is non-zero (i.e. the entire, assumed spherically-symmetric, celestial object where we would like to determine the gravitational field).
<- F(t) = g Mm / d(t)^2 proportional inverse-square distance d
Force is a function of time.
In general, yes; but not necessarily.
Because the law in this form only applies to spherically-symmetric mass distributions, one usually writes "r" (for "radius") instead of "d".
[Another reason is to avoid confusion with the operator for the total derivative which also is "d".]
Newton's Law of Gravitation gives rise to the differential equation
r|e reo (d-#/dt-#) r = G M/r-#
from
F = m a = m (d-#/dt-#) r = G M m/r-#.
Solving this differential equation, one can calculate, for example, the time that it takes for an object to fall in vacuum when the fact that the gravitational acceleration varies with altitude (to begin with) cannot be neglected. This is important, for example, to land space probes safely on comets and asteroids (which has been done).
There is at best a tenuous relation of this to special and general relativity, so F'up2 sci.physics instead.
Ross Finlayson wrote:
F = ma
This is a formulation of Newton's Second Law of Motion.
Originally (in 1686/1687) Newton wrote this law in Latin words that as an equation he would have later written (or has written?)
F = p|c,
(F = dot p, where by a dot above a symbol he defined the derivative of the corresponding quantity with respect to time, a notation that we are still using today), where he had previously defined the equivalent of
p = m v.
He called this [p], what we call "momentum" today, "the quantity of motion", and
defined that it "arises from Velocity [v] and the quantity of Matter [mass
m] together" ("Philosophi|a Naturalis Principia Mathematica": "Definitiones").
[...]
<https://cudl.lib.cam.ac.uk/view/PR-ADV-B-00039-00001/25>
(Cambridge University, Cambridge Digital Library. High resolution digitised version of Newton's own copy of the first edition, interleaved with blank pages for his annotations and corrections. Cited in: <https://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica#Latin_versions>)
One can see that this is equivalent to
F = m a,
considering that the mass is assumed to be constant-|, and that acceleration is the derivative of velocity with respect to time:
F = p|c = m v|c = m a.
Then, lesser-known is the priority dispute between Newton and Hooke over universal gravitation,
or Newton and Kepler over universal gravitation,
then as about the definition of F as mass by acceleration, whether
it's really derivate in definition of power.
Thanks for clarifying G vis-a-vis g, yes, that's what's intended.
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