Riedt's Fc=Fg, a=Fc/mpl and mpl=Fg/a

Dne 21/01/2018 v 08:20 Libor 'Poutnik' StÅ™Ã*ž napsal(a):
Dne 21/01/2018 v 03:08 Peter Riedt napsal(a):
Riedt's Fc=Fg, a=Fc/mpl and mpl=Fg/a

The Centripetal force discovered by Huygens in 1659 is Fc = mpl*v^2/r.

The Gravitational force, after Cavendish discovered in 1798
the gravitational constant G, is Fg=G*(M*m/r^2).

The second law of motion ascribed to Newton is F = ma, a=F/m, m=F/a.

The nine major solar planets connect the different formulas of Huygens,
Cavendish and Newton by Fc=Fg; a=Fc/mpl and mpl=Fg/a:

Only if you approximate the orbits by circles.

As only circular orbits have a single speed value,
and only for circular orbits is gravitation force centripetal,
i.e. perpendicular to motion.


Small related ASCII art illustration

( to be displayed with fixed width font, like Courier New or Consolas )

Legend
O the central mass ( bottom left )
o the body of interest ( upper right )
- and \ the trajectory of a body "o", should be an arc
bending from horizontal on the left to downwards on the right

, the straight line connecting the bodies.
.. The straight line of direction of centripetal force,
perpendicular to the o velocity.
alfa the angle between trajectory and O-o line

o,,,,G the size and direction of the gravitation force
o....C the size and direction of the centripetal force.

o..C::G the sections o..C, C::G and G,,,o form the right angle

triangle, with the right angle at C

Fg = G . M . m / R^2
Fc = m . v^2/ r


Fg the Gravitation constant
M the central mass ( like the Sun )
m the body mass
R the distance of the body to the central mass
v the body speed
r the curvature of the body trajectory

Note that for trajectories other then circular orbits, r R.

The ratio Fg/Fc is equal to the 1/sin(alfa).
You can see Fg/Fc = 1 only for alfa=90 deg,
i.e. only for circular orbits.



--------o
alfa , . \
, . \
, . \
G : . \
, ::::: . \
, :: C \
,
,
,
,
O

--
Poutnik ( The Pilgrim, Der Wanderer )

A wise man guards words he says,
as they say about him more,
than he says about the subject.

Dne 21/01/2018 v 09:17 Libor 'Poutnik' StÅ™Ã*ž napsal(a):

Fc = m . v^2/ r

r the curvature of the body trajectory

Correction,

the curvature *radius* of the body trajectory,

as a curvature is reciprocal to a curvature radius.


--
Poutnik ( The Pilgrim, Der Wanderer )

A wise man guards words he says,
as they say about him more,
than he says about the subject.

On Sunday, January 21, 2018 at 4:17:50 PM UTC+8, Libor 'Poutnik' StÅ™Ã*ž wrote:
Dne 21/01/2018 v 08:20 Libor 'Poutnik' StÅ™Ã*ž napsal(a):
Dne 21/01/2018 v 03:08 Peter Riedt napsal(a):
Riedt's Fc=Fg, a=Fc/mpl and mpl=Fg/a

The Centripetal force discovered by Huygens in 1659 is Fc = mpl*v^2/r.

The Gravitational force, after Cavendish discovered in 1798
the gravitational constant G, is Fg=G*(M*m/r^2).

The second law of motion ascribed to Newton is F = ma, a=F/m, m=F/a.

The nine major solar planets connect the different formulas of Huygens,
Cavendish and Newton by Fc=Fg; a=Fc/mpl and mpl=Fg/a:

Only if you approximate the orbits by circles.

As only circular orbits have a single speed value,
and only for circular orbits is gravitation force centripetal,
i.e. perpendicular to motion.


Small related ASCII art illustration

( to be displayed with fixed width font, like Courier New or Consolas )

Legend
O the central mass ( bottom left )
o the body of interest ( upper right )
- and \ the trajectory of a body "o", should be an arc
bending from horizontal on the left to downwards on the right

, the straight line connecting the bodies.
. The straight line of direction of centripetal force,
perpendicular to the o velocity.
alfa the angle between trajectory and O-o line

o,,,,G the size and direction of the gravitation force
o....C the size and direction of the centripetal force.

o..C::G the sections o..C, C::G and G,,,o form the right angle

triangle, with the right angle at C

Fg = G . M . m / R^2
Fc = m . v^2/ r


Fg the Gravitation constant
M the central mass ( like the Sun )
m the body mass
R the distance of the body to the central mass
v the body speed
r the curvature of the body trajectory

Note that for trajectories other then circular orbits, r R.

The ratio Fg/Fc is equal to the 1/sin(alfa).
You can see Fg/Fc = 1 only for alfa=90 deg,
i.e. only for circular orbits.



--------o
alfa , . \
, . \
, . \
G : . \
, ::::: . \
, :: C \
,
,
,
,
O

--
Poutnik ( The Pilgrim, Der Wanderer )

A wise man guards words he says,
as they say about him more,
than he says about the subject.

The signficance of me Fc comparing to Fg is that Huygens in 1659
was able to already calculate the gravitationl force without knowing G or M.. This was only possible after 1798 when Cavendish discovered G in the laboratory and the final Fg formuala came into being. I do not know by whom this formula was first used but would like to know.

Dne 22/01/2018 v 02:22 Peter Riedt napsal(a):


The signficance of me Fc comparing to Fg is that Huygens in 1659
was able to already calculate the gravitationl force without knowing G or M.

As approximation for circular orbits.

Otherwise he would need for given position
to calculate curvature radius
to calculate speed
to calculate the angle of the force wrt trajectory

This was only possible after 1798 when Cavendish discovered G in the
laboratory and the final Fg formuala came into being. I do not know by
whom this formula was first used but would like to know.

Standard gravitational parameter G.M was known even before that.
Even today it is known with much better relative accuracy
than G or M.

--
Poutnik ( The Pilgrim, Der Wanderer )

A wise man guards words he says,
as they say about him more,
than he says about the subject.

Dne 22/01/2018 v 08:03 Libor 'Poutnik' StÅ™Ã*ž napsal(a):
Dne 22/01/2018 v 02:22 Peter Riedt napsal(a):


The signficance of me Fc comparing to Fg is that Huygens in 1659
was able to already calculate the gravitationl force without knowing G or M.

[...]

This was only possible after 1798 when Cavendish discovered G in the
laboratory and the final Fg formuala came into being. I do not know by
whom this formula was first used but would like to know.

Standard gravitational parameter G.M was known even before that.
Even today it is known with much better relative accuracy
than G or M.

P.S.: As G.M is the proportional constant of the formula

a = ( G.M ) / r^2

--
Poutnik ( The Pilgrim, Der Wanderer )

A wise man guards words he says,
as they say about him more,
than he says about the subject.

Dne 22/01/2018 v 02:22 Peter Riedt napsal(a):


The signficance of me Fc comparing to Fg is that Huygens in 1659
was able to already calculate the gravitationl force without knowing G or M.

He did not need to need G nor M.
It was enough to know their product G.M.

For the centripetal formula and ellipses near to circle,
he can approximate an ellipse as a circle,
setting the radius as the mean distance to Sun,
speed as the mean speed to keep the period,
and than to calculate the centripetal force
as the mean gravitational force.

But it is all just an approximation.

Exact calculation for ellipses by centripetal formula
is possible indirectly, but is MUCH more complex
than a simple calculation from the inverse square law.

BTW, according to Huygens centripetal force formula,
if trajectory is straight, the force is zero,
what is not true.

--
Poutnik ( The Pilgrim, Der Wanderer )

A wise man guards words he says,
as they say about him more,
than he says about the subject.