[McQueen]

As I understand it, at the time, astronomers thought there was just one, unchanging galaxy.

Astronomers still believed that well into the sixties.  The fact that matter; settles into rotating masses like galaxies, solar system and stars, was essential to the shape the Universe eventually took. It also seems to support the view I put forward that the Big Crunch when/if it does come would take place on a time scale practically incomprehensible to us.

Interestingly: If a steel ball with a mass of 50 kg is attracted towards the earth and away from the earth with equal force, the net force exerted on it would be zero and the same would hold good with the feather that weighs just 2 gms. So what is left?  One thing that could be considered is comparative masses.  Since the mass of the earth is significantly larger than either of the objects, what is left is, the acceleration due to the mass of the earth = 9.8 m/s2  so that both objects fall to earth under the same acceleration. 

This would mean that although both objects fall to the earth with an acceleration denoting  they had mass, it is not their mass that is the motivating force but the mass of the earth that exerts a force. This might also explain why the space station, which could be considered to be an object in free fall around the earth, does not experience gravity; or rather does not experience gravity as calculated by its distance from earth.

[Colin2B]

According to Newton, gravity acts according to the density of the object. What this means is that the force of gravity exerted on an object is determined solely by the density of the object and not by any other property.

I think you will find that Newton said “gravity acts according to the mass of the object”
This is also the case for inertial acceleration.

[jeffreyH]

This would mean that although both objects fall to the earth with an acceleration denoting  they had mass, it is not their mass that is the motivating force but the mass of the earth that exerts a force.

The masses of all the objects are important. They all exert a gravitational attraction on each other. They is why the n-body problem is so difficult. Do you actually pause to think before you press the keys or does it all just come flowing out in a rush?

[Halc]

A gravitating source is mainly made of protons, neutrons and electrons.

True at small scales such as a solar system. At galactic scales, the gravity source is composed primarily of things other than these protons and neutrons.

There will be neutrinos flying around but let's ignore them.

Neutrinos move too fast to contribute to a gravity source.  They pretty much all move at far greater speeds than the escape velocity of the system which creates it, so they don't contribute significantly to its gravity, and yes, can be essentially ignored.

Your point in the post holds true despite the actual composition of the mass in a gravity source.

if all the stars are drawn to each other by gravitation, they should collapse into a single point

As I understand it, at the time, astronomers thought there was just one, unchanging galaxy.

Apparently non-rotating as well, or it simply isn't true, as evidenced by the Earth not falling into the sun and other orbital stabilities that you point out.
Still, the Bentley thing doesn't work even for non-rotating systems.  Take 3 point masses, initially at rest in a 3-4-5 triangle configuration.  They will not meet at nor are even attracted to their mutual barycenter, and will in fact eject one of the three from the system leaving the other two in a stable orbit.  Small targets do not easily fall onto one another.

Is the rotation of the galaxy the answer to Bentley's paradox? (Wikipedia doesn't say...)

The universe as a whole doesn't rotate, but the Bentley paradox assumes a bounded collection of mass (the non-rotating galaxy at the time), not the unbounded universe as a whole which has no center to which all the matter might be attracted more than another.
It kind of applies to galactic superclusters which have very little angular momentum to mass ratio, and thus are very much collapsing to a point just like Bentley suggests. Yes, it would be a paradox in a static universe, but it's not a paradox since it is not a stable (static) system and the gathering to a point is happening. We're all being sucked into the Virgo cluster which is itself being pulled into the Great Attractor, and that into the Shapley attractor which is where it all ends.  All the other gravity wells are far away enough that expansion will never allow them to combine with the Shapley attractor.  Everything will indeed get sucked into something. It just takes a lot of time.
This process is not the same as my 3-4-5 example because the objects involved are not point masses and thus they lose kinetic energy to heat as they interact physically.  The motions of the galaxies around these gravity wells is not an orbital relationship despite having angular momentum in relation to any of them.

[alancalverd]

According to Newton, gravity acts according to the density of the object. What this means is that the force of gravity exerted on an object is determined solely by the density of the object and not by any other property.

I think you will find that Newton said “gravity acts according to the mass of the object”
This is also the case for inertial acceleration.

IIRC Newton actually considered density to be the primal factor, mass being the product of density and volume. This is rigorous as density is the general property of a material (iron) and mass is merely the property of a particular lump of said material (cannonball).

[alancalverd]

The accelerating expansion of the universe also shows that the universe is being prevented from collapsing back to the same state as before the big bang.

Not according to Newton's first principle!

A body continues in a state of rest or uniform motion unless a force acts on it.

If the lumps of matter are being dispersed from a point origin, the distance between them increases with time so the gravitational force between them decreases. Nothing is needed to prevent collapse because nothing is inhibiting expansion.

[jeffreyH]

The accelerating expansion of the universe also shows that the universe is being prevented from collapsing back to the same state as before the big bang.

Not according to Newton's first principle!

A body continues in a state of rest or uniform motion unless a force acts on it.

If the lumps of matter are being dispersed from a point origin, the distance between them increases with time so the gravitational force between them decreases. Nothing is needed to prevent collapse because nothing is inhibiting expansion.

The acceleration IS the thing preventing collapse. The gravitational force decreasing does not lead to an accelerating expansion. It would tend toward inertial motion over time. While this would mean the universe would still expand it could also mean an ultimate reversal into a collapsing state. Hence the prevention.

[Colin2B]

IIRC Newton actually considered density to be the primal factor, ........... mass is merely the property of a particular lump of said material (cannonball).

Interesting, I missed that.
Yet at the end of the day the volume doesn’t really play a part if gravity acts as if from a point source, unless the test mass is close to a large surface in which case the volume (distribution of mass) has to be considered.

[alancalverd]

The acceleration

What acceleration? Once the primordial body has exploded, there is no further force acting on the shrapnel. That's the problem with CGI  in space fiction movies: exploding spacecraft don't boil and billow like cumulus clouds, they just expand radially and very quickly.

[Eternal Student]

So I guess you all prefer Chemistry and Biology?
It's not possible to do much with this thread apart from have some fun.

Have you considered this:
    Newton used the idea of density  instead of  mass  in his ideas about gravity because there was some confusion about the fact that a hot-air ballon seems to fall upwards in earths gravitational field?   Maybe he was just searching for a negative gravitational charge that could be repelled instead of attracted. 

I just thought I might as well throw that idea in here.  It's fun.  It's most certainly NOT serious.

[Colin2B]

If they  \(gravitational and inertial mass) are really proportional with constant ratio, the only consequence would be the change of gravitational constant.

Yes.  Well done.   The Gravitational constant AND/OR something else just gets multiplied by a constant.   ................  We can choose constants and/or units to keep the formulae simple.

Aye, there’s the rub.
We can certainly use different formulae and constants, but what of Hamdani’s original question “What's expected to happen if gravitational mass were twice of inertial mass? What if it's only a half?”

Certainly projectile trajectories would be different. A ball thrown from the top of a building would travel a different distance compared to the distance fallen, this implies that orbits would also have a different radius. Changes to Kepler’s laws?

The rate at which a pendulum will swing should change with a new ratio of inertial to passive grav mass.

There would be a different equatorial bulge due to difference between centrifugal and gravitational force.

A plumb bob at a certain latitude would have a slightly different vertical offset, again due to difference between centrifugal and gravitational force.

A mechanical governor would expand to a different circumference, again due to difference between centrifugal and gravitational force.

There must be a lot more, but if that ratio had always been there we wouldn’t think it odd and might be speculating what it would be like if they were equal!

What would we do with the relationship 
.
E ≈ m₀c²+˝m₀v²

Is the energy dependant on whether mass is passive grav or inertial? in the second term it would change if motive was inertial or gravitational. Would we have m₀ as the sum of inertial & gravitational?

Strangely, I always expected that the fact they are the same might be explained by some energy conservation law or, like Pound-Rebka shift, that without it you could build a perpetual motion machine.

Have you considered this:
    Newton used the idea of density  instead of  mass  in his ideas about gravity because there was some confusion about the fact that a hot-air ballon seems to fall upwards in earths gravitational field?   Maybe he was just searching for a negative gravitational charge that could be repelled instead of attracted. 

I just thought I might as well throw that idea in here.  It's fun.  It's most certainly NOT serious.

Well, it might be serious if he was originally trying to include hot air balloons, but then realised this was a flotation thing!

[alancalverd]

Consider the simplest case of a body falling under gravity.

We say that F = mg = ma as if m and g were god-given fundamental constants, but they aren't. If m_g ≠ m_i we would just measure a different value for g.

Fact is that there is no fundamental unit of mass. Until recently it was defined as the ratio of the gravitational force on an object to that of an arbitrary lump of platinum, and its modern redefinition, whilst giving us a more reproducible measurement, still harks back to the arbitrary number we associated with the ISK.

And "big G" doesn't throw any more light on the subject. F = Gm_gM_g/r² would just give us a different value for G because all our measurements of F depend on F = m_ia.

I'm leaning towards the notion that m_g might not necessarily equal m_i, but that the ratio is both fixed and immeasurable.  Can anyone think of an experiment that actually measures the ratio without presuming the answer?   

[Eternal Student]

I'm leaning towards the notion that mg might not necessarily equal mi, but that the ratio is both fixed and immeasurable.  Can anyone think of an experiment that actually measures the ratio without presuming the answer?   

  The experiments and various successors comes to mind.   

I don't think the exact numerical value for the ratio of    is all that important,  just that it is a fixed constant ratio.  Setting the ratio to 1 is more a matter of choosing the right units.

Physicists set  c =1  ;   h = 1   and even  2 = 1   if they want to.  Their only concern is the general behaviour of a system.   The units don't change the fundamental behaviours.  Are you suggesting that we should care if the calculation is actually right?  This isn't the technology section, you aren't talking to Engineers here.

[alancalverd]

Are you suggesting that we should care if the calculation is actually right? 

As an aviator and an engineer, I always care if the calcs are correct. As do my customers and investors.

Anyway, I looked up Eotvos' experiment on Wikipedia and right at the end of the introduction it says

It is sufficient for the inertial mass to be proportional to the gravitational mass. Any multiplicative constant will be absorbed in the definition of the unit of force

which is exactly what this stupid engineer said earlier.

[jeffreyH]

Lots of interesting stuff in here.
https://www.britannica.com/science/gravity-physics/Gravitational-fields-and-the-theory-of-general-relativity#ref210883

[Eternal Student]

I always care if the calcs are correct

    I've read it a few times and said it to myself but I just don't get it....You seem to be using that phrase as if it's a good thing.     

[alancalverd]

If you want to survive your flight or make a profit from your project, it is very important that the prior calculations be correct, and based as far as possible on known physics.

[Colin2B]

I'm leaning towards the notion that m_g might not necessarily equal m_i, but that the ratio is both fixed and immeasurable.  Can anyone think of an experiment that actually measures the ratio without presuming the answer?

I suppose that depends on how we measure inertial mass.
The mass we usually measure is grav mass using say the force on a spring scale. If we use the same force to accelerate the mass horizontally we should get the same value of mass.
If we use a falling weight/pulley to accelerate a mass on a dynamics trolley then, eliminating friction, and weighing the trolley (gravitational mass) we can calculate the inertial mass from the force applied and the motion of the trolley. The 2 mass values should be the same.
Have I missed something here?

[markjhon1223]

if gravitational mass  different from inertial mass it would be a great disaster here

[Colin2B]

if gravitational mass  different from inertial mass it would be a great disaster here
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Let me guess what you are selling - or trying to