Inertia is a consequence of conservation of energy. Consider the
following argument:
-A particle of mass m_0 at rest has zero momentum and energy
E=m_0 c^2
-Quantum mechanically the wave function has frequency
\omega_0 = \frac{m_0 c^2}{\hbar}, and k=0.
-If you Lorentz transform to a moving frame with velocity beta,
the frequency becomes
\omega_1 = \omega_0 \gamma
and the wavenumber becomes
k = \omega_0 \gamma \beta
In other words, to promote an object at rest in one frame to
being at rest in a moving frame, you have to add energy. This
is the experienced as "inertia", the tendency for objects to
remain in constant motion unless disturbed. (By disturbed,
I mean energy is added or subtracted from the object.)

Rich L.

Richard Livingston il 29/10/2023 04:42:09 ha scritto:
> Inertia is a consequence of conservation of energy. Consider the
> following argument:
> -A particle of mass m_0 at rest has zero momentum and energy
> E=m_0 c^2
> -Quantum mechanically the wave function has frequency
> \omega_0 = \frac{m_0 c^2}{\hbar}, and k=0.
> -If you Lorentz transform to a moving frame with velocity beta,
> the frequency becomes
> \omega_1 = \omega_0 \gamma
> and the wavenumber becomes
> k = \omega_0 \gamma \beta
> In other words, to promote an object at rest in one frame to
> being at rest in a moving frame, you have to add energy. This
> is the experienced as "inertia", the tendency for objects to
> remain in constant motion unless disturbed. (By disturbed,
> I mean energy is added or subtracted from the object.)

All this concerns the inertia of a single mass that moves without
having to deal with the inertia of another mass with which it collides.

When in my animation
https://www.geogebra.org/m/mjnqb8vk
body m2 is suddenly hit by m1, does its inertia passively accept the
intrusion or does it rebel and opposes in the opposite direction?

Luigi Fortunati

[[Mod. note -- The problem with phrases like "passively accept" or
"rebel" or "oppose" is that it's hard to pin down their meanings.
For example, how should we operationally define "passive acceptance"?
Without a clear operational definition, it's hard to do a careful
analysis.

Newton's 2nd law is unambiguous: apply a (vector) net force F to
an object, and the object accelerates with a (vector) acceleration.
The acceleration vector is observed to be proportional to the net-force
vector, with a fixed proportionality constant (which we call the
"inertial mass", or just "mass" for short) for any given object.
Each of these phrases has a clear operational definition.
-- jt]]

Luigi Fortunati il 30/10/2023 06:07:04 ha scritto:
> All this concerns the inertia of a single mass that moves without having to deal with the inertia of another mass with which it collides.
>
> When in my animation
> https://www.geogebra.org/m/mjnqb8vk
> body m2 is suddenly hit by m1, does its inertia passively accept the intrusion or does it rebel and opposes in the opposite direction?
>
> [[Mod. note -- The problem with phrases like "passively accept" or
> "rebel" or "oppose" is that it's hard to pin down their meanings.
> For example, how should we operationally define "passive acceptance"?
> Without a clear operational definition, it's hard to do a careful
> analysis.
>
> Newton's 2nd law is unambiguous: apply a (vector) net force F to
> an object, and the object accelerates with a (vector) acceleration.
> The acceleration vector is observed to be proportional to the net-force
> vector, with a fixed proportionality constant (which we call the
> "inertial mass", or just "mass" for short) for any given object.
> Each of these phrases has a clear operational definition.
> -- jt]]

The operational definition according to Newton is exactly the same as what I wrote.

Newton says: <L The vis insita, or innate force of matter, is a power of
resisting, by which every body, as much as it lies, endeavors to
persevere in its present state, whether it be of rest, or of moving
uniformly forward in a right line. This force is proportional to the
body whose force it is; and differs nothing from the inactivity of the
mass, but in our manner of conceiving it. A body, from the inactivity of
matter, is not without difficulty put out of its state of rest or
motion. Upon which account, this vis insita, may, by a most significant
name, be called vis inertiae, or force of inactivity. But a body exerts
this force only, when another force, impressed upon it, endeavors to
change its condition>.

And I also wrote that this inertial force (in my animation) is exerted
*only* when the force of the body m1 tries to accelerate m2 or (which is
the same) when the force of the body m2 tries to slow down m1.

And then Newton continues thus: < And the exercise of this force may be
considered both as resistance and impulse; it is resistance, in so far
as the body, for maintaining its present state, withstands the force
impressed; it is impulse, in so far as the body, by not easily giving
way to the impressed force of another, endeavors, to change the state of
that another. Resistance is usually ascribed to bodies at rest, and
impulse to those in motion; but motion and rest, as commonly conceived,
are only relatively distinguished; nor are these bodies always truly at
rest, which commonly are taken to be so>.

And here too he says what I wrote and that is that (in my animation) the
inertial force of the body m1 is impulse and the inertial force of m2 is
resistance but he warns that the two things are interchangeable.

Luigi Fortunati.

On 10/30/23 4:34 PM, Luigi Fortunati wrote:
> Newton says: <L The vis insita, or innate force of matter, [...]

The translator used a PUN on the word "force".

In Newtonian mechanics, force is a vector while the "vis insita" is a
scalar, which today is called mass.

Tom Roberts

Tom Roberts il 01/11/2023 08:59:12 ha scritto:
>> Newton says: <L The vis insita, or innate force of matter, [...]
>
> The translator used a PUN on the word "force".
>
> In Newtonian mechanics, force is a vector while the "vis insita" is a scalar, which today is called mass.

The "vis insita" and the mass cannot be the same thing.

Newton says: "a body exerts this force (the vis intima) *only* when another force, impressed upon it, endeavors to change its condition".

The vis intima changes over time: it is only there when there is an external force (which it resists and opposes in the opposite direction) and it is not there at all when the external force is not there (because it does not have to resist anything )

Instead, mass is a quantity of matter that always exists and is always the same.

This is why, the "vis insita" and the mass are two different properties of bodies: the first is a non-force (when it does not act) and is a force (when it acts in opposition to an external force), the second is not, because it is never a force.

Luigi Fortunati.