Mathematical definition of inertia

[lxman]

I realize that inertia is the tendency of an object to maintain its present state of motion. Now I am attempting to attach a mathematical definition to this concept. In searching, however, I am finding instances of it being defined variously as kg, kg/m^2, m/s^2, as well as other creative units. Some of the sites I've been to even claim that inertia is unitless.

So, in the world of physics, what is the mathematical definition of inertia?

If I have a 1 kg point mass proceeding in a straight line at 2 m/s, with no other forces acting upon it, what is its inertia?

Feel free to tell me that you think that I'm mixing apples and oranges and that this doesn't apply here, if that is the case. I would just like to find out, mathematically, can you derive a figure which defines how much inertia you have.

 

[jtbell]

If I have a 1 kg point mass proceeding in a straight line at 2 m/s, with no other forces acting upon it, what is its inertia?

1 kg.

Where have you seen the units of inertia defined as kg/m^2 or m/s^2?

There is a related concept called "moment of inertia" or "rotational inertia" which expresses an object's resistance to change in rotational motion. Its units are kg m^2 (the distance being measured from the axis of rotation).

 

[PhanthomJay]

If you are talking about inertia as an inherent property of the body to resist translation as caused by forces, then inertia corresponds to mass units (kg).

If you are talking about inertia as an inherent property of the body to resist rotation about an axis as caused by torques, then mass moment of inertia has units of kg*m^2.

It definitely does not have acceleartion units.

In your example, the inertial mass is 1 kg. The greater the inertial mass, the greater will be the resistance to its acceleration.

 

[OmCheeto]

I realize that inertia is the tendency of an object to maintain its present state of motion. Now I am attempting to attach a mathematical definition to this concept. In searching, however, I am finding instances of it being defined variously as kg, kg/m^2, m/s^2, as well as other creative units. Some of the sites I've been to even claim that inertia is unitless.

So, in the world of physics, what is the mathematical definition of inertia?

If I have a 1 kg point mass proceeding in a straight line at 2 m/s, with no other forces acting upon it, what is its inertia?

Feel free to tell me that you think that I'm mixing apples and oranges and that this doesn't apply here, if that is the case. I would just like to find out, mathematically, can you derive a figure which defines how much inertia you have.

bolding mine


Stand on a scale.

Sorry. Couldn't resist. It's been 25 years since I was a freshman in college so I had to look it up myself.

I see you left a message in the PF library, so you may have seen:

Inertia is the phenomenon that a force is required to cause change of velocity. The amount of inertial mass of an object is measured by measuring how much force it takes to accelerate it. The symbol for inertial mass is m.

But they would probably throw rocks at me if I said inertia was mass.

The Encyclopædia Britannica says the following:

There are two numerical measures of the inertia of a body: its mass, which governs its resistance to the action of a force, and its moment of inertia about a specified axis, which measures its resistance to the action of a torque about the same axis.

Still looks like they are pointing to a mass being equivalent to inertia.

My college text doesn't clarify it much in the linear sense. But they do give another clue in the section on Rotational Inertia where they actually give a mathematical equation:

I = $$\sum$$m_ir_i²

I being the rotational inertia, or moment of inertia.

They then progress to write the kinetic energy of a spinning body as:

K = 1/2 I omega²

to which they follow with:

This is analogous to the expression for the kinetic energy of translation of a body,
K = 1/2 M v².

Since angular speed omega is analogous to the linear speed v, You can see that the rotational inertia I is analogous to the mass, or the translational inertia M.

Ah ha! There. They said it in black and white. So I guess you could rearrange the kinetic energy equation to determine your inertia.

I = M = 2K/(v²)

which would yield what in basic si units? Joules sec²/meter²

But I wouldn't know what to do with a joule sec²/meter², so I'd just stick with kilograms.

 

[lxman]

Wow!

This is my second thread here, and I have to say that you folks do know your stuff. All of your answers are very well defined and adequate for my purposes. Thank you all very much.

Where have you seen the units of inertia defined as kg/m^2 or m/s^2?

It might be a fun exercise one of these days to do a google search for the units of inertia. Then start browsing the results. That's where I got all of those oddball definitions. Apparently there is a good deal of confusion on the subject.

Your answers have defined the term for me clearly, though. Thank you all again.

 

[sophiecentaur]

In my experience, there is a common tendency for people to confuse Inertia with Momentum and that is clearly wrong.
"inertia" is one of those colloquial terms that doesn't seem to have a formal definition. Afaik, there are are very few (if any) occasions in which the term Mass can't be used as a better alternative when the word Inertia is used. (Not sure why it should not be just equal to mass - but the suggestion is that you can't say that?)
The term Inertial Frame implies the possibility of motion (i.e. a constant velocity) but is not "inertia".

My use of the word "inertia" is limited to the state of bone idleness when you can't be arsed to do something. But not a Physics matter, which always gets my immediate attention.

 

[D H]

In my experience, there is a common tendency for people to confuse Inertia with Momentum and that is clearly wrong.

Is it clearly wrong? This dual meaning of inertia is rampant in early scientific use of the term (just past Newton's time). Physicists nowadays largely do not use the term "inertia" precisely because the term has two very different meanings and because each of those meanings has a clear, unambiguous alternative (mass and linear momentum).

Inertia is a fundamental concept in physics that describes an object's resistance to changes in its state of motion. It is one of the key principles described by Sir Isaac Newton's laws of motion. Inertia can be defined as follows:

"Inertia is the property of matter by which an object remains at rest or continues in uniform motion in a straight line unless acted upon by an external force."

This definition highlights several important aspects of inertia:

1. Resistance to Change: Inertia is the tendency of an object to resist any change in its motion, whether that means staying at rest or maintaining a constant velocity.
2. State of Motion: Inertia relates to an object's state of motion, which includes both its speed and direction. An object at rest has inertia that keeps it stationary, while an object in motion has inertia that maintains its current speed and direction.
3. External Force: Inertia can only be overcome or altered by the application of an external force. If no force is applied, an object will remain in its existing state of motion.
4. Mass Dependency: The inertia of an object is directly related to its mass. Objects with greater mass have greater inertia and are more resistant to changes in motion.

Newton's First Law of Motion, often referred to as the law of inertia, formally describes this concept. It states that an object will remain at rest or in uniform motion unless acted upon by a net external force. In practical terms, this law explains why objects tend to stay still when not pushed or why they keep moving at a constant speed unless a force (e.g., friction, gravity, or contact with another object) causes them to accelerate or change direction.

 

[sophiecentaur]

So, using it must be clearly wrong because of the certain risk of total confusion. Best steer clear of that sort of dual meaning. Modern Physics has made an advance, if only in that respect!

 

[Nabeshin]

It seems to me that to say inertia is to imply inertial mass. The reason, then, not to simply refer to mass is the ambiguity between inertial mass and gravitational mass. Although the two appear to be identical, I believe it is still an important distinction to make.

 

[TurtleMeister]

It seems to me that to say inertia is to imply inertial mass. The reason, then, not to simply refer to mass is the ambiguity between inertial mass and gravitational mass. Although the two appear to be identical, I believe it is still an important distinction to make.

Yes. I agree with that. However, I would say proportionally equivalent rather than identical.

 

[Cleonis]

"inertia" is one of those colloquial terms that doesn't seem to have a formal definition.

It seems to me that the word 'inertia' refers to the phenomenon in general, whereas 'mass' refers to a measurable quantity.

For an analogy: 'electrostatic attraction' and 'electric charge' are different concepts. The expression 'electrostatic attraction' refers to the phenomenon in general. The expression 'electric charge' refers to a property of individual objects.

Of course, in the case of electric interaction we have that there are neutral particles as well as charged particles.
But in our universe we have that anything that exists has inertial mass. Even entities without rest mass (such as photons) do have inertial mass, as energy has inertial mass. It seems there are no entities that are free of inertia.


Some people may argue that inertia is irreducible. That is, they may argue that no matter what discoveries are made in the future, we will never learn more about inertia than we already know now. That we should accept inertia as-is.

I don't know, maybe in the future it will be discovered that inertia arises from interaction with a pervasive inertia field. (And that would make passive gravitational mass a measure of how strongly an object couples to the field.)

Well, to keep options open I do not think of inertia and mass as one and the same thing.

 

[ravisastry]

inertia at sub atomic level

please pardon if this was answered earlier..i couldn't find myself.

Inertia should surely apply at sub atomic level as well.
Consider this scenario, a positively charged helium ion is accelerating. the electron cloud in this case should be somewhat oval or elliptical right ?
please let me know if my thinking is correct..

 

[sophiecentaur]

It might depend on what particular force is causing the ion to accelerate?

 

[bahamagreen]

I was just reading last night...

The Hamiltonian for the atom will have operators that only operate on the coordinates of the particular particles, but it also includes motion of the atom as a whole (the center of mass).

So the Schrodinger equation may be separated, and the eigenfunctions connected only through a common separation constant, which includes the kinetic energy of the center of mass of the atom.

So unless the atom is "in a box", the KE can assume any value, so the eigenfunctions are independent of each other, and the internal state of the atom is independent of the motion of its center of mass.

In the event that the separation constant, itself, is quantized; the factors of the eigenfunctions may be only conditionally independent.

Not claiming I understand it, but that's what I read.

 

[ravisastry]

I was just reading last night...

The Hamiltonian for the atom will have operators that only operate on the coordinates of the particular particles, but it also includes motion of the atom as a whole (the center of mass).

So the Schrodinger equation may be separated, and the eigenfunctions connected only through a common separation constant, which includes the kinetic energy of the center of mass of the atom.

So unless the atom is "in a box", the KE can assume any value, so the eigenfunctions are independent of each other, and the internal state of the atom is independent of the motion of its center of mass.

In the event that the separation constant, itself, is quantized; the factors of the eigenfunctions may be only conditionally independent.

Not claiming I understand it, but that's what I read.

someone pls explain this :)...does it concur with what i said earlier ?

 

[OmCheeto]

please pardon if this was answered earlier..i couldn't find myself.

Inertia should surely apply at sub atomic level as well.
Consider this scenario, a positively charged helium ion is accelerating. the electron cloud in this case should be somewhat oval or elliptical right ?
please let me know if my thinking is correct..

I see what you did there. You've introduced quantum physics into the question.

I was just reading last night...

The Hamiltonian for the atom will have operators that only operate on the coordinates of the particular particles, but it also includes motion of the atom as a whole (the center of mass).

So the Schrodinger equation may be separated, and the eigenfunctions connected only through a common separation constant, which includes the kinetic energy of the center of mass of the atom.

So unless the atom is "in a box", the KE can assume any value, so the eigenfunctions are independent of each other, and the internal state of the atom is independent of the motion of its center of mass.

In the event that the separation constant, itself, is quantized; the factors of the eigenfunctions may be only conditionally independent.

Not claiming I understand it, but that's what I read.

I read that 3 times, and I don't understand it either.

someone pls explain this :)...does it concur with what i said earlier ?

I can't explain it.

I read yesterday that photons can take on a rest mass:

In the equations, this effect shows up as a nonzero mass for photons. In short: inside superconductors, photons are heavy.

Ref
The Lightness of Being: Mass, Ether, and the Unification of Forces
Frank Wilczek (Author)
Nobel Prize in Physics in 2004
Herman Feshbach Professor of Physics at MIT

ps. I also post this, because I think photons are just plain weird, and get weirder by the decade.

googling again...

Does light have inertia?

Ah ha! I see someone has been slumming. :tongue:​

So what do fat photons have to do with a mathematical definition of inertia?
Well, this is my illustration of how things are "different" at sub atomic levels.

But let me google "quantum inertia", and see what I can find...

To this end, one can say that as any matter of interpretation at a very general level, the principles of equivalence are open to many opinions and discussions. They have been the beginning of modern physics, and probably they will ever frustrate us in one way or another.

CLASSICAL AND QUANTUM INERTIA: A MATTER OF PRINCIPLES
HARET C. ROSU
Instituto de F ́ısica, Universidad de Guanajuato, Apdo Postal E-143, Le ́on, Gto, Mexico​

hmmm... I have no working knowledge of Quantum Physics, but I interpret the above as; "The experts in the matter have many frustrating, opinionated discussions regarding this matter".

So what chance in hell do I have of contributing to the discussion? None!

-------------------------
Ok to delete. I've saved my thoughts on the matter.

 

[ravisastry]

the reason why i asked this question was, the answer to my question would have repercussions on the explanation for 2nd postulate of special theory of relativity...

 

[OmCheeto]

the reason why i asked this question was, the answer to my question would have repercussions on the explanation for 2nd postulate of special theory of relativity...

the reason why i posted what i did was because this is the general physics section of physics of physics forums and qp questions should probably be posted in the qp section.

 

[bahamagreen]

My reference was Klaus Ziock's book "Basic Quantum Mechanics", 1969 by John Wiley & Sons, Inc., Chapter 5 - The Hydrogen Atom, 5.1 - The Hamiltonian Of The Hydrogen Atom In The Center Of Mass System, p. 76: after going through the math...

"Physically, this means, of course, that the internal state of the atom is independent of the motion of its center of mass."

The separation constant in this case is derived from Coulomb's Law, which is valid for protons down to 10^-13cm and for electrons as far as can be measured - so the separation constant is taken to be not quantized.

Once you begin to ask about the behavior of electrons and protons you certainly are moving into the quantum level... and it looks to me like "inertia" as a classical concept is assigned differently when examining the quantum level...

 

[Dale]

the reason why i posted what i did was because this is the general physics section of physics of physics forums and qp questions should probably be posted in the qp section.

I agree.

ravisastry, please post your question as a new thread in the quantum mechanics sub-forum.