# Difference between inertia and momentum?

1. Jul 8, 2011

### lokifenrir96

Hi, what exactly is the difference between inertia and momentum?

I understand that momentum is F = mv, and is related to F = ma in that you can convert it into Ft = mv. So the momentum will depend on the mass of the object, velocity, and amount of time you apply a certain force...

I'm not very sure about inertia though. How is it calculated and what is its relation to momentum?

Thank you :)

2. Jul 8, 2011

### Crosstalk

I'll assume you're talking solely about linear motion (i.e. not rotational).*

In my understanding, "inertia" is really the property of mass of resisting motion. It is really a result of interpretations of F = ma -- if you don't apply force, the velocity doesn't change (acceleration = 0). If you do apply force, there is a resistance (velocity change isn't instantaneous).

Inertia is not a value that is calculated -- its quantitative equivalent is mass (through F = ma). Do you understand what I'm saying?

Momentum is very much like energy in that it represents how much much, well, momentum a mass has. The difference is that energy--kinetic energy being proportional to an object's velocity--is more useful when various heights are involved (energy increases linearly with respect to height (assuming a uniform gravitational field)), while momentum is a more direct representation (it is proportional to velocity, not velocity squared) that is more useful in other circumstances (most of my knowledge comes from AP Physics, where we used energy far more extensively (we did use momentum _and_ energy together in some collisions calculations, though)).

The equation F = mv is not correct, although Ft = mv is.

* I have to point out that there is a sort of measurement of inertia when it comes to rotational physics. There is the "moment of inertia," which represents a tendency to resist change in motion (about a certain axis of rotation), and is essentially a direct replacement for mass.

I hope this helps clear things up :)

3. Jul 9, 2011

### Philip Wood

The more inertia a body has the less its acceleration for a given applied force. So mass, as it appears in the equation a = F/m, measures a body's inertia. Note that a body's mass doesn't depend on how fast the body is moving. [In Newtonian physics, mass has another quite different role as well; governing the force a body experiences in a given gravitational field.]

Momentum, mv, is a body's mass multiplied by its velocity. Because velocity is a vector, so is momentum. Everyone knows that the more momentum a body has, the harder it is to bring it to rest. The m factor reduces the acceleration for a given force, as we've already said, so we have to apply the force for longer, and of course the larger the body's initial speed the longer we have to apply a given force to reduce the speed to zero.

4. Jul 9, 2011

### lokifenrir96

Oh so momentum is kind of like inertia in motion while factoring in the velocity?

5. Jul 9, 2011

### Philip Wood

That might be a starting point for understanding momentum, but to understand it properly you need to know about the Law of conservation of momentum, the statement of Newton's second law in terms of momentum and so on.

6. Jul 10, 2011

### JDStupi

The way I tend to think about it is as others said, namely that inertia is essentially the necessary concept for Newtonian Dynamics. It postulates a symmetry between absolute rest and uniform linear motion (1st law) and then introduces the dynamical quantity, mass, as a quantitative measure of a body's resistance to change in its state of motion.

Momentum can be thought of as "quantity-of-motion" to a certain extent, namely being the product of a body's inertial mass and it's velocity.

Now, this brings us to discussions about what motion is, namely momentum may have been originally have been conceived of as the absolute measure of the property of a body: its state of motion. However, the conjunction of Newtons first two laws leads to the result that a body's velocity isn't absolute and is measured wrt some reference frame and so our conception of motion has changed. Now instead of speaking in terms of subject-predicate logic "This body is in a state of motion" we say that to truly say something is in motion we need a two valued predicate namely that "body x is in motion with respect to inertial frame y and p=mv is a quantitative measure of its causal effects on other bodies under certain conditions."

I don't claim that any of this is "The right answer" though, the two concepts are complex and inter-related and I would be silly if I thought I could completely characterize the two.

7. Jul 10, 2011

### Drakkith

Staff Emeritus
Both momentum and inertia are a directly related to an objects mass (Except in the case of photons, which have 0 rest mass but still have momentum).

The more mass an object has the more energy it takes to accelerate it to a given speed, or to decelerate it to a given speed. Inertia is used to define how much energy is necessary to do this. Momentum is used when talking about the energy involved in a collision or similar event. I like to think of it as this: Inertia determines how much energy is needed to get an object up to speed, and Momentum is how much energy a moving object posesses.

8. Jul 11, 2011

### Philip Wood

Drakkith's last sentence is, for me, a step too far. Energy and momentum are quite different animals, with different dimensions: [ML2T-2] and [MLT-1]. Momentum is a vector and energy a scalar. They feature non-interchangeably in different laws of nature...

9. Jul 11, 2011

### Drakkith

Staff Emeritus
Don't read too much into it. I'm not saying that momentum = energy.

10. Jul 11, 2011

### Studiot

I think that when we are talking about 'energy' in this context we should realise we mean solely kinetic energy.

However there are difficulties comparing energy, inertia and momentum.

I like the notion that inertia is that quality of a particle which represents its resistance to change of action by outside influence.

I have been careful to avoid the usual formulae for the following reasons

An electron beam in a cathode ray tube has very low inertia, but high energy. We employ the low inertia characteristic to deflect the path to our desire and the high energy to cause light emission by the receiving phosphor.

At absolute zero temperature particles possess zero kinetic energy and zero momentum, but they still possess inertia.

go well

11. Jul 11, 2011

### Drakkith

Staff Emeritus
Exactly Studiot. Momentum is a product of velocity x mass, while inertia is effectively another property of an objects mass. The two are very different in my opinion.

12. Jul 11, 2011

### jambaugh

Inertia is the relationship between rate of motion and momentum.
momentum is the relationship between rate of motion and kinetic energy.

I find it helps to look at other examples to see where things get carried through or distinguished.

An object moving linearly has a mass $m$ (its linear inertia) and a momentum $p=mv$ (the quantity changed and exchanged by forces) and a kinetic energy $T_{lin} = \frac{1}{2} v p = \frac{1}{2} mv^2$.

A rotating object has a moment of inertia $I$ (its rotational inertia) and an angular momentum $L = I\omega$ (the quantity changed and exchanged by torques) and a kinetic energy: $T_{rot} = \frac{1}{2}\omega L = \frac{1}{2}I\omega^2$.

This can be generalized to more complex motions. For more details look up "Lagrangian dynamics" and "Hamiltonian dynamics".

13. Jul 11, 2011

### Studiot

An object has a moment of inertia whether it is rotating or not.

What exactly does that mean?

******************************

A further application of the resistance to change notion is to observe that a system may possess inertia.
So, for instance , we observe the propagation of a wave through different media and notice that the wave propagates at different speeds through different media.
We attribute this to the differernt inertias of the different media. In this case there may be more than mass involved. There may also be intramedia forces.

14. Jul 11, 2011

### jambaugh

Note that we can express a rotation in 3 dimensions with a vector $\vec{\theta}$ with magnitude the angle of rotation and direction the axis of rotation. Its time derivative is the vector angular velocity $\vec{\omega}$. An object's moment of inertia maps vector angular velocity to vector angular momentum $\vec{L}$ and in the most general setting is a matrix $\mathbf{I}$ (called the moment of inertia tensor). Indeed only if the object has a very regular shape such as a sphere or cube is the moment of inertia in all directions the same and thus can be treated as a scalar. What this means is that the angular momentum vector will not always be in the same direction as the axis of rotation. This fact is what causes precession of a freely rotating object.

Kinetic energy then is given in terms of the matrix product:
$$T_{rot} = \vec{\omega}\cdot \mathbf{I}\cdot\vec{\omega}$$
We can pick a set of axes of the object so that the moment of inertia is diagonal (the principle axes) and this allows us to write the moments of inertia for each axis simply as a number. (look up the moments of inertia for a rectangular box to see an example).

Something similar happens when we consider elementary particles transforming though their abstract gauge group as they propagate in time. Their inertia is expressed with a mass matrix and when this is diagonalized we have the physical particles. This is why/how for example "hyper-charge" and "weak isospin" combine to give electrical charge. It is in the inertia of the elementary particles which breaks the gauge symmetry by failing to align with the gauge charges. Along this same vein, neutrino flavor oscillations are analogous to the precession process I mentioned for free rotating objects... the mass matrix is not diagonal in the same basis as is the "flavor matrix".

Ok, maybe that's TMI but some may find it interesting.

15. Jul 11, 2011

### jambaugh

But as I implied later, the moment of inertia is different for different axes of rotation. When I say "rotating" object I also include "rotating at zero angular velocity". The phrase is meant to imply an object free to rotate.
Exactly what it states:
momentum = inertia times the time rate of change of motion

this of course begs the definition of momentum which is why I provided it. Although a better definition is in terms of the derivative of kinetic energy (as a function of velocity) with respect to the velocity. That is what generalizes to abstract systems.

It may sound abstract but understanding forces as rates of change of momenta and then forces times motions equaling work i.e. equaling change in kinetic energy. One can then relate:

$dT = F dx = dp/dt \cdot dx = dx/dt \cdot dp$
Work is force times displacement equals velocity times change in momentum.

"resistance to change" is not specific enough to define inertia. One may for example use this to define friction.

If you want to speak of the inertia of a medium propagating a wave you will need to get into the detailed model of that propagation. The wave is an array of displacements of the medium which then have...
---a time rate of change,
---kinetic energy,
---canonical momenta relating rate of change of the displacement to kinetic energy
---and inertia relating time rate of change to momenta.

Note that waves can propagate through media of different inertias with the same speed if the coupling potential is different in reciprocal fashion. So IMNSHO and with all due respect, I don't think your comment provides any insight into the concept of inertia.

16. Jul 11, 2011

### Studiot

This is not the same statement as the original, and unfortunately it is even more unsatisfactory.

rewriting it as a an equation inertia = momentum/rate of change of motion

leads to division by zero in the case of a particle or system at rest.

17. Jul 11, 2011

### Studiot

You might call it the moment of inertia tensor..

I call it the inertia tensor.

That is because , in general, it contains terms other than moments of inertia.

In another thread here I displayed situations with identical moments of inertia that can only be distinguished by the other terms in the tensor, which are called products of inertia.

18. Jul 11, 2011

### jambaugh

I'm about to be sarcastic but please take it in proper friendly humor...
[begin sarcasm]
Yea, it's a shame we don't have some kind of mathematical thingy... let's call it, oh, a limit which has some way of dealing with indeterminate forms. It would be great we could create a whole mathematics using it, since it would let us calculate things we could call it, something like, oh I don't know, calculus.

Yea! If only we had that then we could actually even define what we mean by instantaneous velocity! But it's a shame we don't so that displacement over interval of time is undefined when considered at an instant. Instantaneous velocity, and acceleration, and power, and so on are just badly defined concepts. And don't get me started on integrals!!!!
[end sarcasm]

In all seriousness, I find your objection without merit. We can define velocity as the relationship between displacement and duration... displacement = duration times velocity...
velocity = displacement/duration. We can take the limit as the duration goes to zero and be precise about instantaneous velocity when that relationship is time dependent. Your objections fall flat.

You can call it "pixilated" if you like but it is called the http://en.wikipedia.org/wiki/Moment_of_inertia#Moment_of_inertia_tensor" in the text books.

Those other terms are still http://en.wikipedia.org/wiki/Moment_(mathematics)" [Broken] of the mass distribution. (mass density being a perfectly good example of a general measure or distribution as those terms are used in mathematics.)

The tensor as a whole is the natural generalization, to vector-tensor forms, of the component moment of inertia in that it's the thingy you multiply the angular velocity by to get the other thingy called angular momentum.

If you want to shorten the name for brevity's sake then that is all fine and good. But I don't think anyone is interested in your choice of abbreviation if that is the case.
Anything you say about components is basis dependent. The elegant and correct statements (which thus also provide good intuition when understood) should be expressed independent of bases. That is especially important in this case where we are trying to clarify for the OP the physical meaning of quantities.

Last edited by a moderator: May 5, 2017
19. Jul 11, 2011

### jambaugh

Just a pedantic correction but zero temperature does not in any way imply zero momentum. Take a chunk of ice at 0deg K and throw it. It doesn't heat up (until it hits something, that is...).

Temperature describes entropic energy and is independent of coherent energy.

That having been said, Yes a stationary object has zero momentum, zero kinetic energy and unchanged inertia. And yet still...

---momentum = inertia times velocity (=0),
---kinetic energy = momentum times velocity (=0).

Another point about your objection to the definition because it "fails" to apply at 0 velocity. To give inertia et al operational meaning i.e. to observe and measure them we must break out of your special case of v=0.

Just because we define something in terms of how we measure it doesn't in any way imply the quantity looses meaning just because someone might choose not to measure it. This is true even in quantum theory. Only the value becomes indeterminate, not the observable.

To measure inertia you have to push on the object in a quantified way and see how fast it ends up going. The push as a force times duration gives us the momentum imparted, and we define the inertia as the the quotient of the --now no-zero-- momentum over the --now non-zero-- velocity. Repeated application of forces shows us this quotient is independent of the amounts of each and we observe the constancy of inertia for such systems. And then there are other systems....

This is the distinction between physics and mathematics. What things mean (in physics) derives from what we actually do in an experiment.

20. Jul 11, 2011

### Studiot

I really don't know how to take these posts so I will just observe two things.

Firstly none of this squabble addresses the original question

What is the difference between inertia and momentum?

So I would just like to emphasize that inertia is a property of a system or individual particle that it always possesses by virtue of the disposition and quantity of mass and sometimes other quantities.

Momentum is a property that a system or individual particle may possess by virtue of its mass in combination with other quantities (velocity).

It is possible for a system or indivdual particle to possess inertia but no momentum.

********************************

As regards the rather ungracious comment towards my use of the inertia tensor.
Since I had never come across the use of the term 'moment of inertia tensor' I thought I would check out in my library to see if there is a transatlantic difference of terminology.

Here are my results

University Texts using the term 'Inertia Tensor'

Sayer University of Bristol
Bones University of Bristol
Synge Dublin Institute of Advanced Physics
Griffith University of Toronto
Fowles University of Utah
Cassidy University of Utah
Shigley University of Mitchigan