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Until today I used to understand and explain anyone the concept of inertia in the following way, but I found a loophole in that explanation.

Explanation : Imagine we take a body in space where there is no resistance from the air or any solid surface. Initially, that body is at rest. I apply a force F to it trying to change its velocity. As I do that, I feel some resistance as if the body applies a force on me as well. This happens with every body that I take and apply a force on. Every time, I feel a resistance in the form of a force that the body applies to me back. So I postulate that, okay, matter has a quality of resisting any changes in velocity by applying a force back to you. That is what you feel as resistance and this quality of matter is called inertia.

Loophole : If we take different bodies in space, and apply the same force on them, by third law all of them apply the same force on me back. So, according to the above explanation, every body offers me the same resistance, because that is what I defined resistance as - the force that the body gives me back when I apply one on it.

So, I believe my interpretation of inertia is wrong. Then, what exactly do we mean by 'resistance' in the definition of inertia?

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  • $\begingroup$ I can't write a full answer at the moment, but to put it bluntly, yes, your notion of inertia was way off. Inertia has to do with this: imagine a compressed spring launching a box in a direction. If the box is 1 unit of mass, it is launched with a certain velocity. If the box is 2 units of mass, it is launched with less velocity. Both boxes are launched with the same force, but we say box #2 has more resistance to change in velocity due to that force. $\endgroup$ Jul 2 at 8:10
  • $\begingroup$ @MaximalIdeal I don't think that is what the OP is asking about. I think the question is this: 'If I assume that the object pushes back with an amount of force that is equal to the amount that I am applying to that object, how can it be that the object starts moving? That is, I tnterpret the words 'same resistance' as: 'pushes back with the same amount of force that I am applying'. $\endgroup$
    – Cleonis
    Jul 2 at 8:22
  • $\begingroup$ @MaximalIdeal Oh, so the resistance in inertia's definition is not the resistance that I'm feeling while applying the force on it, but rather how easy it is to change its velocity. If it is difficult to change velocity, the resistance of that body to change in velocity is more, and it offers larger inertia. Am I right? $\endgroup$ Jul 2 at 13:12
  • $\begingroup$ Not exactly @Cleonis, I used to think that inertia had something to do with the force applied by an object on me. As in, the body tries to resist by saying 'hey you, don't move me, if you move, I will apply a force on you' :P. But that is not the case I learned today. $\endgroup$ Jul 2 at 13:17
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    $\begingroup$ Experiment. Compare throwing a baseball to throwing a bowling ball. Which can you throw faster? What is the difference? $\endgroup$
    – John Doty
    Jul 2 at 22:53

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It is not the resistance you “feel” in response to the same force you apply to the different objects which reflects the inertia of the different objects. It is the responses of the different objects to your force per Newton’s 2nd law that reflects their inertia.

Per Newton’s 3rd law if you apply the same force F to objects with different masses in space they apply an equal and opposite force F on you. But the different objects will have different accelerations of F/m because their masses are different, I.e they will have different resistances to motion due to their different masses.

On the other hand, you will have only one acceleration in the opposite direction based on the force divided by your mass. Your resistance to motion is the same because your mass is the same. So in your case, what you "feel", which is the same for all the masses, does reflect your resistance to motion, i.e., your inertia

Hope this helps.

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  • $\begingroup$ BobD, Inertia is defined in most textbooks as a property of a body to resist changes in its velocity. I think a better definition could be the property of a body to resist changes in its velocity relative to other bodies. I say that because we came up with the idea of inertia only by comparing different bodies under the same force. We can not come up with the concept of inertia with a single body. What do you think? $\endgroup$ Jul 2 at 17:23
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    $\begingroup$ I'm not quite sure what you're getting at. But wouldn't you agree that the greater the mass of a single body the less its acceleration (the greater its resistance to a change in velocity) will be when the same net force is applied to it? To me, that's why inertia is a property of mass. $\endgroup$
    – Bob D
    Jul 2 at 17:39
  • $\begingroup$ @HarshitRajput There are no absolute velocities to begin with so you're just restating things already understood by everyone using those words. $\endgroup$
    – DKNguyen
    Jul 2 at 18:54
  • $\begingroup$ @DKNguyen I guess you're right, I'm overthinking. $\endgroup$ Jul 2 at 20:09
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    $\begingroup$ @HarshitRajput Perhaps you are overthinking. But at least you are thinking! $\endgroup$
    – Bob D
    Jul 2 at 21:56
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I think that you are confusing inertia and mass.

You made the statement When we say inertia is the quality of matter to resist changes in velocity, . . . . . which is correct but your argument concerns mass which is a measure (ie quantifiable) of the amount of inertia a body has.

When you apply the same force that applied you applied to the less massive body to a more massive body the amount by which the velocity changes is decreased.
So you can reason that the more massive body produces a larger resistance to its velocity being changed.

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  • $\begingroup$ Why was the -1 given? $\endgroup$
    – Farcher
    Jul 2 at 22:44
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    $\begingroup$ Not me, but my guess would be because your first two paragraphs are irrelevant to the discussion, as they're just discussing terminology. Physics would work exactly the same if you just used inertia to mean the quantity of resistance to change in velocity. Then the bolded words in your third paragraph pull the focus away from the actual answer to the query brought up in the question. So, at first glance it seems like you're not actually answering the question, but just nitpicking terminology. $\endgroup$
    – Rick
    Jul 3 at 11:35
  • $\begingroup$ @Rick Very many thanks for taking the time to write your suggested reason. $\endgroup$
    – Farcher
    Jul 3 at 12:47
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Inertia is usually associated with Newton's first law

A body remains at rest, or in motion at a constant speed in a straight line, unless acted upon by a force

rather than Newton's third law

If two bodies exert forces on each other, these forces have the same magnitude but opposite directions

One of the best definitions of inertia comes (of course) from Richard Feynman, although he attributes it to his father:

The general principle is that things which are moving tend to keep on moving and things which are standing still tend to stand still, unless you push them hard. This tendency is called 'inertia', but nobody knows why it's true.

Today we might add that we now have a partial understanding of the source of inertia - it comes from the interaction of elementary particles with the Higgs field. However, this only pushes the question back one stage - we do not know why the Higgs field exists or why it acts in the way that it does.

Calling inertia a 'tendency' makes it easier to understand than calling it a 'resistance', as the latter term makes inertia sound active rather than a passive property of objects.

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    $\begingroup$ About the Higgs field: as we know: interaction with the Higgs field confers mass. While the amount of inertia of an object is proportional to the mass, inertia and mass are not the same thing. Example: the inertial mass of protons and neutrons arises predominantly from the kinetic and potential energy of the quarks that make up the proton/neutron. This inertial mass of energy is outside the scope of the Higgs mechanism. See The origin of mass by Jim Pivarski (written while working as postdoc at the LHC) $\endgroup$
    – Cleonis
    Jul 2 at 12:22
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    $\begingroup$ See also: [How hadron mass emerges from the strong force ](physics.stackexchange.com/questions/137127/…) The answer there is written by stackexchange contributor anna v. The general point: we have that energy has a corresponding inertial mass. In the case of hadrons as QCD bound states: the inertial mass corresponding to the energy of that bound state is outside the scope of the Higgs mechanism. This underlines that while the Higgs field accounts for mass in some circumstances, it does not account for inertia. $\endgroup$
    – Cleonis
    Jul 2 at 13:44
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Mass is "resistance" to a change in velocity in the sense that it "opposes" a change in velocity; that is, given a fixed force, an object with a larger mass will "resist" its velocity increasing, and therefore the amount that its velocity increases will be lower. It's similar to "resistance" in electrical theory: electrical resistance is how much a circuit "opposes" current, and so for a fixed voltage, a circuit with higher electrical resistance will have less current.

Mass is a measure of how much force is needed for a particular acceleration, and force is a measure of how much acceleration there will be for a particular mass. Given F= ma (that is, Force = Mass*Acceleration), we have that Acceleration = Force/Mass, i.e. the larger the mass, the smaller the acceleration for a given force.

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A Way to Think About Inertia

I like to think about inertia as how hard it is to shake something. Suppose I have a mass $m$ and I want to shake it, i.e. make it move as $$ x(t) = A\cos\omega t $$ Then the 2nd law says I have to apply force $$ F(t) = m\ddot{x}(t) = -A\omega^2 m \cos\omega t $$ But this depends on time. Ideally I'd like a number that quantifies how hard it is to shake the mass $m$. One way to do that is to calculate the root mean square force $F_{\text{rms}}$, which you can check is given by $$ F_{\text{rms}} = \frac{A\omega^2 m}{\sqrt{2}} $$ Rearranging, we obtain $$ m = \frac{\sqrt{2} F_{\text{rms}}}{A\omega^2} $$ We see that if I hold the amplitude and frequency of my shakes constant, then mass is directly proportional to how much force it takes me to shake the mass.

If I insist on applying the same rms force to a heavier mass as in the loophole, then I will have to decrease my amplitude or frequency. I'll still notice a difference in "resistance" because I won't be able to shake the heavier mass as violently.

So every mass does not offer the same resistance to motion and, indeed, if we keep the motion the same, we find that the resistance to motion is directly proportional to what we call mass.

A Fun Aside

Lest you think this notion of "how hard it is to shake something" is imprecise or fanciful, here is a video of an astronaut's mass being measured with exactly this principle. If you think about it, something like this is really the only way to take the mass of an astronaut in microgravity. The only other method I could think of would be some kind of accounting of the matter that makes up the astronaut's body (i.e. they have this volume of fat, this volume of protein, this volume of bone, etc. which means they have to have this much mass).

A Short Comment on Terminology

If we're being really picky, we should say that inertia is resistance to change in motion in a chosen coordinate system, whereas mass is an intrinsic property of matter that does not depend on coordinates. For example, rotational inertia is resistance to change in motion when our coordinates are angles of rotation about the spatial axes.

Generally speaking, I don't believe in a strict separation of the terms "inertia" and "mass" since we've already muddied the waters so much with commonly accepted physics terminology. For example, you'll hear people speak of mass as "linear" or "translational" inertia.

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the closest analogy to inertia that we have is the phenomenon of inductance.

The standard setup to obtain strong inductance is to roll a conducting wire into a coil. Coiling the wire accumulates the effect.

When current is flowing through a wire there is a magnetic field around the wire. When the wire is looped the magnetic field is strongest inside that loop. A coiled wire has many loops, accumulating the effect.

Electrostatic fields and magnetic fields have a mutual interaction. A changing magnetic field gives rise to an electric field.

Take the following setup: A coiled wire, and a source of electromotive power (such as a battery), to start and maintain a current in the wire.

From the moment the circuit is closed current starts flowing. So that is a changing rate of current. The magnetic field that is induced is a changing magnetic field, and this changing magnetic field acts in opposition to the inducing current. This puts a limit on how rapidly the rate of current can change.

Note especially that the self-induction cannot prevent change of current strength. In order for the induction to occur there must be change of current strength.

As long as presence of electromotive force in the circuit is kept up the current strength will keep increasing.

Increase/decrease symmetry

The same self-induction will oppose a decrease in current strength. Decrease of current strength is change of current strength just as well.

Normal wire has a non-zero resistance to current, so when you stop applying an electromotive force the current strength will start decreasing. This change of current strength induces a changing magnetic field that acts in opposition to the change that induced it. The stronger the self-induction of the setup, the stronger the tendency for current to remain flowing when there is no longer an electromotive force.

Circuits with strong self-induction need to be switched with a heavy duty current switch, because the current will tend to arc.


Inertia

Whatever Inertia may be, it is abundantly clear that Inertia is a responsive phenomenon. In order for Inertia to manifest itself in a forceful way there must be change of velocity.

In order to change the velocity of an object a force is required. The rate of change of velocity is proportional to the applied force.


Summerizing:
Inertia opposes change of velocity, but at the same time: because of its responsive nature it is inherently impossible for inertia to prevent change of velocity.

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Your loophole is correct. If we provided the same force these objects would push back with the same amount of force. The result of having more mass is that, given some force, an object will experience less acceleration.

In day to day life it is 'harder' to push on a massive object because it is easier to do a lot of work on a heavy object. As we push on an object it accelerates away from us. If you pushed with all your might on a pingpong ball it will just fly away which would prevent you from doing any more work on it. If you did that for a heavy boulder you could continue pushing it, which makes you perform a lot of work on that boulder, which makes you tired in the process.

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Technically, the whole talk of resistance is nonsense. In the absence of other forces (notably friction) if you apply a force to an object for a finite amount of time, its velocity will change. Resistance is futile.

So mass produces resistance to change of velocity under application of force only in the same way that thermal capacity produces resistance to change of temperature under application of heat: it occurs in the denominator of a formula. Whether you put a flame to a straw or to a tank of water, in both cases their temperature will rise, just not by the same amount for a given amount of heat; the change of temperature is basically given by the amount of heat transferred divided by the thermal capacity of the object being heated. So the tank (having greatly larger heat capacity) will require a lot more heat to get a given amount of rise in temperature than the straw does, but to call that resistance to getting warmer is just an anthropomorphic expression. (By the way, it is exaclty the same for electrical resistance.)

On the other hand the opposite force exerted by an object is similar to the cooling down of the flame when heat is transferred from it to an object: it just reflects preservation of momentum in the local interaction (as stated in Newton's third law) like the cooling down reflects the preservation of heat energy in a the local heat exchange. To the first order this has nothing to do with the amount a acceleration respectively rise in temperature achieved by the interaction. It is just that if your goal is to achieve a definite change of velocity before you stop pushing, then you will experience the counter-force for a longer period of time for a large mass, just like the flame will experience cooling from the object for a longer time for the tank than for the straw, if the goal is to heat them up by the same temperature difference.

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The best way to visualize inertia is to imagine all the fundamental particles in a mass as individual springs pushing against each other. When you push on one side of a mass, the force will work its way through every individual particle. Each particle also pushes back and when you push harder more springs push back. When pushing at a constant velocity, the springs will eventually expand back to normal as the resistance subsides, leaving the mass with a new velocity. Notice in the image that the mass with five particles would be more resistant than the mass with two particles.

enter image description here

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