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Apparently there is a lack of resources regarding inertia in the web. I have already asked 2 similar questions in this site, but still I have a problem with inertia. My question -

What is value of inertia of a box placed on the floor of an accelerating bus?

I know the standard explanations; but most are qualitative. Please provide with a quantitative solution.

P.S.- As @G. smith suggests, my question is about inertial or fictitious forces.

AccidentalTaylorExpansion
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Eisenstein
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  • There is no equation in physics with a numerical quantity called “inertia”. Are you confused about mass? What do you think the units of inertia are? If kilograms, how do you think inertia differs from mass? – G. Smith Mar 06 '21 at 06:15
  • @G.Smith If by mass you mean inertial mass, then yeah! I am confused about "inertial mass". – Eisenstein Mar 06 '21 at 06:17
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    It’s the $m$ in $F=ma$. What about that confuses you? – G. Smith Mar 06 '21 at 06:18
  • @G.Smith A ball sits on the floor of a bus, which was originally at rest w.r.t the ground. Suddenly it accelerates forward, and we observe the ball moving backwards. In order to be at rest, the ball needs to provide a force $-(F-f)$ where F and f are forces on the bus and friction respectively. I think this force is called inertia. If so, what is the maximum value of this force, and what is the general formula? – Eisenstein Mar 06 '21 at 06:22
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    You seem to be interested in “inertial forces”, also known as “pseudo forces” or “fictitious forces”. If so, you should edit your question to make this clear. Have you read what Wikipedia has to say about them? – G. Smith Mar 06 '21 at 06:32
  • @G.Smith Wow! This makes it clearer. – Eisenstein Mar 06 '21 at 06:38
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    See also https://www.youtube.com/watch?v=YRgBLVI3suM – Deschele Schilder Mar 06 '21 at 08:01

2 Answers2

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There is no way to calculate inertia numerically since inertia is not quantitative property, instead it is a non-numerical qualitative property of an object. We can say that the more mass an object has then the more inertia it has and vice-versa. So the amount of mass an object has can be an indicator of how much inertia it has, or how much it will resist an external force.

Continuing along these lines, the amount of mass of an object, or the amount it resists a force can be calculated using Newton’s second law $$\tag 1 m=\frac{F}{a}$$ where $m$ is the mass, $F$ is the force that causes the acceleration $a$.

In the case of your example, if a bus accelerates with an amount $a$ and the box is free to slide, then according to an observer inside the bus, he/she will measure the mass to be as given in equation (1). This is an example of what is known as an inertial force. But note that an observer on the curb will see that object as being stationary (no friction inside the bus).

Given mass is a measure of how much inertia an object has, and inertia is a measure of an object’s resistance to force, this is the closest we can get to a “numerical value for inertia”. It’s mass.

joseph h
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  • Can we think of inertia as a force opposing motion? – Eisenstein Mar 06 '21 at 07:43
  • We can think of it as a property of matter that opposes motion (if it’s stationary) or a property that opposes a a change in motion. We should not call inertia a force. Cheers. – joseph h Mar 06 '21 at 07:52
  • Then how does the bus-box scenario change with friction? – Eisenstein Mar 06 '21 at 07:56
  • I like this question too - +1 vote! – joseph h Mar 06 '21 at 07:56
  • Well if their is static friction, the box will move along with the bus in the same direction with the same acceleration. Kinetic friction will mean that the object will still accelerate in the other direction but with a value smaller than $a$. – joseph h Mar 06 '21 at 08:02
  • In an earlier comment you propose to think of inertia as a property of matter. Let's look at the following case: a marble is shot into a lump of clay. The marble penetrates into the clay, just like a stick penetrates into clay if we push it. As we know, we cannot suggest that the marble is pushing itself into the clay. That would be like the Baron von Munchhausen story in which he pulled himself out of a swamp by pulling himself upwards by his own hair. We observe the moving marble has leverage to penetrate the lump of clay; we cannot think of that as the marble pushing itself. – Cleonis Mar 06 '21 at 08:17
  • @Joseph What is the time period after which the body accelerates with the bus considering friction? – Eisenstein Mar 06 '21 at 08:22
  • It will do so instantaneously. As soon as the bus moves, so will the object if there is static friction. – joseph h Mar 06 '21 at 08:28
  • @joseph Nope! That's not what happens! Keep a box on a book and apply a force on it (the book). It moves with the object after some time. – Eisenstein Mar 06 '21 at 08:30
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    @Cleonis Don’t confuse momentum with inertia. – joseph h Mar 06 '21 at 08:31
  • Yes if there is kinetic friction. That is why I said static friction. I made that distinction earlier with bus comment. – joseph h Mar 06 '21 at 08:33
  • @josephh Fine, got it. – Eisenstein Mar 06 '21 at 08:34
  • Excellent. Good luck with your studies. Cheers. – joseph h Mar 06 '21 at 08:36
  • @josephh I don't want to waste your valuable time with my stupid questions, but this one is really bugging me. Consider a string attached to the roof of the bus and an object attached to one end of the string. If the bus accelerates, the object tries to remain in it's initial position, hence it appears to move backwards w.r.t an observer in the bus. But the string does not break (since I expect it to remain forever in it's initial position). Why does not the string break? I am having a really hard time trying to answer these questions to myself. Is the definition somewhat vague? Thanks. – Eisenstein Mar 06 '21 at 08:55
  • Actually, it depends on the magnitude of acceleration the bus takes off with. There is a point where acceleration can increase the tension of – joseph h Mar 06 '21 at 10:04
  • Actually there is a point where if the acceleration is large enough that it exceeds the maximum tension in the string where it will break. It would be an excellent new question if you posted and asked how to derive the equation for which the string will break as a function of the force due to the bus accelerating. Please put a comment here when you post this question. Cheers. – joseph h Mar 06 '21 at 10:09
  • As to relation between inertia and momentum. I gather that you use the word 'inertia' for the mass of an individual object, whereas I use the word 'inertia' in a dynamic sense, as to how it is involved in collisions between objects. It could be, I cannot tell from your comment, that you believe that inertia and momentum are independent. – Cleonis Mar 06 '21 at 11:51
  • @josephh I asked the question. – Eisenstein Mar 06 '21 at 13:04
  • Ok Eisenstein I’ll have a look at it soon. Thanks for letting me know. Cheers. – joseph h Mar 06 '21 at 20:56
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While the science of physics has led to explanations for a lot of phenomena, there are things that have to be assumed as is in order to frame a theory at all.


To give context to what I want to say, let me discuss the concept of 'level of description'.

In physics a breakthrough moves the level of description to a deeper level.

Example: Kepler had formulated the three laws of planetary motion.
Newton moved the level of description to a deeper level, showing that when you grant the inverse square law of gravity, and $F=ma$, then Kepler's laws follow from that.

A recurring pattern is this: the physics community can never know whether the current level of description is the deepest possible. Generally when a deeper level of description comes it is unexpected. (Actually, because it has happened multiple times in the history of physics there is now an 'expect the unexpected' expectation.)


In retrospect we see that the ancient greek thinkers thought of motion in terms of friction. The expectation was that any motion, under any circumstances, will come to a stop, unless a continuous force is applied.

As we know: friction correlates with velocity. Roughly speaking friction is proportional to velocity.

It was through the efforts of Galileo Galilei and others that a notion of inertia was developed. As we know, inertia is independent of how fast an object is moving. Instead, inertia opposes change of velocity.

An analogy to inertia is available in the field of electromagnetism: inductance. If you have a coil with self-induction then you get the following property: upon change of voltage the current strength tends to change, but with a coil with self-induction we have that that change of current strength induces a magnetic field, and that magnetic field act counter to that very change of current strength. The stronger the self-induction, the stronger the opposition to change of current strength. Note that inductance is inherently responsive; in order for the opposition to arise there must be a rate of change of current strength to begin with.


Scientists always focus on problems that look to have a decent chance of being solvable. By contrast, let's say you are presented with some puzzle, but you are not given any clues. Then you won't try that puzzle. You can't make bricks without clay.

In the case of inertia: I believe you have not found resources because very, very few physicists allow themselves to wonder about the origin of inertia. There are no viable clues.

To my knowledge it is extremely rare in the physics commmunity to contemplate an attempt at formulating a quantum theory of inertia.

(I have seen proposals of a connection between inertia and the Unruh effect, but to the extent that such proposals get a response at all it is that they are qualified as are a dead end; the idea is regarded as untenable.)

Cleonis
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