Inertia (and, to some extent, circular motion again)

[jds10011]

Yes, I see that now. Well, then I think it all goes back to what we mean when we say "resistance to motion". If you mean resistance to acceleration, as Orodruin explained in post 2, then everything seems to be just fine. If you mean something else, then it may not.

I'm specifically asking about the ball's tendency to pull harder on the string as it is revolved in both the case where its mass is increased and in the case where its speed is increased. The first we typically attribute to inertia. The second I often hear attributed to inertia, but I cannot see how this can be correct. It seems like in both cases the ball's behavior is explained by the same root characteristic, which doesn't track with inertia, but seems to be something else (which may be its structural integrity?). The followup is whether this means that its tendency to continue in straight-line motion is equally hampered in both cases, in which case we have a problem with the definition of inertia as both mass and tendency to continue in straight-line motion (but here I think I'm the one who's confused).

 

[Drakkith]

The second I often hear attributed to inertia, but I cannot see how this can be correct. It seems like in both cases the ball's behavior is explained by the same root characteristic, which doesn't track with inertia, but seems to be something else (which may be its structural integrity?).

I can't see how structural integrity comes into this and I really don't know what else I can say. I personally don't see any problem with the way inertia is defined.

 

[jds10011]

I can't see how structural integrity comes into this and I really don't know what else I can say. I personally don't see any problem with the way inertia is defined.

Does it seem correct to use inertia to explain why revolving the ball faster means it pulls harder on the string? If so, please help me to understand how inertia explains this. If not, please help me to understand what about the ball causes it to behave in this manner (without just saying it behaves in accordance to Newton's laws, which I don't dispute, but it doesn't explain the physical cause of the behavior).

I was bringing up structural integrity as it was the answer to the other situation where I felt inertia was misapplied, that of dropping an object onto the table and trying to explain why a higher drop height will result in the object applying a higher contact force to the table. One might say that due to inertia, the object tries to maintain in straight-line motion and therefore applies a force to the table. Yet, clearly the inertia is unchanged when dropping from the higher height, but the object seemed to have an increased "desire" to go through the table (the contact force was larger) , just as it would have been if the object's mass were increased (which would track with the definition of inertia). A better explanation is that the object's structural integrity is more challenged by encountering the table having fallen from the higher height, and therefore the object exerts a larger force on the table. I do rather wonder whether a similar explanation is the root cause of the behavior that we're describing as inertia when the mass is increased rather than the drop height, also. Further, I think it might be really helpful to elucidate why the fact that the object hits the table with greater force is "evidence" for a greater tendency to continue in a straight-line motion, but only in the scenario where the mass is increased, not in the scenario where the drop height is increased -- or to tell me why it is incorrect/correct in both cases.

 

[Drakkith]

Does it seem correct to use inertia to explain why revolving the ball faster means it pulls harder on the string? If so, please help me to understand how inertia explains this. If not, please help me to understand what about the ball causes it to behave in this manner (without just saying it behaves in accordance to Newton's laws, which I don't dispute, but it doesn't explain the physical cause of the behavior).

I disagree. I think Newton's laws explain the cause just fine.

A better explanation is that the object's structural integrity is more challenged by encountering the table having fallen from the higher height, and therefore the object exerts a larger force on the table.

Again, I disagree. Think back to my electron example from earlier. I believe the increased force arises because the atoms get closer together during the collision, which increases the repulsive force. The force itself would depend solely on the distance between the object and the countertop, regardless of what the mass of the object was. A larger mass causes a larger force simply because it is more difficult to accelerate and will thus get closer to the atoms in the countertop before being stopped..

 

[jds10011]

I disagree. I think Newton's laws explain the cause just fine.

By saying it must occur as a reaction to being pulled as per N3? Or by saying it must occur for F=ma to be true? How does this explain the cause of the ball's action of pulling on the string? This is why I was discussing the snowball, which doesn't behave in the appropriate manner (at high speeds or at high masses), and asking if this meant the root cause of the ball pulling was that non-snowballs are capable of something the snowball isn't, and asking what property of the ball this would be attributed to?

Again, I disagree. Think back to my electron example from earlier. I believe the increased force arises because the atoms get closer together during the collision, which increases the repulsive force. The force itself would depend solely on the distance between the object and the countertop, regardless of what the mass of the object was. A larger mass causes a larger force simply because it is more difficult to accelerate and will thus get closer to the atoms in the countertop before being stopped..

Which atoms get closer together? The ones in the falling object? How would you differentiate this from something related to the object's structural integrity being challenged? I'm also not sure I'm clear on why this doesn't happen with the increased mass case.

 

[jbriggs444]

By saying it must occur as a reaction to being pulled as per N3? Or by saying it must occur for F=ma to be true?

Both. The force of string on ball must be enough to explain the circular motion that you have specified. That's Newton's second law. The force of ball on string is then equal and opposite. That's Newton's third law.

If you wish to explore structural integrity as an explanation then it is fairly straightforward. Suppose that the force of string on ball were more than $\frac{mv^2}{r}$. Then the ball would be accelerated inward at more than $\frac{v^2}{r}$ and its distance to the center of rotation would decrease over time. The string would go slack and its tension would be reduced. Suppose that the force of string on ball were less than $\frac{mv^2}{r}$. Then the ball would be accelerated inward at less than $\frac{v^2}{r}$ and its distance to the center of rotation would increase over time The string would stretch and its tension would be increased.

There is a stable equilibrium situation with tension in the string given by $\frac{mv^2}{r}$ so that the ball remains in uniform circular motion.

The word "inertia" never enters in. The mass of the ball does.

 

[jds10011]

Both. The force of string on ball must be enough to explain the circular motion that you have specified. That's Newton's second law. The force of ball on string is then equal and opposite. That's Newton's third law.

I know both of these things are true. The point I was making was that we are not really explaining a root cause of the behavior by saying it is "explained" by Newton's laws. This looks like the same thing as if someone from Newton's time observes a car accelerating along a road, and asks what the cause of this behavior is. Would a satisfying explanation be that the cause of this is a force according to Newton's second law? Would they not be curious about what might be inside the car?

If you wish to explore structural integrity as an explanation then it is fairly straightforward. Suppose that the force of string on ball were more than $\frac{mv^2}{r}$. Then the ball would be accelerated inward at more than $\frac{v^2}{r}$ and its distance to the center of rotation would decrease over time. The string would go slack and its tension would be reduced. Suppose that the force of string on ball were less than $\frac{mv^2}{r}$. Then the ball would be accelerated inward at less than $\frac{v^2}{r}$ and its distance to the center of rotation would increase over time The string would stretch and its tension would be increased.

There is a stable equilibrium situation with tension in the string given by $\frac{mv^2}{r}$ so that the ball remains in uniform circular motion.

I can't tell if you're saying the structural integrity explanation has any valid basis in fact, or not.

The word "inertia" never enters in. The mass of the ball does.

The property of inertia is defined as being the same as the object's mass. The behavior of the object with larger mass is said to be a larger inertia.

 

[jbriggs444]

I know both of these things are true. The point I was making was that we are not really explaining a root cause of the behavior by saying it is "explained" by Newton's laws. This looks like the same thing as if someone from Newton's time observes a car accelerating along a road, and asks what the cause of this behavior is. Would a satisfying explanation be that the cause of this is a force according to Newton's second law? Would they not be curious about what might be inside the car?

That's a bit like the children's game of asking "why" in an infinite loop. The car accelerates because there is a force on the car. If you want to explain how there comes to be a force on the car, that's a different question. We might then talk about the engine, the drive train or the fact that the driver is late for work.

I can't tell if you're saying the structural integrity explanation has any valid basis in fact, or not.

I meant the explanation in all seriousness. If you want to explain how there comes to be a particular tension in the string then a useful explanation may be in terms of the relaxation of the system (a ball on a string with a specified tangential velocity) toward an equilibrium state with just the right string tension to maintain uniform circular motion at that speed.

The property of inertia is defined as being the same as the object's mass. The behavior of the object with larger mass is said to be a larger inertia.

If you want to use the term "inertia" as a synonym for "inertial mass" or just plain "mass", that's fine. Please do so. You can then stop worrying about what "is said" about inertia and concentrate on what the laws of mechanics say about mass.

 

[Khashishi]

I suggest not using the term inertia to refer to a quantity that can be measured. Inertia refers to a concept, not a quantity. It's fine to use the term to describe the concept (basically the conservation of momentum). It doesn't have a precise definition when used as a quantity. The quantities most commonly associated with inertia are mass and momentum. These have more precise definitions and can be used in equations.

You wouldn't say, "How much conservation of momentum do I have when spinning in a gravotron?", so you shouldn't say how much inertia do I have. (Historically, inertia was perhaps used as a quantity in some contexts, but we have moved beyond that point.)

 

[jds10011]

I suggest not using the term inertia to refer to a quantity that can be measured. Inertia refers to a concept, not a quantity. It's fine to use the term to describe the concept (basically the conservation of momentum). It doesn't have a precise definition when used as a quantity. The quantities most commonly associated with inertia are mass and momentum. These have more precise definitions and can be used in equations.

You wouldn't say, "How much conservation of momentum do I have when spinning in a gravotron?", so you shouldn't say how much inertia do I have. (Historically, inertia was perhaps used as a quantity in some contexts, but we have moved beyond that point.)

This is the crux of the original post. What you have written is inconsistent with the accepted definition of inertia, as far as I know; I often hear it misapplied this way. For example, when increasing the speed of an object, the force required is certainly not associated with its present velocity -- this has nothing to do with its tendency to continue in straight-line motion at a constant speed, which is described by its mass. Yet, in the gravitron, we often do hear people say that that if the ride goes faster, the fact that you hit the wall harder is because your tendency to continue in straight-line motion has increased. I don't think a different definition of inertia should apply here.

 

[jbriggs444]

This is the crux of the original post. What you have written is inconsistent with the accepted definition of inertia, as far as I know; I often hear it misapplied this way. For example, when increasing the speed of an object, the force required is certainly not associated with its present velocity -- this has nothing to do with its tendency to continue in straight-line motion at a constant speed, which is described by its mass. Yet, in the gravitron, we often do hear people say that that if the ride goes faster, the fact that you hit the wall harder is because your tendency to continue in straight-line motion has increased. I don't think a different definition of inertia should apply here.

If we cannot agree on what "inertia" even means then we should avoid using the term. It is irrelevant to the question you ask in any case.

 

[jds10011]

If we cannot agree on what "inertia" even means then we should avoid using the term. It is irrelevant to the question you ask in any case.

I don't see how we can use Newton's laws if we don't think inertia is a valid agreed upon concept.

To be specific, how about the "tendency to maintain straight-line motion" in this case. I've written this before without receiving an answer. I'm in the gravitron. The ride spins at a constant speed. I keep smashing into the wall. I now put on a heavy backpack. I smash into the wall harder. I don't think it is disputed that my tendency to maintain straight-line motion has increased. Now, instead of the backpack, the ride is made to spin at a faster constant speed. I again smash into the wall harder. Has my tendency to maintain straight-line motion again increased?

 

[jbriggs444]

I don't see how we can use Newton's laws if we don't think inertia is a valid agreed upon concept.

Newton's laws make quantitative predictions about operationally measurable quantities. None of which need any concept of "inertia".

To be specific, how about the "tendency to maintain straight-line motion" in this case. I've written this before without receiving an answer. I'm in the gravitron. The ride spins at a constant speed. I keep smashing into the wall. I now put on a heavy backpack. I smash into the wall harder. I don't think it is disputed that my tendency to maintain straight-line motion has increased. Now, instead of the backpack, the ride is made to spin at a faster constant speed. I again smash into the wall harder. Has my tendency to maintain straight-line motion again increased?

If you have an quantitative operational procedure to measure "tendency to maintain straight-line motion" then we can talk. Otherwise not.

 

[Drakkith]

This is the crux of the original post. What you have written is inconsistent with the accepted definition of inertia, as far as I know; I often hear it misapplied this way. For example, when increasing the speed of an object, the force required is certainly not associated with its present velocity -- this has nothing to do with its tendency to continue in straight-line motion at a constant speed, which is described by its mass. Yet, in the gravitron, we often do hear people say that that if the ride goes faster, the fact that you hit the wall harder is because your tendency to continue in straight-line motion has increased. I don't think a different definition of inertia should apply here.

Anyone who says that your tendency to continue in a straight line motion has increased is wrong. Nowhere in physics is inertia actually described that way. Not correctly at least.

 

[Dale]

The property of inertia is defined as being the same as the object's mass

its tendency to continue in straight-line motion at a constant speed, which is described by its mass.

It seems like you have all of the pieces, but are just having trouble putting them together. Based in your own comments any time that you see the word "inertia" or the phrase "tendency to continue ..." you can simply substitute the word "mass". So:

<Edits from original indicated>

Now, instead of the backpack, the ride is made to spin at a faster constant speed. I again smash into the wall harder. Has my tendency to maintain straight-line motion mass again increased?

Clearly the answer as revised is "no". Your mass has clearly not increased. So by your own reasoning this also implies that your "inertia" has not increased nor has your "tendency to maintain..." increased.

 

[jds10011]

It seems like you have all of the pieces, but are just having trouble putting them together. Based in your own comments any time that you see the word "inertia" or the phrase "tendency to continue ..." you can simply substitute the word "mass". So:Clearly the answer as revised is "no". Your mass has clearly not increased. So by your own reasoning this also implies that your "inertia" has not increased nor has your "tendency to maintain..." increased.

I don't disagree with these statements. I am not disputing that inertia is just mass (though others are, here and otherwise), nor that it is the tendency to maintain straight-line motion. I am saying that I often hear it mis-applied or mis-represented in other scenarios (see my very first post).

(1) Perhaps you could help me to understand what differentiates the two cases (the increased mass as I ride the gravitron and the increased speed in the same ride) such that it makes intuitive sense that despite smashing into the wall harder in both cases, only one is a change in my tendency to maintain straight-line motion. I understand this concept from a definitional standpoint (tendency to maintain straight line motion is inertia, inertia is just mass, mass hasn't changed, ergo my tendency to maintain straight-line motion hasn't changed), but not really from a standpoint of understanding the physics behind this assertion.

(2) I have asked this question a number of times without really getting an answer: When the ride speed is increased, and I smash into the wall harder, is there no property of myself that helps to explain this behavior? Must I just say that I am smashing into the wall harder because of the actions of the wall? Most everyone says yes, but this seems a bit unsatisfactory. If it is yes, then why?

(3) The followup to this question was the similar example of the ball being revolved on the string. Suppose the string is tied to a fixed rod (no one is there). The ball is set in motion by a person, but they don't interact further with the setup. The ball will pull harder on the string if a higher mass is used OR if a higher initial speed is used. Must we explain the force that the ball exerts on the string only by N3, and positing the rod pulling as the action and the ball pulling as the reaction? It seems like it should be equally valid from the perspective of the ball as the action and the rod as the reaction, but how then to explain why the ball is pulling? Additionally, why are we willing to explain that it pulls harder based on an innate property of the ball only in the case of the increased mass? Is there not a property of the ball responsible for this in both cases? I had suggested that perhaps it is illuminating to think of revolving a snowball, which would fly apart at high speeds, rather than revolving -- in other words, it would be incapable of pulling on the string appropriately.

 

[Drakkith]

I don't disagree with these statements. I am not disputing that inertia is just mass (though others are, here and otherwise), nor that it is the tendency to maintain straight-line motion. I am saying that I often hear it mis-applied or mis-represented in other scenarios (see my very first post).

How would you revise these explanations? Or is there a different issue here? Thanks!

I'd say the answer is to use mass instead of inertia and reformulate the questions to state everything in terms of Newton's laws.

 

[Dale]

I often hear it mis-applied or mis-represented

Yes, but that is true of most subjects. All we can do is help straighten things out the best we can

what differentiates the two cases

You already know exactly what differentiates the two cases. You are the one who designed the cases. In the first case $m$ increases and $a$ is constant. In the second case $m$ is constant and $a$ increases. Since you designed them this way how can you say you don't know what differentiates them?

this seems a bit unsatisfactory.

Why is this unsatisfactory? You have not only correctly differentiated between increasing $m$ and increasing $a$, but you have also correctly applied $f = m a$ to correctly predict that $f$ must increase in both cases. What can be more satisfactory in physics than a simple formula correctly applied to predict the actual behavior of the world?

Overall it seems that your issue is not a physics problem but an unfounded emotional reaction of dissatisfaction.

Must we explain the force that the ball exerts on the string only by N3, and positing the rod pulling as the action and the ball pulling as the reaction?

The labels "action" and "reaction" are completely arbitrary. You can always swap the designation with no change in the physics.

Also, I don't think that there is ever only one way that you must describe something. For instance, you might choose to describe them from a rotating reference frame, or using Lagrangian mechanics, or using Hooke's law, or ...

Additionally, why are we willing to explain that it pulls harder based on an innate property of the ball only in the case of the increased mass? Is there not a property of the ball responsible for this in both cases? I had suggested that perhaps it is illuminating to think of revolving a snowball, which would fly apart at high speeds,

Sure. For a complete treatment you would need to consider the material properties. You can make typical idealizations such as an inextensible string and a rigid ball. Or you could use Hooke's law and more realistic material properties.

I am sure that from your specification of the problem that most people assume the typical idealizations are what you intended. If you want to discuss Hooke's law then you probably should specify the relevant parameters clearly and directly.

 

[jds10011]

You already know exactly what differentiates the two cases. You are the one who designed the cases. In the first case $m$ increases and $a$ is constant. In the second case $m$ is constant and $a$ increases. Since you designed them this way how can you say you don't know what differentiates them?

Why is this unsatisfactory? You have not only correctly differentiated between increasing $m$ and increasing $a$, but you have also correctly applied $f = m a$ to correctly predict that $f$ must increase in both cases. What can be more satisfactory in physics than a simple formula correctly applied to predict the actual behavior of the world?

Overall it seems that your issue is not a physics problem but an unfounded emotional reaction of dissatisfaction.

What I was asking was what differentiates the ball's behavior in the two cases. I other words, when we increase the mass, and the ball pulls harder on the string, we are typically willing to say that its inertia has increased, and therefore it tugs harder on the string in a larger effort to maintain straight-line motion than when it had a lower mass. We still haven't exactly explained what about the physical nature of the ball at a root level has caused this, but it's a start. However, when we instead set the ball in motion with a larger speed, the ball again pulls harder on the string, but I have yet to hear anyone give any explanation from the standpoint of what the ball is doing, but instead will fall back on a description of Newton's laws (which we could have done for the increased mass case -- oh, we must pull harder on the string to maintain the required acceleration according to N2, therefore the ball responds in accordance with N3).

The labels "action" and "reaction" are completely arbitrary. You can always swap the designation with no change in the physics.

I agree. If you read back in the thread, most were falling back on describing the ball's behavior solely as a reaction. I was asking how to explain it as the action, which, as you say, should be entirely possible, but I haven't heard it yet.

Also, I don't think that there is ever only one way that you must describe something. For instance, you might choose to describe them from a rotating reference frame, or using Lagrangian mechanics, or using Hooke's law, or ...

Sure. For a complete treatment you would need to consider the material properties. You can make typical idealizations such as an inextensible string and a rigid ball. Or you could use Hooke's law and more realistic material properties.

I am sure that from your specification of the problem that most people assume the typical idealizations are what you intended. If you want to discuss Hooke's law then you probably should specify the relevant parameters clearly and directly.

What I am starting to wonder here is whether a point mass, were there really such a thing, would actually behave in these ways, or whether we actually need to talk at least in vague terms about the properties of the material. For example, when I drop a brick onto a surface, and then two bricks stacked together onto the surface, I often hear a hand-wavy explanation about how the contact force was larger because the top brick kept going for an instant, the bricks compressed a bit, etc. Yet, in comparing two point masses, one of a larger mass, performing the same experiment (at least as gedanken), we certainly wouldn't be able to explain it this way...

 

[Dale]

instead will fall back on a description of Newton's laws

What is wrong with that? That is the proper way to do physics: describe a situation and apply the laws of physics! This objection doesn't make sense. How can you possibly complain that the answer to a question in physics is to apply the laws of physics?

agree. If you read back in the thread, most were falling back on describing the ball's behavior solely as a reaction. I was asking how to explain it as the action, which, as you say, should be entirely possible, but I haven't heard it yet.

The explanation is precisely the same. Newtons laws make no distinction between action and reaction. That is why you are able to swap them as desired.

Take any explanation in terms of reaction, change all occurrences of the word "reaction" to the word "action", and you have the desired explanation.

 

[jds10011]

What is wrong with that? That is the proper way to do physics: describe a situation and apply the laws of physics! This objection doesn't make sense. How can you possibly complain that the answer to a question in physics is to apply the laws of physics?

There is a difference between blindly repeating statements, even if they are correct, and actually doing physics. Doing physics is not just remembering which statements to recall when, even if answering textbook questions may well be solved that way. This is especially the case when asserting the contrapositive of these statements (and yes, I know the contrapositive is also always true), e.g. "the object must behave this way, because if it didn't behave this way, it wouldn't obey Newton's laws". This doesn't provide any explanation whatsoever, just asserts that one exists.

The explanation is precisely the same. Newtons laws make no distinction between action and reaction. That is why you are able to swap them as desired.

Take any explanation in terms of reaction, change all occurrences of the word "reaction" to the word "action", and you have the desired explanation.

OK, then, what is the answer to my question? A ball is revolved on a string attached to a rod. It pulls harder on the string when either its mass is increased or its initial speed is increased. When the mass is increased, we say its inertia has increased and it is pulling harder on the string because it is harder for it to maintain straight-line motion. This isn't the explanation for the same increased pulling force when the initial speed is increased. And please don't say that we categorize this behavior as an increased centripetal acceleration, which requires an increased centripetal force, or else N2 isn't true, and therefore it behaves in accordance with N2. I'm not disputing that, but it's just as logical as explaining an accelerating car by saying that N2 is obeyed and therefore greater force must be exerted on it, rather than explaining to me that there's an engine whose behavior has changed (i.e. not wrong, but also not illuminating).

 

[Dale]

There is a difference between blindly repeating statements, even if they are correct, and actually doing physics

There is also a difference between asking a physics question and trolling. Telling a bunch of experts in physics that correctly using the laws of physics isn't "actually doing physics" is rude and pointless.

This doesn't provide any explanation whatsoever, just asserts that one exists

Nonsense. In any theory of physics there is a set of statements (often called postulates or laws) that serve as the basis for all of the predictions of the theory. All of the conclusions of that system are derived from those laws. So not only does showing how a conclusion follows from the laws provide an explanation, it is the only form explanation allowed by the theory. Experimental tests of the predictions are then taken as evidence that the laws are valid explanations.

And please don't say that we categorize this behavior as an increased centripetal acceleration, which requires an increased centripetal force, or else N2 isn't true, and therefore it behaves in accordance with N2.

That IS a valid explanation! The law of physics, $f=ma$, says that $f$ increases if either $m$ increases or $a$ increases. By construction $a$ increases, therefore $f$ increases as explained by the law.

but it's just as logical as explaining an accelerating car by saying that N2 is obeyed and therefore greater force must be exerted on it, rather than explaining to me that there's an engine whose behavior has changed (i.e. not wrong, but also not illuminating)

The only other law involved in your scenario is Hooke's law. This was also explained earlier.

Since your question has been answered in terms of the laws of physics, there is no point in continuing this discussion. Please be aware for future questions that the established laws of physics are always considered to be valid explanations here on PF.