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

[jds10011]

I often hear inertia used as an explanation in areas where it seems to make intuitive sense, but appears to me to be inconsistent with the definition of inertia as just depending on an object's mass. I offer three examples (they're very similar):

Example 1: An elevator
Suppose an elevator begins at rest and then accelerates upward. The rider has naturally brought a bathroom scale to stand on in the car. Of course, the scale indicates that the normal force on the rider has increased from just the magnitude of the rider's weight. Often this is explained by saying: "The rider was at rest, and therefore due to inertia had a tendency to remain at rest. By attempting to remain at rest, the rider exerted a greater-than-usual force on the floor of the car (scale). You can see this also by the fact that the rider's knees buckled slightly as the elevator started to move. In fact, we can see that it is a result of the person's inertia by substituting a more massive person -- for the same acceleration of the car, the more massive person pushes down harder on the scale, since they have more inertia." The issue that I have with this explanation is that if the elevator were now given a greater acceleration, the person would exert greater force on the floor/scale. However, their inertia has not changed, so it seems like a poor explanation for the phenomena.

Example 2: A ball on a string
Suppose a person whirls a ball on a string around in a circle at a constant speed. There is tension in the string, so clearly the person is pulling inward on the string, and the ball is pulling outward on the string (yes, even though there is no outward force on the ball, there is on the string). Many are surprised that the ball has reason to pull outward on the string, or that the person must pull inward. Often this is explained by saying: "The ball has inertia -- the tendency to continue in straight-line motion. In order to make it go around in a circle, rather than continue in a straight line, the person must use the string to change the direction of the ball's motion, which the ball resists as it tries to go in a straight line. Again, we can see this is the result of the ball's inertia by substituting a ball of greater mass -- for the same speed of revolution, the person and the ball must pull harder on the string." As before, the issue that I have with this explanation is that if the ball were now given a greater speed, the person and the ball would both pull harder on the string. However, the ball's inertia has not changed, so it seems like a poor explanation for the phenomena.

Example 3: A person in a gravitron ride (or a towel in the clothes dryer):
Suppose the ride travels at a constant speed. A rider (inside) is against the outer wall of the drum. The person is surprised that they seem to be pushing against the wall (in fact, they can stand horizontally on the wall if the ride goes fast enough). Often this is explained exactly as with the ball on the string -- "The person's inertia means they want to continue in a straight line, so they keep running into the wall, which then responds to this contact force by exerting its third law pair, the normal force, on the person. Again, we can see this is the result of the rider's inertia by substituting a rider of greater mass -- for the same speed of revolution, the new rider pushes harder on the wall, and the wall responds with a greater normal force." As before, the issue that I have with this explanation is that if the ride were now given a greater speed, the person would exert greater force on the wall (and the wall would exert greater force on the rider). However, their inertia has not changed, so it seems like a poor explanation for the phenomena.

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

 

[Orodruin]

Inertia is the resistance to acceleration. The force is proportional to the inertia only if you keep the same kinematic setup. This is well understood. If you change acceleration the force will also increase in proportion to the inertia.

 

[vela]

As Newton told us, $F = ma$. The force doesn't depend on the mass alone.

 

[jds10011]

As Newton told us, $F = ma$. The force doesn't depend on the mass alone.

I agree. Hence, the reason that I don't see the concept of inertia as fully explaining these scenarios. The same mass offers greater "resistance" to greater accelerations, but this isn't called inertia.

 

[Orodruin]

The same mass offers greater "resistance" to greater accelerations, but this isn't called inertia.

This statement makes no sense. The ratio between the force and the acceleration is the inertia in all of the cases you mentioned.

 

[jds10011]

This statement makes no sense. The ratio between the force and the acceleration is the inertia in all of the cases you mentioned.

So, for example, the ball on the string. If I try to whirl it around faster, the ball pulls harder on the string, i.e. it resists the faster circular motion more than it did the slower one. Yet, its inertia is unchanged.

 

[Drakkith]

Get away from using "normal" language and try to look at it from a standpoint of the equations. The ball isn't resisting the change in its direction of motion any more or less than at any other time. In all cases its mass determines how quickly it is acceletated under a force, or how much force must be applied for any given acceleration. There is little ambiguity there.

 

[PeroK]

So, for example, the ball on the string. If I try to whirl it around faster, the ball pulls harder on the string, i.e. it resists the faster circular motion more than it did the slower one. Yet, its inertia is unchanged.

Circular motion at constant speed, unlike linear motion at constant speed, requires constant acceleration (towards the centre of the motion). So, the "faster" the circular motion the greater the force that is required. Unlike linear motion where once you have reached a certain speed it takes essentially no force to maintain that speed.

By the way, "inertia" is not a term I've ever used. The term "mass" does the job.

 

[Orodruin]

If I try to whirl it around faster, the ball pulls harder on the string, i.e. it resists the faster circular motion more than it did the slower one.

Yes, because by increasing the speed you have increased the acceleration according to $a = v^2/r$ which holds for any circular motion at constant speed. Therefore you need more force - all in accordance with $F = ma$.

 

[jds10011]

Get away from using "normal" language and try to look at it from a standpoint of the equations. The ball isn't resisting the change in its direction of motion any more or less than at any other time. In all cases its mass determines how quickly it is acceletated under a force, or how much force must be applied for any given acceleration. There is little ambiguity there.

I agree with most of this. And yes, I am specifically trying to get at an issue of language here. It seems like substituting the more massive ball is analogous to whirling the less massive ball faster -- in both cases, as I hold the end of the string, I can feel an increased tension, which I would attribute to the ball's resistance to changes in its motion (inertia). Yet, substituting the more massive ball is clearly changing the inertia (I don't think this is disputed), whereas increasing the rotation speed isn't (I don't think this is disputed). However, would you not say that in both cases the ball's resistance to changes in its motion has increased? If I want to break the string, let's say, I can accomplish this either by using such a massive ball that the string is incapable of sufficiently changing its motion even at a low speed, or I can do this by whirling the smaller mass so fast that the string is again incapable of sufficiently changing its motion. It still seems like the former is clearly a result of inertia, and the latter is clearly not.

 

[Orodruin]

However, would you not say that in both cases the ball's resistance to changes in its motion has increased?

No. We are saying that in the case when you spin it faster the ball's change in motion has increased. If you can change the motion more by applying the same force, clearly you have less inertia. Again, inertia tells you how much force you need for a given rate of change in the motion.

 

[jds10011]

No. We are saying that in the case when you spin it faster the ball's change in motion has increased. If you can change the motion more by applying the same force, clearly you have less inertia. Again, inertia tells you how much force you need for a given rate of change in the motion.

So, the fact that the ball applies more force to the string when it is spun faster is not an indication that it is providing more resistance to its speed changing? In other words, suppose I am holding the string blindfolded when the ball is set in motion by a friend. I will feel some tension force. Now the setup is changed by either giving the ball a larger speed or by changing the mass of the ball and keeping the same speed. I will now feel a larger tension force. As I am blindfolded, I don't know which has occurred. Are you saying I can't tell based on the increased tension force that the ball's resistance to changes in its motion has changed?

Or, in the gravitron (drum) ride, if a rider asks why they are slamming into the outer wall harder the faster the ride goes, the answer is not related to the fact that they are more resistant to changes in their speed at higher speeds? (And yes, I know they "shouldn't" be, but bear with me, and I appreciate the help.)

I guess what I keep going back to is the old trick question about a stick being swung to hit a small 100 gram block on a frictionless horizontal surface. The question asks "If I apply a 1000N horizontal contact force to the block, by N3 does the block respond to me with 1000N?" The answer is "No, because you'll never be able to apply a 1000N force in such a scenario. The tiny mass doesn't resist the change in motion sufficiently for you to be able to remain in contact with it beyond a certain applied force -- far less than 1000N. Whatever force you do succeed in applying will be applied back to you by the block in accordance with N3."

It seems that the ball on the string is responding to the string pulling it inward with an equally large outward force on the string in accordance with N3, and if I increase the speed by increasing the inward pull, it responds by increasing its outward pull on the string. This is what I am interpreting as an increase in resistance, and I think where I am tripping myself up. I think you are saying that its resistance to changes in its motion is solely based on its mass (inertia), and therefore to accomplish a larger change in motion means a larger force (N2). I understand this from the perspective of the person applying the force. I think the issue is just that when I think of the ball pulling outward on the string, I am at a loss to explain why it is doing that harder at higher speeds from its perspective.

I think I'm getting close, and I appreciate your help and patience.

 

[PeroK]

So, the fact that the ball applies more force to the string when it is spun faster is not an indication that it is providing more resistance to its speed changing? In other words, suppose I am holding the string blindfolded when the ball is set in motion by a friend. I will feel some tension force. Now the setup is changed by either giving the ball a larger speed or by changing the mass of the ball and keeping the same speed. I will now feel a larger tension force. As I am blindfolded, I don't know which has occurred. Are you saying I can't tell based on the increased tension force that the ball's resistance to changes in its motion has changed?
It seems that the ball on the string is responding to the string pulling it inward with an equally large outward force on the string in accordance with N3, and if I increase the speed by increasing the inward pull, it responds by increasing its outward pull on the string. This is what I am interpreting as an increase in resistance, and I think where I am tripping myself up. I think you are saying that its resistance to changes in its motion is solely based on its mass (inertia), and therefore to accomplish a larger change in motion means a larger force (N2). I understand this from the perspective of the person applying the force. I think the issue is just that when I think of the ball pulling outward on the string, I am at a loss to explain why it is doing that harder at higher speeds from its perspective.

I think I'm getting close, and I appreciate your help and patience.

Your confusion, in general, is due to using an inexact concept of inertia as "resistance to motion" and attributing a characteristic of the motion to inertia of the object (a property of the object itself). The two examples here are

1) rate of acceleration is a characteristic of the motion (not of the object): increase the rate of acceleration and you increase the required force (but you do not change the inertia of the object).

2) Speed of circular motion is a characteristic of the motion: increase the circular speed and you increase the required centripetal force (but you do not change the inertia of the object).

You could add a third, which is an object moving on a rough surface. It's harder to increase the speed on the rough surface, so you could attribute this to increased inertia, But, it's easier to reduce the speed of the object, so you could attribute this to decreased inertia. In this way, you would have a variable inertia that depends on whether you are trying to increase or decrease the speed of the object.

This would also pass your blindfold test. Blindfolded you might interpret the friction as an increase or decrease in inertia of the object.

In an extreme case, someone might glue the object to the surface and you'd attribute this to the ball having gained inertia. This example highlights the problem of using an airy-fairy notion of inertia instead of mass. The mass of the object is the same, it's just glued to a surface. But, in a woolly way, by being glued down its resistance to motion has increased.

 

[jds10011]

Your confusion, in general, is due to using an inexact concept of inertia as "resistance to motion" and attributing a characteristic of the motion to inertia of the object (a property of the object itself). The two examples here are

1) rate of acceleration is a characteristic of the motion (not of the object): increase the rate of acceleration and you increase the required force (but you do not change the inertia of the object).

2) Speed of circular motion is a characteristic of the motion: increase the circular speed and you increase the required centripetal force (but you do not change the inertia of the object).

You could add a third, which is an object moving on a rough surface. It's harder to increase the speed on the rough surface, so you could attribute this to increased inertia, But, it's easier to reduce the speed of the object, so you could attribute this to decreased inertia. In this way, you would have a variable inertia that depends on whether you are trying to increase or decrease the speed of the object.

This would also pass your blindfold test. Blindfolded you might interpret the friction as an increase or decrease in inertia of the object.

In an extreme case, someone might glue the object to the surface and you'd attribute this to the ball having gained inertia. This example highlights the problem of using an airy-fairy notion of inertia instead of mass. The mass of the object is the same, it's just glued to a surface. But, in a woolly way, by being glued down its resistance to motion has increased.

This is a really helpful set of explanations. Much appreciated. I certainly can see that a block on a surface with friction would resist changes in motion based on the amount of friction AND its mass. I guess the question then becomes that I can easily see the friction force as causing this, and would look to a free-body diagram to explain it. With the ball on the string, though, I don't have any other forces (we can assume the experiment is done in a vacuum, if needed) on the ball other than its weight and the tension on the string. Thus, I don't really see why a faster speed causes the ball to pull outward on the string harder than it would for a slower speed. This is where I was saying that the concept of inertia seems to be misapplied to explain this.

 

[PeroK]

Thus, I don't really see why a faster speed causes the ball to pull outward on the string harder than it would for a slower speed. This is where I was saying that the concept of inertia seems to be misapplied to explain this.

You're the one misapplying inertia by insisting it's a woolly concept separate from mass. If it's not mass, what is it? It can't be mass x acceleration, as that's force. The beauty of Newton's 2nd law is that all motion boils down to $F = ma$, a simple relationship involving three fundamental quantities. There is no room in this equation for "inertia", "resistance", "recalcitrance" or "temporary lassitude". Those are things exhibited by students, not moving objects!

 

[vela]

With the ball on the string, though, I don't have any other forces (we can assume the experiment is done in a vacuum, if needed) on the ball other than its weight and the tension on the string. Thus, I don't really see why a faster speed causes the ball to pull outward on the string harder than it would for a slower speed. This is where I was saying that the concept of inertia seems to be misapplied to explain this.

Why are you consistently ignoring the acceleration of the ball? When the ball moves faster around the circle, its velocity changes at a greater rate, which requires a bigger force to cause it.

 

[Drakkith]

This is a really helpful set of explanations. Much appreciated. I certainly can see that a block on a surface with friction would resist changes in motion based on the amount of friction AND its mass. I guess the question then becomes that I can easily see the friction force as causing this, and would look to a free-body diagram to explain it. With the ball on the string, though, I don't have any other forces (we can assume the experiment is done in a vacuum, if needed) on the ball other than its weight and the tension on the string. Thus, I don't really see why a faster speed causes the ball to pull outward on the string harder than it would for a slower speed. This is where I was saying that the concept of inertia seems to be misapplied to explain this.

How so? As you increase the ball's speed, the force required to keep it moving in a circle increases as well. So the resistance to motion hasn't changed, but the rate of change of its motion has. The same "resistance", but a greater rate of change, and hence a larger force.

 

[jds10011]

You're the one misapplying inertia by insisting it's a woolly concept separate from mass. If it's not mass, what is it? It can't be mass x acceleration, as that's force. The beauty of Newton's 2nd law is that all motion boils down to $F = ma$, a simple relationship involving three fundamental quantities. There is no room in this equation for "inertia", "resistance", "recalcitrance" or "temporary lassitude". Those are things exhibited by students, not moving objects!

100% in agreement. I began this discussion by saying I have often been told these concepts are explained by inertia, and yet, it doesn't seem to be consistent with "just mass" to me.

When I whirl the ball on the string faster, the ball pulls outward on the string harder. Given that it is not inertia that causes this additional "resistance", what is it? And yes, with the blindfold test I do believe we've established that I feel the same "resistance" in both the case of the added mass OR the case of the increased velocity.

 

[Drakkith]

There is no increased resistance in your example, there is just a greater force since the "rate of change" of the motion is greater. Increased force is not the same thing as increased inertia.

 

[jds10011]

There is no increased resistance in your example, there is just a greater force since the "rate of change" of the motion is greater. Increased force is not the same thing as increased inertia.

I agree 100% that inertia has not increased. Yet, the tension increases, just as it would if the mass (inertia) were increased rather than the speed. It appears that in the case of the increased mass, we would explain the tension increase by saying that the object has more of a tendency to continue in a straight line, and is therefore "straining at the leash" more than before. However, when we increase the speed instead, we again see the tension increase ("straining at the leash"), but we now seem to fall back on just saying, well, F=ma, so since a increased, F increased (and, yes, this is not disputed). In other words, we seem not to have a property of the object to ascribe its tendency to "strain at the leash" more in this situation. Are we saying that the mass SOLELY pulls outward on the string as an N3 reaction to being pulled by the string? Why is it ok to ascribe its outward pull on the string to a property of the object (inertia) ONLY if we aren't discussing why the outward pull increases when the rotation speed is increased?

This is what is bugging me -- let's go back to the gravitron (drum) ride. If I'm in the ride, smashing against the wall, and you ask me why I keep hitting the wall, it's OK to say "Well, I am in motion, so inertia means I have a tendency to keep moving in a straight line. The wall keeps getting in my way." Now, if you give me a heavy backpack to wear, and I'm hitting the wall harder, it's OK to say "See, now my inertia has increased due to the added mass. Now I have even more tendency to go in a straight line, so I smash into the wall harder." Yet, if instead of the backpack, the ride's rotation speed is increased, and again I start hitting the wall harder, it would clearly be incorrect to ascribe this to my (unchanged) inertia, despite the fact that the effect on me is the same -- I smash into the wall harder. If you now ask me why I'm smashing into the wall harder, must I say "actually, now it is the wall smashing me harder in an attempt to get me to revolve faster"? Can I not explain my action by any means other than as an N3 reaction?

 

[Drakkith]

You can certainly explain why you are forced into the wall harder by saying that your inertia has somehow increased, but I expect that this concept would turn out to be much more complicated to fully explain, much like how you can develop the fundamental laws of motion from a rotating frame of reference. You can do it, but it makes things so much more complicated and far less intuitive than our current laws, which are developed with inertial frames in mind.

 

[jds10011]

You can certainly explain why you are forced into the wall harder by saying that your inertia has somehow increased, but I expect that this concept would turn out to be much more complicated to fully explain, much like how you can develop the fundamental laws of motion from a rotating frame of reference. You can do it, but it makes things so much more complicated and far less intuitive than our current laws, which are developed with inertial frames in mind.

I'm not convinced this is a reference frame issue. This seems like the same thing as the ball on the string, which certainly pulls harder on the string in either the case of the increased mass of the ball or in the case where the ball's speed is increased. The force with which the ball pulls on the string appears analogous to the force with which the rider in the drum hits the wall. I don't see why this force (ball-on-string) isn't validly observed from the (inertial) reference frame of the person holding the string. Yet, increasing the mass of the ball resulting in an increased tension is OK to explain by saying the ball's inertia has increased, but increasing the speed of the ball resulting in an increased tension is not (and I don't think this is disputed). It seems that there should be an explanation for why the ball should behave in this way ("straining at its leash") that refers to an innate property of the ball in both cases.

 

[Drakkith]

I can understand why you'd think that, but I'm betting the explanation using Newton's laws is probably the easiest way to analyze the problem that is both self consistent and easy to generalize to many other situations. While I lack the experience and knowledge to show this, I know I've seen similar questions as yours before and they always end up being most easily explained by the standard method of Newton's laws.

Perhaps someone with more experience can show this.

 

[jds10011]

I can understand why you'd think that, but I'm betting the explanation using Newton's laws is probably the easiest way to analyze the problem that is both self consistent and easy to generalize to many other situations. While I lack the experience and knowledge to show this, I know I've seen similar questions as yours before and they always end up being most easily explained by the standard method of Newton's laws.

Perhaps someone with more experience can show this.

I don't dispute that Newton's laws solves the problem. What I'm saying is that I often hear inertia used as an explanation for something that clearly isn't inertia. I'm trying to get at what the actual physics is of the situation. For example, I have often been asked about dropping an object onto a table. Why should the normal force from the table be greater than the weight of the object (as it is when sitting on the table)? While it would be ok to say that due to the object's inertia, it tries to keep going, and therefore applies a large force to the table, it's a bit of a poor explanation as clearly if the object is dropped from higher up, the object hits the table harder, but its inertia is unchanged. Here, I actually have an almost passable (though hand-wavy) answer -- the object has some amount of structural integrity, as shown by it not flying to bits when hitting the table (which would probably render it unable to exert such a large force on the table). When it impacts the table at higher speeds, its structural integrity can be thought of as meaning that it undergoes greater internal stresses that cause it to exert a larger force on the table -- at a high enough speed, it will fly apart.

In the case of the ball on the string, I'm not sure what property of the object to which to ascribe the fact that it pulls harder on the string at faster speeds. Is it also its structural integrity? Is it all the bits of the object undergoing internal stresses and trying not to fly apart? Are these internal stresses greater at greater speeds, resulting in it applying a greater force to the string? If I had a terrifically strong string, at high speeds would the object fly apart rather than breaking the string, as it ordinarily would?

 

[jbriggs444]

I don't dispute that Newton's laws solves the problem. What I'm saying is that I often hear inertia used as an explanation for something that clearly isn't inertia. I'm trying to get at what the actual physics is of the situation. For example, I have often been asked about dropping an object onto a table. Why should the normal force from the table be greater than the weight of the object (as it is when sitting on the table)? While it would be ok to say that due to the object's inertia, it tries to keep going, and therefore applies a large force to the table, it's a bit of a poor explanation as clearly if the object is dropped from higher up, the object hits the table harder, but its inertia is unchanged. Here, I actually have an almost passable (though hand-wavy) answer -- the object has some amount of structural integrity, as shown by it not flying to bits when hitting the table (which would probably render it unable to exert such a large force on the table). When it impacts the table at higher speeds, its structural integrity can be thought of as meaning that it undergoes greater internal stresses that cause it to exert a larger force on the table -- at a high enough speed, it will fly apart.

In the case of the ball on the string, I'm not sure what property of the object to which to ascribe the fact that it pulls harder on the string at faster speeds. Is it also its structural integrity? Is it all the bits of the object undergoing internal stresses and trying not to fly apart? Are these internal stresses greater at greater speeds, resulting in it applying a greater force to the string? If I had a terrifically strong string, at high speeds would the object fly apart rather than breaking the string, as it ordinarily would?

It is not clear what you are actually asking about.

If you want to know how an object can be subject to (and, by Newton's third law, produce) a higher force when it goes around in a circle more rapidly, that's just Newton's second law. F=ma. Increase the a while holding the m constant and you have a higher F.

If you are asking how the same string attached to the same object can be subject to a higher force then yes, that's because seemingly rigid objects and seemingly inextensible strings deflect and stretch under stresses. The higher the stress, the greater the deflection. In some cases, a stable equilibrium can be achieved. The Christmas tree ornament hits the floor and stops. In some cases, the stresses are too much. The Christmas tree ornament hits the floor, shatters and a different equilibrium is attained by the resulting pieces.

 

[jds10011]

It is not clear what you are actually asking about.

If you want to know how an object can be subject to (and, by Newton's third law, produce) a higher force when it goes around in a circle more rapidly, that's just Newton's second law. F=ma. Increase the a while holding the m constant and you have a higher F.

If you are asking how the same string attached to the same object can be subject to a higher force then yes, that's because seemingly rigid objects and seemingly inextensible strings deflect and stretch under stresses. The higher the stress, the greater the deflection. In some cases, a stable equilibrium can be achieved. The Christmas tree ornament hits the floor and stops. In some cases, the stresses are too much. The Christmas tree ornament hits the floor, shatters and a different equilibrium is attained by the resulting pieces.

Newton's laws match/model the behavior patterns of objects in various situations. I am asking what is actually happening when I have a ball on a string and revolve it faster that causes the ball to pull harder on the string. Is it a fundamentally different thing that happens in comparison to substituting a more massive ball and keeping the same speed? In the case of the more massive ball, we're all willing to say (I think) that the ball has a property called inertia, which increases when its mass is increased. We then use this property to say that when the mass is increased, the ball has an increased resistance to being revolved on the string, and therefore exerts a greater force on the string. In the case of the faster speed with the same mass, we can't say it is the ball's inertia that has increased, because the mass has remained the same; yet, it still exerts a greater force on the string. I'm asking (1) what property of the ball is it that is responsible for the behavior of exerting the greater force on the string when the speed is increased? (I know it exhibits this behavior, and I know it is modeled by Newton's laws, by I don't feel that gets at the physical cause of the behavior.) And (2) Is this a fundamentally different property of the ball than what we are explaining by inertia in the case of exerting a greater force on the string when the mass is increased? After all, while mass is a clear property of the ball, inertia according to N1 is really a behavior model rather than an intrinsic property of the object. Are both of these really a consequence of the ball's structural integrity? Would a snowball be equally prone to flying apart rather than revolving whether its mass were increased or whether its speed of revolution was instead increased? I feel as if it would.

Separately, many folks seem to be saying that despite the fact that I have more trouble whirling the ball around in both cases (increased mass and increased speed), this doesn't mean the mass has more of a tendency to maintain straight-line motion in both cases (and thus preserving the argument that inertia is unchanged). I think I could use a bit more explanation on that point as well, but here I am probably just being dense.

 

[jbriggs444]

I am asking what is actually happening when I have a ball on a string and revolve it faster that causes the ball to pull harder on the string.

The ball is pulling harder on the string because you are pulling harder on the string. You are pulling harder on the string because you are trying to maintain the ball in a trajectory that has a higher acceleration.

 

[jds10011]

The ball is pulling harder on the string because you are pulling harder on the string. You are pulling harder on the string because you are trying to maintain the ball in a trajectory that has a higher acceleration.

Sure, but this seems like a cop-out. The ball has an ability to resist my pulling, even when I pull harder. There must be a physical explanation for the ball's end of the bargain. Just as when I was discussing dropping the object onto the table, I could have just said that the object hits the table harder from a higher height because the table uses a larger force to stop it, but this would not be very satisfying (despite being true). Or, in the case of the increased mass in the ball-on-string, we clearly point to the increased inertia of the ball to explain its pulling harder on the string, rather than just attributing its behavior to a reaction to you pulling (despite also being true).

This is why I was discussing the trick question of the small block on the frictionless surface a few posts ago -- If i swing a bat at a fairly small object on a frictionless surface, and I ask what happens if I apply 1000N with the bat to the object, does it really respond back with 1000N, the answer is actually I'm incapable of applying the 1000N in this manner, because the block will move out of contact with the bat when I have applied far less than 1000N. A significantly more massive block would be required (or, a significant amount of friction) in order for it to be able to resist a 1000N force while remaining in contact. N3 is not a guarantee that all objects respond with equal and opposite reaction forces in all scenarios -- there first has to be a physical reason that the force can actually be applied, and then it will be appropriately responded to as modeled by N3, but there will still be a physical explanation for the mechanism by which the reaction force is supplied.

Similarly, if I ask what happens if I use a 1000N pulling force on the string (and let's assume I've got a steel cable or something that will really allow me to use such a force without itself breaking), and I try to whirl around a snowball, I suspect the answer is also that I actually can't do this, because a snowball will fly to bits with a much smaller force than 1000N, and the reaction force will only be up to the amount of force I'm actually capable of applying. Yet, if I try the same trick with a metal ball bearing or something, I probably can apply the 1000N, and it probably can respond with the 1000N. So, is it a consequence of the ball's structural integrity that it successfully responds to me via N3?

 

[jbriggs444]

The ball has an ability to resist my pulling, even when I pull harder. There must be a physical explanation for the ball's end of the bargain

The wording, "resist my pulling" is problematic. The ball is a passive participant. It responds to your pulling by accelerating. The slowness of the resulting acceleration is given by a = F/m.

What sort of thing would you accept as a physical explanation? A little man with his finger on the string monitoring tension and yelling commands at the navigator to move in some pattern based on the observed force?

 

[Drakkith]

I am also unsure about what kind of answer you would accept. Inertia is a manifestation of an object's mass and describes the observation that an object with more mass requires a larger force to accelerate at the same rate as an object with less mass. Newton's laws are just about the most fundamental explanation you can find when it comes to describing motion. Things accelerate in response to forces, and the rate of acceleration depends on the amount of force applied and the object's mass.

The ball has an ability to resist my pulling, even when I pull harder. There must be a physical explanation for the ball's end of the bargain.

There is. It has mass and obeys Newton's laws. I feel you're reaching for some property that is already rolled into an object's mass/inertia.

Just as when I was discussing dropping the object onto the table, I could have just said that the object hits the table harder from a higher height because the table uses a larger force to stop it, but this would not be very satisfying (despite being true).

Yes, but the question would be why the force is larger, and that question would probably require looking at the microscopic interactions between the molecules and atoms of the object and the table, along with some other complicated analysis. An easier example would be two electrons moving towards each other. First with a slow velocity and again with a larger velocity. One can easily show why the repulsive force and the maximum acceleration increases with increasing closing velocity (it's because they get closer to each other in the latter example).

Similarly, if I ask what happens if I use a 1000N pulling force on the string (and let's assume I've got a steel cable or something that will really allow me to use such a force without itself breaking), and I try to whirl around a snowball, I suspect the answer is also that I actually can't do this, because a snowball will fly to bits with a much smaller force than 1000N, and the reaction force will only be up to the amount of force I'm actually capable of applying. Yet, if I try the same trick with a metal ball bearing or something, I probably can apply the 1000N, and it probably can respond with the 1000N. So, is it a consequence of the ball's structural integrity that it successfully responds to me via N3?

Technically you can apply almost any force imaginable if you do so for only a very small amount of time. But yes, the basic idea in this example is that the forces holding the snowball together are much weaker than the forces holding the ball bearing together. When the tension exceeds the forces binding the object together, the object begins to break apart.

That being said, in both cases Newton's third law applied. You can certainly apply 1000 N of force to your string when you have a snowball on the end of it. There's absolutely nothing stopping you from doing so. You'll just end up with pieces of snowball everywhere and a rapidly accelerating string. The reaction force from the string on your hand is exactly equal to whatever force you're applying to the string, regardless of what happens to the snowball or ball bearing.