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

1. Jan 2, 2017

### 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!

2. Jan 3, 2017

### Orodruin

Staff Emeritus
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.

3. Jan 3, 2017

### vela

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

4. Jan 3, 2017

### jds10011

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.

5. Jan 3, 2017

### Orodruin

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

6. Jan 3, 2017

### jds10011

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.

7. Jan 3, 2017

### Drakkith

Staff Emeritus
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.

8. Jan 3, 2017

### PeroK

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.

9. Jan 3, 2017

### Orodruin

Staff Emeritus
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 $F = ma$.

10. Jan 3, 2017

### jds10011

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.

11. Jan 3, 2017

### Orodruin

Staff Emeritus
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.

12. Jan 3, 2017

### jds10011

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.

13. Jan 3, 2017

### PeroK

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.

14. Jan 3, 2017

### jds10011

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.

15. Jan 3, 2017

### PeroK

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!

16. Jan 3, 2017

### vela

Staff Emeritus
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.

17. Jan 3, 2017

### Drakkith

Staff Emeritus
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.

18. Jan 3, 2017

### jds10011

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.

19. Jan 4, 2017

### Drakkith

Staff Emeritus
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.

20. Jan 4, 2017

### jds10011

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?