I Galilean relativity in terms of homogeneity

[gionole]

TL;DR Summary

Landau's book

I have a question related to Landau's book. In that, he says:

If we were to choose an arbitrary frame of reference, space would be in- homogeneous and anisotropic. This means that, even if a body interacted with no other bodies, its various positions in space and its different orienta- tions would not be mechanically equivalent. The same would in general be true of time, which would likewise be inhomogeneous; that is, different in- stants would not be equivalent. Such properties of space and time would evidently complicate the description of mechanical phenomena. For example, a free body (i.e. one subject to no external action) could not remain at rest: if its velocity were zero at some instant, it would begin to move in some direc- tion at the next instant.

It is found, however, that a frame of reference can always be chosen in which space is homogeneous and isotropic and time is homogeneous. This is called an inertial frame. In particular, in such a frame a free body which is at rest at some instant remains always at rest.

As an example, I'd like to bring a car and the ball hung inside the car and we can look at it from 2 different frames of reference.

Frame of reference is me(I'm inside the car): If car moves with constant speed, nothing happens to the ball, but if car accelerates, ball starts to swing back or forward(depending on acceleration forward or backward). You could say that in acceleration mode, space inside the car is inhomogeneous, anistropic as ball behaves differently.

When we choose a frame or reference as the person outside the car, to him, wouldn't the ball inside car behave the same way as observed from my reference frame ? It's like things are kind of mixed up in my head and can't put them in order.

Would appreciate the complete, good explanation of Landau's thoughts(I think he is repeating Galillei's relativity, but still) in terms of my example.

 

[PeroK]

Would appreciate the complete, good explanation of Landau's thoughts(I think he is repeating Galillei's relativity, but still) in terms of my example.

It seems a slightly complicated way of putting things. The idea is that the underlying space is the same everywhere. Space, of course, isn't a vacuum and is filled with galaxies, stars, planets and other smaller objects. So, it's more of a theoretical point that there is no inherent difference in the space itself.

Likewise with time, there is no inherent difference between time today and time in the past or future. Systems evolve (galaxies, stars, planets, living organisms), but time itself (in some fundamental way) does not change.

The irony is that with General Relativity and modern Cosmology, this is not quite true. The expanding universe means that spacetime itself does change. But, while you are studying Newtonian physics, space and time and homogeneous and isotropic.

Also, about your example of the car:

In Newtonian physics, we have the concept of an inertial reference frame: where objects subject to no external force remain at rest or moving with constant velocity. These are "special" frames in which the laws of Newtonian mechanics apply directly: in particular, Newton's second law.

In any other frame (e.g. the frame of an accelerating car), you cannot simply apply Newton's second law. To do Newtonian mechanics, you must include so-called inertial or fictitious forces (to take account of the acceleration of the reference frame itself).

 

[gionole]

@PeroK

Newton's second law is F = ma.

These are "special" frames in which the laws of Newtonian mechanics apply directly: in particular, Newton's second law.

If we have inertial frame, then acceleration is 0, F = 0, so how does 2nd law apply ? It's pointless, no ?

In any other frame (e.g. the frame of an accelerating car), you cannot simply apply Newton's second law. To do Newtonian mechanics, you must include so-called inertial or fictitious forces (to take account of the acceleration of the reference frame itself).

If we imagine the ball hung in a car and write down F = ma (m is the mass of the ball). Are you saying that we can't apply F=ma because we don't know the force acting on it since it's fictitious ? If we look at it from a different frame(outside the car), how can we apply it then ?

 

[PeroK]

@PeroK

Newton's second law is F = ma.

Yes.

If we have inertial frame, then acceleration is 0, F = 0, so how does 2nd law apply ? It's pointless, no ?

There's often a debate about whether Newton's first law is necessary. Isn't is just a special case of the second law?

Newton's first law can be interpreted, as I mentioned above, as implying the existence of inertial reference frames.

My opinion, from looking at Newton's original work, is that the first law was so radical that he needed to emphasise it. It went against previous physical theory, so he wanted to spell it out before stating his second law.

If we imagine the ball hung in a car and write down F = ma (m is the mass of the ball). Are you saying that we can't apply F=ma because we don't know the force acting on it since it's fictitious ? If we look at it from a different frame(outside the car), how can we apply it then ?

You're confusing spatial locations with frames. The car's frame extends across all of space (and time). In the car's frame a lampost accelerates, but there is no force on the lampost. In that frame, you need to add a fictitious force to explain the acceleration of objects not physically bound to the car. A ball in the car might be another example. You have to be clear about what "real" forces are acting on something - and whether the real forces explain the motion. If they don't, then you must be using a non-inertial frame.

Note that I used the word "using", not "in" a frame. Everything is in all frames. A reference frame is not a thing, but a system of coordinates that you use to make measurements (of time and position).

 

[gionole]

@PeroK

You're confusing spatial locations with frames. The car's frame extends across all of space (and time). In the car's frame a lampost accelerates, but there is no force on the lampost. In that frame, you need to add a fictitious force to explain the acceleration of objects not physically bound to the car. A ball in the car might be another example. You have to be clear about what "real" forces are acting on something - and whether the real forces explain the motion. If they don't, then you must be using a non-inertial frame.

Q1: I don't think I'm. I said that in a car's frame, lampost accelerates for sure, but in order to write F=ma for the lampost, we need to know the force which we don't know that easily since the force is fictitious. So here, we don't know F and we don't know a(acceleration) as well, right ? but wouldn't lampost accelerate with the same magnitude of the car's acceleration but in opposite direction ? What is the problem that we're trying to solve here and we can't from the car's frame ? I think this is bugging me. Would you care to explain it a little bit further what the problem is here ? Is the exercise to define the motion of lampost and we can't do it because we don't know F ?

Q2: What I meant by "outside the car's frame" is that how is it a better frame(than car's frame) to deal with the problem in my Q1 ?

 

[Herman Trivilino]

If we have inertial frame, then acceleration is 0, F = 0, so how does 2nd law apply ? It's pointless, no ?

No. The inertial frame is a mathematical device. Objects can accelerate in that inertial frame.

 

[Dale]

If we have inertial frame, then acceleration is 0, F = 0, so how does 2nd law apply ? It's pointless, no ?

An inertial frame does not mean that acceleration is 0. An inertial frame means that any acceleration of an object is the result of a real force. I.e. it means there are no fictitious forces.

When we choose a frame or reference as the person outside the car, to him, wouldn't the ball inside car behave the same way as observed from my reference frame ?

No. The biggest difference is that in the non-inertial frame the point where the ball is attached is stationary, while in the inertial frame that point is accelerating. In the non-inertial frame the presence of the fictitious force is required in order to explain the motion of the ball. In the inertial frame the real forces are sufficient.

 

[A.T.]

If we have inertial frame, then acceleration is 0, F = 0

You are confusing the acceleration of a reference frame with the acceleration of a mass.

 

[PeroK]

@PeroKQ1: I don't think I'm. I said that in a car's frame, lampost accelerates for sure, but in order to write F=ma for the lampost, we need to know the force which we don't know that easily since the force is fictitious. So here, we don't know F and we don't know a(acceleration) as well, right ?

We never know force, mass or acceleration unless we measure them.

but wouldn't lampost accelerate with the same magnitude of the car's acceleration but in opposite direction ?

Yes.

What is the problem that we're trying to solve here and we can't from the car's frame ? I think this is bugging me. Would you care to explain it a little bit further what the problem is here ? Is the exercise to define the motion of lampost and we can't do it because we don't know F ?

The issue is that for the lampost the real force $F = 0$, but the acceleration is non-zero. Newton's second law is not directly obeyed in a non-inertial reference frame.

Q2: What I meant by "outside the car's frame" is that how is it a better frame(than car's frame) to deal with the problem in my Q1 ?

You can solve the same problem using difference frames of reference. If you choose a non-inertial frame you have to include the relevant fictious force(s).

 

[gionole]

An inertial frame does not mean that acceleration is 0. An inertial frame means that any acceleration of an object is the result of a real force. I.e. it means there are no fictitious forces.

No. The biggest difference is that in the non-inertial frame the point where the ball is attached is stationary, while in the inertial frame that point is accelerating. In the non-inertial frame the presence of the fictitious force is required in order to explain the motion of the ball. In the inertial frame the real forces are sufficient.

@Dale

As far as I understood, car frame reference is the non-inertial frame because lampost is accelerating, but no real force is acting on it. For the outside person, reference frame is inertial as you say and how does the person explain the motion of the lampost ? Since it's inertial, he should explain it caused by real force. What's the "real force" then ?

 

[PeroK]

@Dale

As far as I understood, car frame reference is the non-inertial frame because lampost is accelerating, but no real force is acting on it. For the outside person, reference frame is inertial as you say and how does the person explain the motion of the lampost ? Since it's inertial, he should explain it caused by real force. What's the "real force" then ?

In the ground frame of reference, the lampost is not accelerating.

 

[gionole]

In the ground frame of reference, the lampost is not accelerating.

I mean how ? I stand on the ground, I look at the car moving, and car accelerated which caused the acceleration of the lampost as it moved backward. Can you explain why I'd not see the lampost accelerating ?

 

[PeroK]

I mean how ? I stand on the ground, I look at the car moving, and car accelerated which caused the acceleration of the lampost as it moved backward. Can you explain why I'd not see the lampost accelerating ?

Because it's fixed to the ground. It can't even move in the ground frame!

 

[gionole]

Ah, by lampost, we mean different things. Let me re-repeat my question that I asked to Dale above. I meant ball hung in a car instead of lampost.

An inertial frame does not mean that acceleration is 0. An inertial frame means that any acceleration of an object is the result of a real force. I.e. it means there are no fictitious forces.

No. The biggest difference is that in the non-inertial frame the point where the ball is attached is stationary, while in the inertial frame that point is accelerating. In the non-inertial frame the presence of the fictitious force is required in order to explain the motion of the ball. In the inertial frame the real forces are sufficient.

As far as I understood, car frame reference is the non-inertial frame because ball is accelerating, but no real force is acting on it. For the outside person, reference frame is inertial as you say and how does the person explain the motion of the ball ? Since it's inertial, he should explain it caused by real force. What's the "real force" then ?

 

[PeroK]

I stand on the ground, I look at the car moving, and car accelerated which caused the acceleration of the lampost as it moved backward.

The acceleration of the lampost is relative to the car.

 

[PeroK]

As far as I understood, car frame reference is the non-inertial frame because ball is accelerating, but no real force is acting on it. For the outside person, reference frame is inertial as you say and how does the person explain the motion of the ball ? Since it's inertial, he should explain it caused by real force. What's the "real force" then ?

Please clarify the scenario:

What is the motion of the ball a) in the car frame and b) in the ground frame?

 

[gionole]

Please clarify the scenario:

What is the motion of the ball a) in the car frame and b) in the ground frame?

Well, by reading all of your answers, I understood that the inertial frame is when acceleration of the object can only be explained by "real force".

In the car frame, ball that is hung, accelerates because the car accelerates, but this acceleration can not be explained by real force as nothing external acted on the ball - i.e we got fictitious force.

Now, if you agree with me till now, then what do we call "the outside frame of reference"(imagine observer looks at car movement from outside - I think we can call this ground frame) ? We call it inertial frame(I derived this from Dale's answer), but we know in inertial frame, acceleration must be explained by the real force, not fictitious. So from this inertial frame, with what "real force" do you explain the ball's movement inside the car ?

 

[PeroK]

In the car frame, ball that is hung, accelerates because the car accelerates, but this acceleration can not be explained by real force as nothing external acted on the ball - i.e we got fictitious force.

The ball is accelerated by the string - that's a real force. If it wasn't attached to something inside the car, then it wouldn't move until something inside the car hit it.

 

[gionole]

The ball is accelerated by the string - that's a real force. If it wasn't attached to something inside the car, then it wouldn't move until something inside the car hit it.

Correct, but the real question is: why didn't we say that from car frame reference that ball was accelerated by the string and there must have been fictitious force?

 

[PeroK]

Correct, but the real question is: why didn't we say that from car frame reference that ball was accelerated by the string and there must have been fictitious force?

I think you need to draw a diagram of this scenario with the real forces marked.

If the ball is attached to the car, then in the car frame it is not accelerating. But, there is a measurable real force on it. And, again, Newton's second law is not obeyed: force but not acceleration. So, we add a fictitious force to the ball to cancel out the real force and we have restored Newton's second law: net force is zero and acceleration is zero.

 

[gionole]

Basically, from car frame, we say fictitious force happened, and acceleration of the ball was caused by that fictitious force(hence non-inertial frame), while from the road reference, acceleration of the ball was caused by real force(string) hence inertial frame.

Why do we say that from car frame reference ball was accelerated by fictitious force, but from road frame, by string force ? why couldn't we even have said ball was accelerated by string force even in the car frame ?

@PeroK

 

[PeroK]

Basically, from car frame, we say fictitious force happened, and acceleration of the ball was caused by that fictitious force(hence non-inertial frame), while from the road reference, acceleration of the ball was caused by real force(string) hence inertial frame.

Why do we say that from car frame reference ball was accelerated by fictitious force, but from road frame, by string force ? why couldn't we even have said ball was accelerated by string force even in the car frame ?

@PeroK

You're replying to these posts so quickly that you are clearly taking only a few seconds to think. I suggest you get a cup of coffee and have a good think about everything in this scenario.

 

[gionole]

I mostly understand what you mean and I will definitely start thinking, but before I do, could you quickly mention one thing ? you said: "If the ball is attached to the car, then in the car frame it is not accelerating", why is that ? If I'm in a car and car accelerated, ball will definitely move to the opposite direction. Why is this not an acceleration of the ball to me(person sitting in a car) ? @PeroK

 

[Nugatory]

If we have inertial frame, then acceleration is 0, F = 0, so how does 2nd law apply ? It's pointless, no ?

Not at all. We can use an inertial frame to analyze the motion of objects, including objects that are subject to forces, and Newton's second law is essential for that. You may be conflating the notions of "inertial frame" and "inertial motion" - objects may be moving inertially or not no matter what frame we are using to describe that motion and no matter whether that frame is inertial or not.

 

[Herman Trivilino]

I stand on the ground, I look at the car moving, and car accelerated which caused the acceleration of the lampost as it moved backward.

Do you really believe that a lamp post moves to a new location every time a car passes by!? The lamp posts in my neighborhood have remained in the same place since they were constructed decades ago.

 

[PeroK]

I mostly understand what you mean and I will definitely start thinking, but before I do, could you quickly mention one thing ? you said: "If the ball is attached to the car, then in the car frame it is not accelerating", why is that ? If I'm in a car and car accelerated, ball will definitely move to the opposite direction. Why is this not an acceleration of the ball to me(person sitting in a car) ? @PeroK

The ball accelerates initially until the string gets a chance to pull it along with the car. That's an irrelevant complication. It just means the real force on the ball acts slightly after the real force on the car.

 

[Herman Trivilino]

"If the ball is attached to the car, then in the car frame it is not accelerating", why is that ? If I'm in a car and car accelerated, ball will definitely move to the opposite direction.

If the ball is "attached" to the car we take that to mean the ball doesn't move relative to the car.

 

[PeroK]

The ball accelerates initially until the string gets a chance to pull it along with the car. That's an irrelevant complication. It just means the real force on the ball acts slightly after the real force on the car.

To add to this. If you want to set up a scenario where the ball is essentially a pendulum inside an accelerating car, then that's a little more complicated. The same principles apply. In an inertial frame, the motion of the ball is explained by the real forces from gravity and the string. In the frame accelerating with the car, you have to add the fictitious force to explain the ball's motion relative to the car.

Given your conceptual difficulties with things like lamposts bolted to the ground, I would leave the more complex scenarios until you have grasped the basics.

 

[Dale]

In the car frame, ball that is hung, accelerates because the car accelerates, but this acceleration can not be explained by real force as nothing external acted on the ball - i.e we got fictitious force.

In the car frame (assuming uniformly accelerating car) the ball does not accelerate. It hangs at rest at an angle. The tension should make it accelerate horizontally, but it does not. So the fictitious force must be pulling it backwards to prevent it from accelerating.

in inertial frame, acceleration must be explained by the real force, not fictitious. So from this inertial frame, with what "real force" do you explain the ball's movement inside the car ?

In the inertial frame the ball is accelerating horizontally. The real force is the tension acting on the ball.

Correct, but the real question is: why didn't we say that from car frame reference that ball was accelerated by the string and there must have been fictitious force?

In the car's frame the ball is not accelerating.

"If the ball is attached to the car, then in the car frame it is not accelerating", why is that ? If I'm in a car and car accelerated, ball will definitely move to the opposite direction.

One problem is that you are not clear about the setup you are proposing. I am assuming for simplicity that the car is accelerating uniformly. You have not stated clearly what acceleration profile you are considering.

In a uniformly accelerating car, in the car's frame, a ball hanging does not accelerate. It simply hangs there at an angle from the vertical. The ball is hanging from a pivot point, that pivot point is attached to the car so it does not move in the car's frame, so the distance to the pivot point does not change and in the case of uniform acceleration the angle does not change either. The effect of the fictitious force is to allow the ball to not accelerate while the tension is pulling it forward.

The rest frame of a non-uniformly accelerating car is MUCH more complicated to analyze. I think you should avoid it for now. I would not participate in that discussion as it is more effort than it is worth.

 

[gionole]

It seems like something is confusing to me, but I don't know what. If you can bear with me, I will explain how I understand the whole situation.

Car is moving with constant velocity 20m/s and then accelerates in uniform mode.

From ground frame: everything in the car was moving with the same 20m/s velocity and when car accelerated, the following happened: ceiling accelerated instantly, which caused the forward force(the force in the same direction as the car) to act on the upper point on the string, but force transfer through the string is not instantenous and there's a delay because of which the following occurs: the car moves with updated velocity, but ball itself still moves with old velocity(force didn't arrive yet through the end of the string where the ball is attached), so we got a difference in velocities - car moves faster than the ball which ends up in swing mode of the ball backwards. In reality, it's not truly that ball moves backward, it's just our perception, but in reality, it's just car moves faster than ball and thats why swinging perception of the ball occurs. We say this obeys in newton's law, because ball actually continued motion with the old velocity(if no force acted on it, object continues moving in the same speed and it's true due to force delay).

From the car frame: I got a problem here. I sit in a car. I understand that to me, when car moves with constant speed at first, everything is stationary - i look at ball or anything for that matter, and nothing moves. Now acceleration happened, but to me, ceiling is still stationary - to me, car/ceiling still feel like stationary. Now, if to me, if i feel that everything is still stationary, why did the ball swing back ? I think this is the scenario we try to explain and can't explain with the same logic as we did in ground frame. Since ceiling to me didn't accelerate, it couldn't have caused force in the tension the same way as in ground frame, but ball still swang backwards, so if no force acted on it, why did it move backwards ? It seems like newton's law broke - if no force, it should have stayed at rest or moving with the same speed, but the ball accelerated. The way you explain the swing of ball backwards is(i.e it's acceleration - because to me, it's not stationary during its backwards movement), some fictitious force must have happened.

Where would you say I'm wrong ? Would appreciate to point out the exact things that I wrongly said.

Thanks already so much for bearing with me. Physics in terms of intuitivity is lot harder than I imagined.

 

[PeroK]

From the car frame: I got a problem here. I sit in a car. I understand that to me, when car moves with constant speed at first, everything is stationary - i look at ball or anything for that matter, and nothing moves. Now acceleration happened, but to me, ceiling is still stationary - to me, car/ceiling still feel like stationary. Now, if to me, if i feel that everything is still stationary, why did the ball swing back ? I think this is the scenario we try to explain and can't explain with the same logic as we did in ground frame. Since ceiling to me didn't accelerate, it couldn't have caused force in the tension the same way as in ground frame,

The ceiling didn't accelerate in the car frame, but the ball did. I don't think it's a good idea to start talking about "cause and effect" in a non-inertial frame, because it creates confusion.

but ball still swang backwards, so if no force acted on it, why did it move backwards ?

It moved backwards because of the fictitious force on it. The same fictitious force that applied to everything else in the car and cancelled out the real "accelerating" force.

It seems like newton's law broke - if no force, it should have stayed at rest or moving with the same speed, but the ball accelerated. The way you explain the swing of ball backwards is(i.e it's acceleration - because to me, it's not stationary during its backwards movement), some fictitious force must have happened.

The fictitious force applies as soon as you use the accelerating reference frame.

Where would you say I'm wrong ? Would appreciate to point out the exact things that I wrongly said.

Thanks already so much for bearing with me. Physics in terms of intuitivity is lot harder than I imagined.

With non-inertial frames you need to rely more on solid technique and mathematics. Your inituition is letting you down and you are drifting from one misconception to another. I suggest you start using free-body diagrams. In an non-inertial frame you add the fictitious force to everything.

Note that, in fact, you can combine the fictitious force with the "real" gravity to get an overall virtual-gravity in the non-inertial frame. In the car frame this acts at an angle, down and backwards. The ball, therefore, exhibits the motion of a simple pendulum under this virtual gravitational force.

 

[gionole]

@PeroK

Q1. Was everything correct in ground frame analysis of mine ?

Q2:

The ceiling didn't accelerate in the car frame, but the ball did. I don't think it's a good idea to start talking about "cause and effect" in a non-inertial frame, because it creates confusion.

Isn't this what I said ? Could you re-read my "car frame reference" analysis ?

Q3:

It moved backwards because of the fictitious force on it. The same fictitious force that applied to everything else in the car and cancelled out the real "accelerating" force.

I asked the question: "why ball moved backwards" to convey the problem, but I answer it by myself in there. I think re-reading my analysis would be so appreciated.

 

[Dale]

Car is moving with constant velocity 20m/s and then accelerates in uniform mode.

This is precisely the non-uniform acceleration that I wished to avoid. Changing acceleration from 0 to a constant is non-uniform acceleration. I would do this as two separate scenarios

1) Car is moving with constant velocity
2) Car is accelerating uniformly

I am not getting involved in this one as it is, but good luck maybe others like @PeroK can help. There are too many complications for having a non-uniform acceleration that it is not worth it for me.

 

[gionole]

This is precisely the non-uniform acceleration that I wished to avoid. Changing acceleration from 0 to a constant is non-uniform acceleration. I would do this as two separate scenarios

1) Car is moving with constant velocity
2) Car is accelerating uniformly

I am not getting involved in this one as it is, but good luck maybe others like @PeroK can help. There are too many complications for having a non-uniform acceleration that it is not worth it for me.

hey Dale. You got it wrong. I didn't mean to bring non-uniform acceleration. The reason I brought 20m/s was I wanted to convey the idea that before accelerating, with constant speed of car, ball also was moving with 20m/s and when acceleration happened, my analysis of why ball didn't instantly receive the force due to force transfer delay caused the ball to continue moving with 20m/s and car with higher speed which caused the perception of the ball moving backwards. I don't want to complicate this even more, believe me. If 1) car is moving with constant velocity, I don't know what we need to discuss at all. I want to discuss uniform acceleration. I know you mean when car has a = 2 all the time which I agree, but as I. said I only brought 20m/s in the beginning to convey the speed difference in the ball/car due to delay. with uniform acceleration, my analysis should still be right as ball receives the updated speed at a delay due to force delay while car instantly increases speed by 2 hence swinging.

Having said that, would it be hard for you to let me know in your opinion what I've explained wrongly in my above analysis ? @Dale

 

[Dale]

I didn't mean to bring non-uniform acceleration. The reason I brought 20m/s was I wanted to convey the idea that before accelerating, with constant speed of car, ball also was moving with 20m/s and when acceleration happened,

When there is a "before accelerating" and a "when acceleration happened" then the acceleration is non-uniform. I don't want to deal with the transient mess of the swinging ball. Good luck.

would it be hard for you to let me know in your opinion what I've explained wrongly in my above analysis ?

Yes. That is why I don't want to do non-uniform acceleration. It is MUCH harder and provides no instructional benefit.

 

[gionole]

Thanks for the help. You don't have to answer, but I will still mention the below.

When there is a "before accelerating" and a "when acceleration happened" then the acceleration is non-uniform. I don't want to deal with the transient mess of the swinging ball. Good luck.

I didn't know that it was that complicated. That means my analysis of force delay thing is wrong. That's just great. Were you describing the case when the ball is already swung backwards ?

In the car frame (assuming uniformly accelerating car) the ball does not accelerate. It hangs at rest at an angle. The tension should make it accelerate horizontally, but it does not. So the fictitious force must be pulling it backwards to prevent it from accelerating.

I believe, you mean, looking at it from inside the car, the ball is stationary. It's tilted, but it doesn't move, just stays tilted.

In the inertial frame the ball is accelerating horizontally. The real force is the tension acting on the ball.

Wouldn't the ball still stay tilted at the angle all the time the same way here as well ? it's just from the ground frame, the tilted string ball for sure moves past the observer because the car is moving.

I think the distinction is, from looking outside(ground frame), we see the ball which moves, because the car moves and movement of the tilted string/ball is because there's acceleration on it by the tension. but from inside the car, we look at the ball, we see it's stationary and you say that if it's stationary, why is it tilted backwards ? in that case, ok, fictitous force causes this, but why do we even ask this question: "why the ball is tilted backwards" - it is because previous event caused it or the question we're asking is why does it stay tilted ? well if it's stationary, then it will remain at rest then until force acts on it so what's the correct question we should be asking that newtonian law fails to explain ?

 

[PeroK]

To be honest, this is not difficult. In an linearly accelerating reference frame you add a fictitious force to everything. The fictitious force represents a common acceleration equal and opposite to the "real" acceleration of the frame. That's it. You just pretend there is a second gravitational force acting on everything, and apply Newton's laws.

It's no more complicated than that.

 

[gionole]

To be honest, this is not difficult. In an linearly accelerating reference frame you add a fictitious force to everything. That's it. You just pretend there is a second gravitational force acting on everything, and apply Newton's laws.

It's no more complicated than that.

Yes, but I still have hard time intuitively. I'd love to ask you last favor.

In the above analysis(#30), I use the constant velocity and then acceleration to explain the backward motion of the ball in both of the frames. At least would be happy to know what exactly is wrong in that analysis. @PeroK you pointed out some things, but I believe I had already written them that way as you pointed out. check #32.

Then for uniform acceleration, from inertial frame, I get the idea. We see the ball which moves, because the car moves and movement of the tilted string/ball is because there's acceleration on it by the tension and tension is because the ceiling is accelerating, so ceiling -> tension -> ball. But ball stays tilted all the time in uniform acceleration. For non-inertial frame, I'm not sure what is the question I should be asking. Why do we say newton law doesn't hold true and we have to bring frictitious force ? what is it that doesn't hold ? is it that because from inside the car, tilted string/ball seems stationary, but we should be able to explain why it's tilted backward - is this the question ? or what else ?

 

[Dale]

Were you describing the case when the ball is already swung backwards ?

Yes. All of my posts were assuming a uniformly accelerating frame. The acceleration is constant over time, and the ball is not swinging but just hanging at an angle.

You can easily draw a free body diagram for this scenario.

 

[gionole]

Yes. All of my posts were assuming a uniformly accelerating frame. The acceleration is constant over time, and the ball is not swinging but just hanging at an angle.

You can easily draw a free body diagram for this scenario.

@Dale And what’s the question we ask that netwonian laws fail to answer in car frame ? Is it :’why the ball and string stay tilted if no force acts on it ?’

I understand that from car frame, ceiling does not accelerate which means it does not give force to the string which means there is now no tension force. Also, the ball itself is tilted and is stationary. Since there is no tension force, what causes it to stay tilted and stationary ? and thats where we bring fictious force to say that this fictitous force is what holds it stationary and tilted ?

And in ground frame, we didnt have this problem because we did not have to explain why the ball stays tilted/stationary. In there, we had to explain why the ball is accelerating and we explain it such as ‘tension causes it to accelerate’.

Am I right ?

 

[Dale]

what’s the question we ask that netwonian laws fail to answer in car frame ? Is it :’why the ball and string stay tilted if no force acts on it ?’

No. The question is why does it not accelerate due to the real forces acting on it? Draw the free body diagram and you can see the real forces are unbalanced. And yet it remains stationary in the car’s frame.

it does not give force to the string which means there is now no tension force

The tension is a real force. It is directly measurable with a force gauge. It must exist in all frames, inertial or non-inertial.

 

[Ibix]

Here's a diagram of the forces on the ball while the car is under steady acceleration (and any oscillations have died out). There is tension $T$ at angle $\theta$ to the vertical and weight $mg$

Obviously we can resolve components and there's an unbalanced force $T\sin\theta$ to the left, so the ball ought to accelerate to the left.

In an inertial frame that's fine. The ball does indeed accelerate to the left along with the car.

In the non-inertial frame where the car is at rest, we want to say that the ball is not accelerating. But there's still an unbalanced force! The solution is to introduce a so-called "inertial" or "fictitious" force pointing to the right. Its magnitude is $m\alpha$, where $\alpha$ is the acceleration of the car, and this will turn out to be equal to $T\sin\theta$ if you work through the maths. Thus the pendulum hangs at an angle to the ceiling because the "inertial force" balances the horizontal component of the tension.

Mathematically, what happens is that in the inertial frame we have a net force causing an acceleration: $F_\mathrm{net}=m\frac{d^2x}{dt^2}$. Then we switch to a coordinate frame accelerating to the left. If its coordinate is denoted $x'$ then we have $x'=x+\frac 12\alpha t^2$ where $\alpha$ is the car's acceleration. That means that $\frac{d^2x'}{dt^2}=\frac{d^2x}{dt^2}+\alpha$ and therefore $F_\mathrm{net}=m\frac{d^2x'}{dt^2}-m\alpha$. But if we want to keep our intuitive idea that any unbalanced force causes an acceleration we can't have that second term on the right. So we just shift it over to the left and call it a "inertial force": $F'_\mathrm{net}=m\frac{d^2x'}{dt^2}$, where $F'_\mathrm{net}=F_\mathrm{net}+m\alpha$ adds the inertial force to the real ones.

 

[gionole]

Thanks so much. Now it is finally clear. Though, alternatively, I could now ask that in an inertial frame, why the string/ball stays tilted if they are accelerating and if Tsin0 acts as horizontal force in the car’s movement direction? Why is it not getting back to complete equilibrium position ?

In a non inertial frame, I get it, you say that horizontal force is balanced out by fictious and it stays tilted. But how do you explain it in an inertial frame ? The only explanation I could come up with car accelerates faster than ball due to force delay transfer so car’s speed increases faster than ball’s and because of that ball stays tilted/behind. Is not this the right way to look at it ?

 

[Ibix]

Why is it not getting back to complete equilibrium position ?

Because both the car, to which the top of the string is attached, and the ball are always moving at the same (increasing) speed. That's the point of the equilibrium position - it's where the horizontal component of $T$ causes the ball to accelerate at the same rate as the car. So it never catches up nor falls behind, and the string stays at that angle.

 

[PeroK]

Thanks so much. Now it is finally clear. Though, alternatively, I could now ask that in an inertial frame, why the string/ball stays tilted if they are accelerating and if Tsin0 acts as horizontal force in the car’s movement direction? Why is it not getting back to complete equilibrium position ?

In a non inertial frame, I get it, you say that horizontal force is balanced out by fictious and it stays tilted. But how do you explain it in an inertial frame ? The only explanation I could come up with car accelerates faster than ball due to force delay transfer so car’s speed increases faster than ball’s and because of that ball stays tilted/behind. Is not this the right way to look at it ?

I would question to your whole approach to this subject. You seem to be treating physics as an inituitive, philosophical subject, where wordy arguments are used to justify things. Perhaps that's Landau's style. I don't know his books. Perhaps someone who does can comment.

We have 44 posts on this thread, where you have repeatedly posted the same questions over and over. You have never used any sort of formal analysis of these questions, by using a free-body diagram or by doing specific calculations. This is a bad sign.

If you learned mechanics from, say, Kleppner and Kolenkow, you would be doing physics by drawing free-body diagrams and using force-based calculations. If you attach a string to a ball and pull the string, then a tension in the string arises and the ball gets pulled. That would be an "obvious" starting point for most students, I would say. I wouldn't expect a page of supporting text to explain what's going on.

That said, if forces and motion are really something of a mystery to you, then classical mechanics is going to be tough to learn.

It's possible that you should put to one side the whole concept of non-inertial reference frames until you have a grasp of Newton's laws and have solved all different problems, static and dynamic, involving forces, acceleration and equilibrium - using inertial reference frames.

 

[vanhees71]

In this case the description of motion in an accelerated frame is pretty simple, if we assume the car moves along a straight line, i.e., that its motion is described by an arbitrary function $\vec{x}(t)=x_0(t) \vec{n}$ with $\vec{n}=\text{const}$ wrt. an inertial reference frame. An observer at rest wrt. the car will then describe the motion of an arbitrary body by the position vector $\vec{r}'(t)$. The position vector with respect to the inertial frame then obviously is
$$\vec{r}(t)=x_0(t) \vec{n} + \vec{r}'(t),$$
and in this frame of reference the usual Newtonian equations of motion hold true,
$$m \ddot{\vec{r}}=\vec{F}(\vec{r})$$
Now
$$m \ddot{\vec{r}}=\ddot{x}_0(t) \vec{n} + \ddot{\vec{r}}' \; \Rightarrow \; m \ddot{\vec{r}}'=\vec{F}(x_0 \vec{n}+\vec{r}')-m \ddot{x}_0(t) \vec{n},$$
i.e., in addition to the "true force" you have an inertial force (what some posters here call fictitious force, which can be kind of misleading, because these inertial forces are not fictitious, but the point is that you can transform them away by going the way backwards to the original inertial frame, and they are thus not "true forces" in the sense of some real interactions like the electromagnetic or gravitational interaction).

 

[A.T.]

I could now ask that in an inertial frame, why the string/ball stays tilted if they are accelerating and if Tsin0 acts as horizontal force in the car’s movement direction? Why is it not getting back to complete equilibrium position ?

It must accelerate with the car, so it needs the horizontal component of T, which is 0 if the string is vertical.

 

[gionole]

Everything looks clear now. I want to thank everyone included in the discussion for your big help. It's not hard to be honest and drawing free-body diagram is easy. The whole idea of physics being hard is sometimes it's a game of playing words. Well, I understood Lagrange and euler quicker than this, so it's full of surprises. Some topics are easy, some are not. It's all relative to the person.

 

[Dale]

I could now ask that in an inertial frame, why the string/ball stays tilted if they are accelerating and if Tsin0 acts as horizontal force in the car’s movement direction? Why is it not getting back to complete equilibrium position ?

Draw the free body diagram in the non-tilted position. What is the acceleration?

It's not hard to be honest and drawing free-body diagram is easy.

Then why didn’t you do that? FYI, it is kind of rude to ask people to answer 50 posts and then say “it’s easy”. Please start your next question with your best personal effort, particularly when the important part is easy for you.

 

[gionole]

Draw the free body diagram in the non-tilted position. What is the acceleration?

Then why didn’t you do that? FYI, it is kind of rude to ask people to answer 50 posts and then say “it’s easy”. Please start your next question with your best personal effort, particularly when the important part is easy for you.

Well, Dale, you misunderstood my part. The easy part I meant was about free body diagram. I drew it before even I asked the question, but the hardship I had was somehow I couldn't make myself sure why acceleration in car frame of the ball was 0 and everything just added it up to this and I couldn't even distinguish what i understood and what not.

I don't think anything that I have done is rude. The physics sometimes is a play of words and once I understood all words in a good way, then I finally understood. Drawing free body diagram is again easy for this problem and I needed all of your help regarding the mixed up things in my head which you all guys solved it and I now get it.

Thanks again for you help