Inertial and non-inertial frames

Question

Firstly, I want to say that I realise that there are a lot of questions on inertial and non-inertial frames of reference, but the question I had in mind, no one seemed to have asked.

I've just started to study about Newton's Laws of Motion, and I was studying inertial and non-inertial frames. In the book I was studying, they gave an example of a lift's cord breaking, thus causing the lift to fall. Now, the lift had a person and a lamp inside it, and so the person looked at the lamp, saw that it was not accelerating, and thus concluded that $a=0$ and hence $F=0$. So, the tension in the string of the lamp balanced the weight of the lamp, and we got $T=mg$.

Then, the book said that a person standing on the ground saw the lift falling, and concluded $a=10$, hence $F$ is not $0$. Hence, the book concluded, one frame was a bad frame, and one should not apply Newtons First Law on it.

But, this is what I don't understand. Why was the person in the lift in a bad frame? He applied Newtons Frist Law correctly and got the correct answer. With respect to him, the lamp was at rest, and so he concluded correctly that $F=0$. The person has no way of knowing that the lift had in fact broken and was accelerating. To him, it was at rest.

Then, the book went on to say that the Earth was approximately an inertial frame. Again, what's the point? If tomorrow, scientists found out that the entire universe is accelerating at $1000m/s^2$, who would care? We cant notice it, and to us, the universe is at rest. Hence, to us, Newton's First Law holds.

Any answer would be appreciated. I realise that I'm wrong in by thought process, but I just wanted to make sure everyone understands my reasoning.

Answer 1

The non inertial frame is not really "bad." Sometimes, using a non inertial frame is quite useful. The important thing is to realize you have fictitious (pseudo) forces in it.

Let's consider the free falling elevator.

I) Inertial frame. Pretty simple; the net force is simply the force of gravity. $$\Sigma F_y = mg$$

II) Non inertial frame moving with elevator. From the point of view of whoever is in the elevator, everything is at rest. Since gravity is applied, there must be a force countering it such that there is no acceleration: $$\Sigma F_{y, \; \rm non-inertial} = mg - F$$

Since $\Sigma F_{y, \; \rm non-inertial} = 0$, we get that $$F=mg$$

So in case II), we get the same result as in case I), it's just we must consider the effects of the fictitious force. You may now ask, "Why is the force $F$ fictitious?" A: Because it only shows up in the non inertial frame, there is no "equal and opposite reaction to it," so by Newton's 3rd law, it's not a real force. You only feel this force in non inertial frames of reference.

Answer 2

You are correct, applying Newton's laws in the frame of the lift is valid. In fact it is valid whether the lift is hanging still or freely falling. The reason is, it is customary to fix which forces are present, to conserve validity of Newton's laws in the local region (it can't be done globally).

If the lift is hanging still, gravity force is present in lift's frame and the person inside has to take it into account when applying the second law to bodies in the lift. The first law cannot really be verified in the lift, because all bodies with no exception experience external force due to gravity, but the first law is still assumed valid in the sense that if gravity force is exactly balanced by other force (thrust force due to propeller, jet engine, or forces of mechanical contact or cord tension), the body will move rectilinearly.

If, on the other hand, the lift is freely falling, no gravity force can be experienced by observer or instruments measuring motion of bodies inside the lift. In lift's occupant description, no gravity force is introduced and bodies move rectilinearly. First law then does apply in that frame to all bodies inside. We preserved the law by not introducing the gravity force into our considerations.

There is an idea that gravity force is "definite" or "real" and thus in the second case, the lift occupant should include it in their description and conclude that Newton's laws are not valid in their frame, since gravity is present but it is doing nothing inside. But this idea does not work well, because the "real gravity force" of ours is completely arbitrary vector, as can be seen by considering what direction and magnitude it has when judged from various existing reference frames (centered in celestial bodies, the frames must not rotate). Yes, in Earth's frame there is single definite gravity force acting on the lamp, but in Moon's frame the gravity force is different, and in Sun's frame it is still different from those two. And in freely falling lift's frame, it is zero.

In other words, mutually accelerating frames, such as those centered to celestial bodies, can be considered local inertial frames for the purposes of motions local to those bodies. However, these reference frames are not global inertial frames. Distant bodies (other planets) do not behave in accordance with Newton's laws in those frames.

Global inertial frame would mean that there is a group of bodies that are all very far from other heavy bodies, such as some solar system, and we can associate inertial frame to that group in the sense, all motions are in agreement with 2nd Newton's law that takes into account all gravity forces. In this global inertial frame, the group center of mass moves rectilinearly. But even this is arbitrary, because there may be still a larger group of bodies with its own inertial frame. So the inertial frame to use for analysis is always chosen from infinity of options, usually in the simplest possible way that is accurate enough for the purpose. For launching artificial satellites, frame associated with system Earth-Moon is inertial enough. For launching probes to Mars, Sun and Mars and possible Jupiter have to be taken into account. And so on, the bigger the region of inertiality needed, the more bodies have to be included in the definition of the inertial frame.