# Why do we need Newton's First law? And how the First law works?

This must have been asked before. But never mind here I ask it again to convince myself.

Newton's Second law says $\vec{F}$ = m$\vec{a}$
Now if we put $\vec{F}$ = 0 here we get $\vec{a}$ = 0 which is Newton's First law.
So why do we need to state First law as a separate law?

Before I asked this I did little bit of searching. And what I got is - First law is necessary to define the Inertial reference frame on which Second law can be applied.

But why can't we just use Newton's second law to define Inertial frame? We can say $\vec{F}$ = m$\vec{a}$ is the Second law. So if $\vec{F}$ = 0 but $\vec{a}$ $\neq$ 0 (or vice versa) then the frame is non inertial.

One can say (can one?), we cannot apply Second law to define Inertial reference frame because the Second law in valid only in Inertial frame. Thus unless we know in advance that a frame is Inertial, we cannot apply the Second law.
But then why this is not a problem for first law? We don't need to know in advance if a frame is inertial to apply first law because we take First law as definition of inertial frame. Similarly, if we take Second law as the definition of Inertial frame, it should not require to know if a frame is Inertial or not to apply Second law (to check if the frame is inertial).

So, what's the reason for First law to exist?

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A.T.
How does Newton's First law defines Inertial frame?
An "inertial frame" is a frame where any object without external forces either remains at rest or continues to move at a constant velocity.

sophiecentaur
Gold Member
I think Newton's First Law is very relevant because of its historical context. Before he formalised it, people thought that things naturally slow down (as they do - but for a good reason) yet could not square this with what went on with astronomical objects. N1 is a statement that unifies what goes on down here and up there. Well worth writing, IMO.

An "inertial frame" is a frame where any object without external forces either remains at rest or continues to move at a constant velocity.
Thanks but I know what an Inertial reference frame is. But why it's the First law which is credited for the definition of Inertial frame?

I think Newton's First Law is very relevant because of its historical context.
So Newton's first law exists as a separate law just for its historical context?

olivermsun
Thanks but I know what an Inertial reference frame is. But why it's the First law which is credited for the definition of Inertial frame?
Well, most basically because there are other frames in which the First law does not hold. That is, an object is observed to accelerate in the absence of any force defined in that frame.

So Newton's first law exists as a separate law just for its historical context?
Or you could say that it tells you the exact set of conditions under which the second law is claimed to be true.

sophiecentaur
Gold Member
Thanks but I know what an Inertial reference frame is. But why it's the First law which is credited for the definition of Inertial frame?

So Newton's first law exists as a separate law just for its historical context?
It will be a long time, if ever, before they stop teaching Newtonian Mechanics as a first step for students and I don't thing that the historical context should be neglected. When you get down to it, N1 is only a special case of N2 but it's still worth keeping 'if only' for the historical context.
It's easy to forget that the notion of a 'frame of reference' is quite a struggle for a beginner. Using an assumed Earth frame for your first mechanics experience is surely acceptable.

UltrafastPED
Gold Member
Thanks but I know what an Inertial reference frame is. But why it's the First law which is credited for the definition of Inertial frame?
Newton's First Law of Motion also provides a physical test; if there are no net forces, then one sees inertial motion. If one does not see inertial motion, then there must be unbalanced forces.

So Newton's first law exists as a separate law just for its historical context?
Newton's version is an improvement upon the work of Galileo - see "Two New Sciences" - and Descartes. But the first law is the "setup" for the application of Newton's Second Law of Motion.

Some modern presentations prefer to start with symmetry principles; when this is done the "laws of mechanics" are expressed in much different form. This is convenient when the subject is analytical mechanics, i.e., Lagrangian and Hamiltonian mechanics.

Newton's First Law of Motion also describes the motion in the _absence_ of net forces: it continues as before (in this inertial reference frame), moving in a straight line, with unchanged speed.

This is why it is called the Law of Inertia. The expression "F=ma" is consistent with this. Newton constructed geometric proofs from the laws and definitions provided; being one of the world's great mathematicians and geometers, he would not have introduced an axiom or law which he felt to be unnecessary.

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Newton's First Law of Motion also provides a physical test; if there are no net forces, then one sees inertial motion. If one does not see inertial motion, then there must be unbalanced forces.
But can't the Second law provide the same thing?

But why can't we just use Newton's second law to define Inertial frame? We can say $\vec{F}$ = m$\vec{a}$ is the Second law. So if $\vec{F}$ = 0 but $\vec{a}$ $\neq$ 0 (or vice versa) then the frame is non inertial.

Nugatory
Mentor
I think Newton's First Law is very relevant because of its historical context. Before he formalised it, people thought that things naturally slow down (as they do - but for a good reason) yet could not square this with what went on with astronomical objects. N1 is a statement that unifies what goes on down here and up there. Well worth writing, IMO.
This point about historical context is important. It is not at all unusual for the modern formulation and presentation of a theory to be very different (mathematically cleaner, more crisply formulated axioms, all neatly trimmed with Ockam's razor) than the stumbling path that that the pioneers followed. It's a lot easier to plot a direct path when you already know your destination. (Both quantum mechanics and special relativity also followed this pattern).

UltrafastPED said:
But the first law is the "setup" for the application of Newton's Second Law of Motion.
It is, and in the 17th century that setup was much more necessary than it is in the 21st century. In the 350 years in between, we've come to accept the second law to such an extent that it feels natural to start with it and watch the first law fall out of the ##F=0## case, rather than following Newton's path. Back then... not so much.

Although science and the history of science are deeply entwined (you cannot study the latter without understanding the former; and although you can use physics to solve problems without knowing the history, it is very difficult to make new contributions to physics without knowing the history) they are different disciplines.

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• 1 person
But the first law is the "setup" for the application of Newton's Second Law of Motion.
It is, and in the 17th century that setup was much more necessary than it is in the 21st century. In the 350 years in between, we've come to accept the second law to such an extent that it feels natural to start with it and watch the first law fall out of the F = 0 case, rather than following Newton's path. Back then... not so much.
I see that historical context is really crucial here. But then just to be sure, can we say that Newtonian mechanics can be formulated by using just his Second and Third law? ...While the First law is a reminder of the intellectual leap that had to be taken at Newton's time.

Hmm... Still, a great mind like Newton obviously noticed that his first law is a special case of his second law. But he kept it as a separate law anyway.

Well, so the first law stays. :tongue2:

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D H
Staff Emeritus
You're looking at it wrong, Adjoint. Just because you get a constant velocity from the second law with zero net force does not mean the first law is a special case of the second. The first law stands by itself. The other two laws must necessarily be consistent with that first law (and they are). It is that need for consistency that makes the first law appear to be unnecessary.

The modern view is that the Newton's first law establishes the context in which Newton's other two laws are valid. A rotating observer can invent fictitious forces to make Newton's second law appear to be valid, but now there's a new problem: There are no equal but opposite forces on some other object to those fictitious forces.

Thanks DH.

In the example you gave - I see that the inconsistency appears because the reference frame in non-inertial. That was certainly your point, right?
But I am not completely clear how the First law (and only the first law) can rescue us from here?

As far as I understand, the first law would say, the particle here (in a rotating frame) is changing its velocity or accelerating without any force acting on it. So the reference frame in non-inertial.

But cant the second law say the same thing? The second law can say, well here a ≠ 0 but F = 0, so this is a non-inertial frame.

I don't understand why an observer would invent a fictitious force when he applies the second law but won't invent a fictitious force when he applies the first law?

Would you explain a bit more please?

olivermsun

How would you determine that F = 0? Normally you would measure m and a and infer F, right? So if you're in a rotating frame and you measure a nonzero a for an object which is actually in inertial motion, then what? Is there an F acting on that body?

sophiecentaur
Gold Member
You're looking at it wrong, Adjoint. Just because you get a constant velocity from the second law with zero net force does not mean the first law is a special case of the second. The first law stands by itself. The other two laws must necessarily be consistent with that first law (and they are). It is that need for consistency that makes the first law appear to be unnecessary.

The modern view is that the Newton's first law establishes the context in which Newton's other two laws are valid. A rotating observer can invent fictitious forces to make Newton's second law appear to be valid, but now there's a new problem: There are no equal but opposite forces on some other object to those fictitious forces.
Just imagine a humanoid society that had evolved on a massive wheel, set in motion by some previous, extinct civilisation. They develop 'a Mechanics'. Whilst it would get to the same stage as ours, eventually, would they have got there (or not) with their own version of N1, on the way?

olivermsun
On a massive wheel, like... the rotating Earth? A.T.
An "inertial frame" is a frame where any object without external forces either remains at rest or continues to move at a constant velocity.
Thanks but I know what an Inertial reference frame is. But why it's the First law which is credited for the definition of Inertial frame?
The definition above is just a reformulation of the First law.

The definition above is just a reformulation of the First law.
You are pulling me back to my first question. I understand that the first law gives the definition of Inertia. What I don't understand is why we can't say that the second law does the same?

olivermsun

How would you determine that F = 0? Normally you would measure m and a and infer F, right? So if you're in a rotating frame and you measure a nonzero a for an object which is actually in inertial motion, then what? Is there an F acting on that body?
For me (in a non inertial frame) it would seem so... Well now I am asking myself that how can I identify a frame as non inertial if I am in that frame? Certainly I will feel so; because of my inertia. But really, if I am in a non inertial frame, how can I tell that if an object is moving because a force is acting or because it is in a non-inertial frame?

EDIT: Provide the answer please. Last edited:
sophiecentaur
Gold Member
On a massive wheel, like... the rotating Earth? No. I meant like Arthur C Clarke's in 2001. :tongue: i.e. where the rotational forces play a big part in their everyday lives. It took a long time for anyone to measure any rotational effect on Earth.

But really, if I am in a non inertial frame, how can I tell that if an object is moving because a force is acting or because it is in a non-inertial frame?
Use the third law.

I am sorry to say, but it seems like I don't know enough about inertial and non-inertial frames and rotational motion to continue following this discussion properly (though I started this thread )
For example: I cant simply figure out How one person in a non-inertial frame can measure force on an object properly? (I mean avoiding fictitious forces.)

But thank you all for your replies. Meanwhile I should go and do some study! To use the third law, I think you mean, I need to find if that accelerating object is giving any reaction force...?

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To use the third law, I think you mean, I need to find if that accelerating object is giving any reaction force...?
Yes, if there are forces without counter forces than the frame of reference is non-inertial (or you missed interactions).

Suppose I put a ball on a table and the ball starts moving. It's certainly moving without a counter force. So I decide my frame is non inertial.
But what if there was actually a counter force. Suppose someone with an almost invisible string attached to the ball was moving it around?
So unless I can be sure about all the forces acting on the ball and by the ball, how can I tell if the frame was inertial or not?
And in a certain situation it might not be possible for me to make sure about all the forces acting on and by an object.

As you said too -
or you missed interactions
It's likely to miss one or two interactions. So in practical case how can we be sure if a frame is inertial or non-inertial?

Also it seems like Newton's first law is not enough to check for an Inertial frame. Because to exclude the fictitious forces We also need the Third law. What's happening?

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Jano L.
Gold Member
So why do we need to state First law as a separate law?
Apart from the issue of setting up context for the 2nd and 3rd law, the 1st law stands apart of the 2nd law also for another reason: in its original formulation, apart from being consistent with 2nd law, it says also something in addition that cannot be derived from the 2nd law.

How so?

Let the position of the particle at time ##t_0## be ##x_0##. Now imagine the force ##F## is a function of position ##x## only and ##F(x_0) = 0##. What will happen to the particle put at ##x_0##?

Based on the 1st law, we conclude the particle will stay at ##x_0## forever, because there is "no impressed force to compel the particle to change its state of rest".

Based on the 2nd law only, we cannot make such conclusion. This is because in its modern formulation, the 2nd law is this statement: mass times second derivative of position equals net force impressed:

$$m\frac{d^2 x}{dt^2} = F.$$

This equation connects force only to the second derivative of the position and disregards any other characteristics of motion. Due to this, in this case it is not sufficient to determine what will happen to the position of the particle: depending on the function ##F(x)##, there may be infinity of solutions distinguished by the time ##t_1## the particle begins to move.