# Newton's Law of motion

## Main Question or Discussion Point

Hi,

Newton's 2nd Law can study the motion of the objects in the case of zero net force:

F = ma

if F = 0 then a = 0 then v = constant, then the object is either at rest (v=0) or moving in uniform rectilinear motion. (Bold quantities are vectors).

What is the point of Newton's first law as it is nothing more than a special case of Newton's 2nd Law?

I am sure there is a good reason that Newton himself stated his first law as a separate law, but I can not see this reason.

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But Newton didn't say that! Your statement is a modern rearrangement used for calculative convenience.

The alteration of the motion is ever proportional to the motive force
In modern terms a = kF where k = 1/m is the constant of proportionality.

Newton's First law tells us that

A body continues in its state of rest or uniform motion in its right line unless it is compelled to change its state (of motion) by forces impressed upon it.
Note this does not require the proportional relationship spelled out the the second law.

What would happen if we ever found an alteration of motion that had some other relationship than direct proportionality to the forces impressed?

What is the point of Newton's first law as it is nothing more than a special case of Newton's 2nd Law?
The first law is the qualitative definition of force: It is the reason for a change of the state of motion.

The second law is the quantitative definition of force: It is proportional to the change of momentum. (Today the constant of proportionality has been set to 1.)

What would happen if we ever found an alteration of motion that had some other relationship than direct proportionality to the forces impressed?
We already found it in non-inertial frames of reference and nothing unusual happened.

OK, seems good.

If we have a book on a table at rest. The book is submitted to two forces whose net is null.
What do we apply here to conclude that table at rest, Newton's first or second law?

russ_watters
Mentor
Neither. At rest and in motion are results of a choice of reference frame, arbitrarily selected before you start to do any analysis.

Neither. At rest and in motion are results of a choice of reference frame, arbitrarily selected before you start to do any analysis.
OK, Lets choose the frame of reference coincides with the table.

Hi,

Newton's 2nd Law can study the motion of the objects in the case of zero net force:

F = ma

if F = 0 then a = 0 then v = constant, then the object is either at rest (v=0) or moving in uniform rectilinear motion. (Bold quantities are vectors).

What is the point of Newton's first law as it is nothing more than a special case of Newton's 2nd Law?

I am sure there is a good reason that Newton himself stated his first law as a separate law, but I can not see this reason.
Check out:
https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_%281846%29/Axioms,_or_Laws_of_Motion

https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_%281729%29/Definitions#Def2

Now, you are right that from the second law follows that without a force, the state of motion continues. However, the first law describes more precisely what that state of motion is. In order to fully include the first law into the second, he would have had to write:

The alteration of a state of rest, or of uniform motion in a right line, is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

So, it appears that he chose to split it up in order to phrase, as a separate law on its own right, what this alteration is relative to. In particular, the second law is not relative to orbital motion.

See: http://www.4physics.com/phy_demo/newton/newton_1.htm

russ_watters
Mentor
OK, Lets choose the frame of reference coincides with the table.
In the reference frame of the table, the book is not moving.