Newton's 1st Law: Incl in 2nd Law? Reasons/Ex

[riddler5280]

Is included in the second law? Give reasons with examples.

 

[Dick]

Quote Newton's first law. Quote Newton's second law and apply to the case where the external applied force is zero.

 

[neelakash]

I can help you for understanding purpose.
First law gives the definition of inertial frame while second law says in that frame F=ma is valid.
You may verify 1st law from the 2nd law.But,you cannot prove it.
Whenever you are assuming F=ma,you are also assuming you are in an inertial frame.

 

[becko]

The second law states:
(d^2 x)/(dt^2 )=F/m

I will describe an example of the necessity of the first law.

Consider a particle whose position is given by a function of time x=f(t). Suppose that this function is not differentiable (imagine a particle that teleports from one place to another, making the function f(t) discontinuous). Since the second derivative of f(t) is not defined, the second law would tell us nothing about the motion of the particle. But if we take into account Newton’s first law, this kind of motion becomes impossible. The first law states (taken from The Feynman Lectures on Physics):

"…principle of inertia: if an object is left alone, is not disturbed, it continues to move with a constant velocity in a straight line if it was originally moving, or it continues to stand still if it was just standing still."

By being alone and non-disturbed we understand that no force acts on the body. This means that either a body is standing still, moving with constant velocity in a straight line, or is being acted upon by a force. If it is being acted upon by a force, then the second law tells us that the motion of the body will be differentiable. Together, the two laws forbid the motion described above. This means that the first law is not completely contained in the second law.

A different argument could say that the equality

(d^2 x)/(dt^2 )=F/m

implies that x=f(t) is differentiable, since for any value of the force, we could compute the value of the second derivative (we assume that m≠0). For this to be true, we assume that the force always has a value, that is, it is always defined.

For example, consider the motion described above. We could just say that for this scenario, the force is not defined. For example, we could suppose that the body is under the action of a force of gravity (F=(m1*m2)⁄r^2 ), and r=0. Then the force wouldn’t be defined (division by zero), and the second law would only tell us that the motion of the body is not differentiable. Again, with the help of the first law, we can forbid this situation. The first law tells us that the body is standing still, moving with constant velocity (in this two cases F=0), or it is being acting upon by a force (which means that the force has a value different from zero). In other words, the force always has a value.

The modern view of Newton’s laws is to take the first law as the definition of an inertial frame of reference. That is, any frame of reference in which the first law holds is called an inertial frame of reference. Then we say that the other two laws hold in all inertial frames of reference. This is the view adopted by Einstein when he stated his Special Theory of Relativity.

 

[lpfr]

In the modern view, we take the first law as the definition of inertial frames of reference.

Newton Laws are not enigmas given to men by Gods and subject to interpretations by prophets, priests or physicist.
The meaning of these laws is the one given by Newton himself and no other one. There is not such thing as "modern view" of these laws, no more than Freudian or Hegelian view.

But, why not take the second? We could define an inertial frame of reference as a frame in which the second law holds.

Yes. This is how things have been done for some centuries. The Newton Second Laws works in inertial frames, and, an inertial frame is a frame where Newton Second Laws works.

See my post here:
https://www.physicsforums.com/showthread.php?t=165317

Yes, I did read your post and this is why a posted a Feynman citation.

Consider a particle whose position is given by a function of time x=f(t). Suppose that this function is not differentiable (imagine a particle that teleports from one place to another, making the function f(t) discontinuous). Since the second derivative of f(t) is not defined, the second law would tell us nothing about the motion of the particle.

If I see a particle whose position or speed is a non differentiable function of time and/or discontinuous, I won't conclude that I am in a non-inertial frame. I will conclude that I leaved the physical world and that I am inside a scene of a science-fiction film as "Stars War". In non-quantum physics all position and speed functions are continuous and differentiable.
And this comes not from the First Law which do not quantify things, but from the second: a discontinuous value of speed or position will ask for an infinite force and we know that this is physically impossible.

First Law is a particular case of Second Law put there for historical reasons.