# Newton’s second law as one complete law

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• rudransh verma
The comma is after "equation" and it's talking about the word "these" which are implied by the equation, not the equation itself.

#### rudransh verma

Gold Member
https://www.feynmanlectures.caltech.edu/I_08.html Page 9-3. It says “In these terms, we see that Newton’s second law, in saying that the force is in same direction as the acceleration,is really three laws, in the sense that the components of force in the x, y,z directions is equal to the mass times the rate of change of corresponding components of velocity:
##F_x=m\frac{dv_x}{dt}=m(\frac{d^2x}{dt^2})=ma_x##
…..and so on”
I want to ask I can see there is second law and first law. First law because without force there is no change in velocity. But where is third law here?

Just below that there are three components of forces. Everyone of them includes ##\cos##. Is it right? Shouldn’t it be cos, sin, etc That is how we resolve vectors into components.

On page 9-2 he says in the first paragraph ”Now there are several points to be considered. In writing down any law such as this we use many intuitive ideas………we say, it does not change.”

So he is basically saying Newton thought mass is independent of velocity and therefore he came of this law but later it was shown mass do depend on velocity. So his law is actually wrong.

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rudransh verma said:
But where is third law here?

Feynman didn't mean that all three Newton's laws are included in the second one, he said that the second law can be considered as three "second laws", each one for each component. He explicitly stated that: "(...)is really three laws, in the sense that the components of force in the x, y,z directions is equal to the mass times the rate of change of corresponding components of velocity".

rudransh verma said:
Shouldn’t it be cos, sin, etc

These are cosines of three different angles between force and three axes. Check out this link:
https://en.wikipedia.org/wiki/Direction_cosine

weirdoguy said:
These are cosines of three different angles between force and three axes.
Can we also write direction cosines for velocity vector and displacement too?
weirdoguy said:
Feynman didn't mean that all three Newton's laws are included in the second

rudransh verma said:
Can we also write direction cosines for velocity vector and displacement too?

Yes, we can do that for any vector.

rudransh verma said:
So he is basically saying Newton thought mass is independent of velocity and therefore he came of this law but later it was shown mass do depend on velocity. So his law is actually wrong.

Feynman for some reason uses relativistic mass in his lectures, which is generally considered to be an outdated concept, so you shouldn't pay much attention to it. What is really the core of what he says is that Newton's laws are just approximations of more general special relativistic equations. Also, saying they are wrong is a little bit wrong This insight articles is a good summary of that:
https://www.physicsforums.com/insights/classical-physics-is-wrong-fallacy/

weirdoguy said:
re general special relativistic equations. Also, saying they are wrong is a little bit wrong This insi
But he did say”These ideas were of course implied by Newton when he wrote his equation, for otherwise it is meaningless.”

Yes, and? Mass does not depend on velocity i Newtonian mechanics.

weirdoguy said:
Yes, and?
I think it’s not wrong or meaningless if it applies and describes successfully to the real world.

weirdoguy said:
But it does not in Newtonian mechanics, and Feynman himself explicitly gives an examply of why it does not. Please read carefully.
I said :
rudransh verma said:
I think it’s not wrong or meaningless if classical mechanics applies and describes successfully to the real world.

And what it has to do with what Feynman wrote? He wrote that it's meaningless to treat mass in Newtonian mechanics as velocity-dependent, because it does not fit with what we observe.

weirdoguy said:
And what it has to do with what Feynman wrote? He wrote that it's meaningless to treat mass in Newtonian mechanics as velocity-dependent, because it does not fit with what we observe.
Oh! I thought eqns are meaningless! So Newton said mass didn’t depend on velocity as was observed. His law was and is good even now.

Yes.

weirdoguy said:
Yes.
It’s not physics but can you tell in “These ideas were of course implied by Newton when he wrote his equation, for otherwise it is meaningless.” what does comma mean after “equation”. Isn’t the words written after comma about the word just before it?

rudransh verma said:
Isn’t the words written after comma about the word before it?

Well, they are. He said that Newtons equations are meaningless for velocity-dependent mass. Sorry, I streched his words a little bit.

weirdoguy said:
Well, they are. He said that Newtons equations are meaningless for velocity-dependent mass. Sorry, I streched his words a little bit.
Happens to me all the time.
So Newton did not knew about the relation between mass and velocity. He said they don't change and it doesn’t in observable world. So Newton was approximately right and the CM holds true.

rudransh verma said:
So Newton did not knew about the relation between mass and velocity.
The relationship is not between mass and velocity. It is between momentum and velocity. In the modern nomenclature, mass is defined in a way that makes it independent of velocity.

The correct relativistic relationship is ##p=m\gamma v## where ##\gamma= \frac{1}{\sqrt{1-v^2/c^2}}##.

As long as speeds are low, ##\gamma## is very nearly one and ##p = mv## is a decent approximation.

Although it is tempting to roll the ##\gamma## factor into mass, the result does not preserve the correctness of ##F_x=ma_x##, ##F_y=ma_y## and ##F_z=ma_z##. The difficulty is that the "relativistic mass" in one direction can depend on velocity in another. You get things like "transverse relativistic mass".

• sysprog
rudransh verma said:
https://www.feynmanlectures.caltech.edu/I_08.html Page 9-3. It says “In these terms, we see that Newton’s second law, in saying that the force is in same direction as the acceleration,is really three laws, in the sense that the components of force in the x, y,z directions is equal to the mass times the rate of change of corresponding components of velocity:
##F_x=m\frac{dv_x}{dt}=m(\frac{d^2x}{dt^2})=ma_x##
…..and so on”
I want to ask I can see there is second law and first law. First law because without force there is no change in velocity. But where is third law here?

Just below that there are three components of forces. Everyone of them includes ##\cos##. Is it right? Shouldn’t it be cos, sin, etc That is how we resolve vectors into components.

On page 9-2 he says in the first paragraph ”Now there are several points to be considered. In writing down any law such as this we use many intuitive ideas………we say, it does not change.”

So he is basically saying Newton thought mass is independent of velocity and therefore he came of this law but later it was shown mass do depend on velocity. So his law is actually wrong.
There are only 3 components if the force/acceleration vector ISN'T along the same line as the reference axis. Why would anyone do that unless there are other forces/vectors/axes involved ?

jbriggs444 said:
The relationship is not between mass and velocity
Feynman doesn’t explain anything accurately. He talks like he is reciting a story.
The only thing that is clear to me in this is that a law is meaningless if the theorist doesn’t know what he is talking about. Like how mass varies with velocity, if Newton didn’t knew then his law is meaningless. That’s it!
I have heard a lot about Feynman. How great is a teacher he was. That’s why I picked up his Feynman lectures Vol.1. Maybe I am not doing it right. So any advice would be very helpful.

rudransh verma said:
Maybe I am not doing it right.

Maybe it is a language barrier? I mean, it took me quite a lot of time to really read english textbooks freely.
Besides, I'm in a minority with this, but I really don't think that Feynman's lectures are as good as most of people say, that is, I don't like his style of teaching. His books are of course full of insights and non-standard approaches, but his style...

rudransh verma said:
I have heard a lot about Feynman. How great is a teacher he was. That’s why I picked up his Feynman lectures Vol.1. Maybe I am not doing it right. So any advice would be very helpful.
Feynman's lectures were intended for the best students in introductory physics, so he discusses subtleties that may be glossed over in a typical intro physics class. These points are often not discussed because they would simply confuse the average student more than helping them learn physics at the intro level. So it turns out the Feynman Lectures isn't the best textbook for most students to learn from initially. Once you've learned the basics, though, his lectures can provide insights into physical concepts, which is why many physicists like the books.

In this particular instance, Feynman is simply pointing out that the law ##\vec F = d(m\vec v)/dt## has many assumptions baked in, though not explicitly stated. As an example, he notes we assume that the mass ##m## is constant based on our everyday experience. Without these assumptions, an equation like ##\vec F=d(m\vec v)/dt## doesn't really say anything useful.

That's how it starts—with assumptions and perhaps some unclear notions about what the different quantities actually represent. Later, others may refine the concept of mass, questioning the assumptions and asking, "Is the mass of an object really constant?" There may be some mistakes made along the way, like Einstein initially suggesting mass varies with velocity, and then as we learn more about the universe, we realize that mass is indeed a constant.

• • Lnewqban and berkeman
vela said:
mass m is constant based on our everyday experience. Without these assumptions, an equation like F→=d(mv→)/dt doesn't really say anything useful.
Ok! So if mass varies or if mass and velocity were inversely proportional then the law doesn’t make any sense. Change is a constant is a force doesn’t make sense.
vela said:
ften not discussed because they would simply confuse the average student more than helping them learn physics at the intro level.
Thank you for telling me as average. vela said:
That's how it starts—with assumptions and perhaps some unclear notions about what the different quantities actually represent. Later, others may refine the concept of mass, questioning the assumptions and asking, "Is the mass of an object really constant?" There may be some mistakes made along the way, like Einstein initially suggesting mass varies with velocity, and then as we learn more about the universe, we realize that mass is indeed a constant.
well it does help to read the book. Maybe not alone but with some help even the average can understand what he’s trying to say.
He said ones ”I am a ordinary guy who studied very hard”.

I think the Feynman lectures are among the best introductory theory (!) books ever written, and I think it's readable for any student. It's perhaps misleading to think about it as a real "freshman text", for which it was originally intended. I think, given the decline in the quality of the coverage of the STEM subjects in high school nowadays (at least in Germany), one first needs a more elementary book as used in the introductory experimental-physics lectures like Tipler or Halliday.

vanhees71 said:
one first needs a more elementary book as used in the introductory experimental-physics lectures like Tipler or Halliday.
I am doing the same thing. First Halliday ,then some numericals ,then Feynman.

What do you mean by "numericals"?

vanhees71 said:
What do you mean by "numericals"?
Problems To solve the problems is of course always of utmost importance. If you can't solve the problems, you don't have understood the material. So you need to do the problems to see, whether you have to learn more about a subject of if you have really understood it and you can go on to the next one.

Oh for the days when you could go into the back yard. Use levers, wheelbarrows, hammers, shovels, bricks, blocks and tackle and gain schooling at Father's feet. This seems to provide an understanding that mere books do not transfer to some students.

• A.T., Doc Al and vanhees71
rudransh verma said:
Change is a constant is a force doesn’t make sense.
Neither does that sentence. ;)

Thank you for telling me as average. 