# Can someone explain me why f=ma ?

1. Aug 3, 2012

### vansh123

hey guys i had a doubt can someone please explain me why force=mass x acceleration?

2. Aug 3, 2012

### Pengwuino

That's how the concept of force is defined. If you apply larger amounts of net force on an object, you'll find that the acceleration increases linearly and the property that decides how quickly the acceleration increases is the mass.

3. Aug 3, 2012

### RobinSky

A quicky my teacher once said:

" I don't like the way the original equation is written because it brings up misconceptions to many new students, instead think of it this way, f/m=a. A force acting on a body will cause it to accelerate". Then you only also have to remember one of the laws of newton: that a body will stay at rest (or uniform motion in a straight line) as long as no net force is applied to the body. And when suddenly we have a net force, we have an acceleration.

Not an expert in this subject but maybe that will help.

Best Regards

4. Aug 3, 2012

### Jolb

There's a fun quote by Arthur Eddington which touches on this:
F=ma is sometimes looked at as a definition (especially by mathematicians), where it constitutes the definition of force based on measurable quantities mass and acceleration. But according to Thornton and Marion, really when Newton stated this, he meant something physical. I think what Newton really meant, more specifically, was that from an inertial reference frame, objects continue in straight lines unless acted upon by a net force--but the concept of an inertial reference frame isn't defined when you begin to state Newton's laws--so it's kind of a built-in assumption that's hard to avoid. That's why the mathematicians just ignore the physical meaning and take it as a definition.

5. Aug 3, 2012

### Staff: Mentor

Usually Newton's first law is considered to be a definition of inertial frames and Newton's second law is considered to be a definition of forces. Once those terms are defined, then Newton's third law is the one that contains the actual physics.

6. Aug 3, 2012

### Jolb

I beg to differ, DaleSpam. Newton's third law applies only to interactions between two (or more) bodies--if that law contained all the actual physics, then how is it possible that there is a whole class of problems that deal with a single particle under the influence of some "external" force or field? For example, a particle on a spring (SHO), a massive particle in an external gravitational field, and a charged particle in an EM field all are nontrivial physical problems which can be solved using only the first two laws--not the third.

7. Aug 3, 2012

### Staff: Mentor

Sure, but in all of those cases there is something besides Newtons three laws defining the physics, the force law. You mentioned Hookes law, Newtons law of gravitation, and Maxwells equations. Newtons first two laws are still generally considered to be definitions in the scenarios you mentioned.

8. Aug 3, 2012

### Jolb

Well, Hooke's law isn't exactly a physical law of the universe--it just is a force function. You could treat Newton's gravitational force law as just some random "given" force with a certain functional form, and likewise, you could ignore Maxwell's equations and simply give the Lorentz force law. If you want some even more trivial examples, you could have F=constant, F(t)=at+b, F(x) = cos(x), or whatever. They all are just given "external" forces. All problems of this form do not require the third law.

9. Aug 3, 2012

### Staff: Mentor

Agreed, but again, the physics is not contained in the first two laws, they are simply definitions. The physics is contained in whatever force law you propose or in the third law.

All problems involving only the first two laws are simply problems designed to help students learn the definitions.

10. Aug 3, 2012

### Jolb

Well, I believe we are still at odds here. I think the problems which involve only the first two laws do contain actual physics--they tell you how a particle actually moves under the influence of a force. And though many of the examples I gave do seem like problems you'd only give to help students learn the definitions, I think the simple harmonic oscillator and a particle under the Lorentz force law are quite a bit beyond just learning the definitions. They give nontrivial predictions on trajectories.

Even in well-respected textbooks, you'll find admissions that the scenario is not as clear-cut as what you've argued. There are alternate ways of answering the question, and they amount to different perspectives on these laws.

Thornton and Marion's "Classical Dynamics of Particles and Systems," Fifth Edition, treats this question with a few paragraphs (pp.49-50), and it includes this footnote on page 50.
Your view is certainly a valid way of looking at things, but it is not the only possible way.

Last edited: Aug 3, 2012
11. Aug 3, 2012

### mikeph

I'm sure the answer is that it's an empirical law. If you apply double the force, you observe the object accelerates twice as much. Do this experiment in enough ways and it's soon fairly convincing that F=ma and not m/a or ma^2 or something weird.

12. Aug 3, 2012

### voko

This is not how Newton gave them. See http://en.wikisource.org/wiki/The_M...l_Philosophy_(1846)/Axioms,_or_Laws_of_Motion

"LAW II.

The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subducted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both."

Definitions were given before the laws:

http://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/Definitions

"DEFINITION IV.

An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line."

It can also be seen that his definition of force refers to inertial frames, but perhaps not entirely satisfactorily (the translator may be to blame, though).

This repeats Newton's own words almost verbatim! In the scholium that follows the Laws, he refers to multiple experiments, particularly by Galileo, that had established these laws.

13. Aug 3, 2012

### Staff: Mentor

Correct, it is not how Newton gave them. It is how modern physicists interpret them with the advantage of several hundred years of hindsight.

14. Aug 3, 2012

### Staff: Mentor

Excellent, then it seems we are not at odds.

Which is why I qualified my statements with the word "usually".

Btw, the reason I prefer the usual approach over alternatives is the difficulty in defining what a force is without using f=ma. It can be done, usually via an experimental prototype force, but force prototypes are not very reproducible.

15. Aug 3, 2012

### tkjtkj

Actually, it is an 'observation' .. not really a definition: and the observation is simply : what features of an object are directly proportional to the 'force needed to move it'.... and what features, if any, are found to be INVERSEly proportional to that force ...

So, experiment shows that only two items affect the ability of a force in a friction-less vacuum to move any object <mass> ... and both of them are DIRECTLY proportional to the force required ...
Ie, if the MASS is bigger, the force must be bigger <other things being eqaual> .. and if the required motion is to start, then the force also is directly <NOT: 'inversely'> proportional to how fast the mass is made to move up to any velocity <ie, it's 'accelleration'>

Mathematicly, we state such a condition this way:
In english: for a given force, giving it a velocity is DIRECTLY proportional to the mass of the object, and also Directly proportional to how much you choose to accellerate it. ...

In math: F <the force required> = M <the mass> TIMES any other 'directly propportional' item .. in this case, 'A' , how much we choose to make it go faster.... with a 'Constant of Proportionality' < 'K'>
conveniently chosen to be '1' , or:
F = kMA , where k = 1 for our chosen units

If there were any 'INVERSELY Proportional entity, say <some 'vacuum-like' theoretical thing called 'Z' which would HELP accellerate, then the formula would be:
F = kMA <times> 1/Z , or: F=kMA/Z

because the BIGGER Z is, the SMALLER F can be ... ie, they are INVERSELY proportional.

This is how many relationships are described ... another example woujld be 'Ohm's Law' where:
E = I x R .. I'll leave that to yu to sort out ;)

and remeber: I MUST = E/R , atain from the observation.
<simply dividing both sides of a known 'equality by the same item, in this case: 'R' ..

E / R = (I x R) /R ..and the R's cancel out on the right side..

Hope this helps ...

j.a. , mech-eng, m.d.

16. Aug 4, 2012

### Philip Wood

The problem is: How do we know that we've doubled the force? How else do we measure the force except using $F = kma$ itself? In other words, we're brought back to regarding $F = kma$ as true by definition.

I've just been a bit of a devil's advocate, though, because things aren't quite as simple as I've made out. For example, if we use two identical, equally stretched springs, pulling in the same direction on an object, instead of just one of the springs, we find twice the acceleration (if resistive forces are negligible). It's hard to quarrel with the idea that doubling the springs must double the force (so giving us an independent way of knowing that we've doubled the force), but I've known people who believe that it's making an unacceptable assumption.

Last edited: Aug 4, 2012
17. Aug 4, 2012

### Staff: Mentor

Yes, this is exactly what I meant by "an experimental prototype force".

18. Aug 4, 2012

### vansh123

hey thanks guys for your help
i had solved so many questiond on newtons laws of motion but i didnt get the correct meaning of his second law

19. Aug 11, 2012

### Jolb

I find it exceptionally beautiful that even the most basic laws of physics have such nuanced philosophical interpretations. The arguments presented here really show how hollow "definitions" can be, and how ambiguous "empirical laws" can be.

20. Aug 11, 2012

### GarageDweller

Personally, I look at newtons laws as instructions for how to construct other equations based on observations.
The third law is just a statement of the conservation of momentum, nothing much to see there.
You have to take note that, although a law might be named after someone, the law that we write down nowadays can be drastically different from the one originall derived.
Originally, newton defined force as the rate of change of momentum.
This way he essentially described a model for how the universe could function.
Every particle has a momentum vector assigned to it (newton used the words motion and momentum interchangingly), it's time derivative is equal to the outside interactions which he calls force.
Position, velocity and acceleration can be defined from this.
To summarize, I see newtons first and second laws as describing the arena in which all of classical mechanics takes part in, given the initial conditions and outside forces, solve for the trajectory of the particle.