# Newton's Second Law

1. ### bigerst

57
I'm just uncertain (haha), what IS newton's Second law exactly?
F=ma, or F=dp/dt, aren't these defining equations for F itself? what is a force? what would you mean a "force" acts on a object, would it be equally probable to define say, d^3Q/dt^3 = m dx^3/dt^3?
does newton's second law have anything to do with the fact that coulomb, gravity etc can be expressed as a second order differential equation? is it a definition, or an accident?

2. ### WannabeNewton

5,862
It is empirical if you want to look at the second law as being the fundamental element. If you look at the principle of stationary action and energy as more fundamental elements then it comes right out of varying the action.

3. ### rahulpark

10
Newton's laws were written after some experiments. force is actually a word for 'interaction'.
Newton observed that a body could move with uniform velocity without any support or interaction form another object in a particular direction. So any interaction would cause a change in velocity , thus F=m*(dv/dt). (forces are abstract)
Forces can't be seen, but the changes due to it can't be clearly seen. The object's changes help calculate forces.
How can the physical laws be an accident ? birth of human s may be an accident, but physical laws,No. Coulumb's law and Theory of gravitation defines force caused due to charge and mass interactions. Well it's interaction, but without any contact. So they are forces irrespective of their birth.

4. ### jbriggs444

1,935
While we may not be able to "see" a force, we can feel how hard we have to push on a box to move it across the floor. We can readily suppose that two bricks exert twice as much force on a scale as one brick. We can even measure this by noticing that the deflection of a spring supporting two bricks is twice that of a spring supporting one brick.

So physical forces are measurable. Further, they are measurable without referencing Newton's second law. Or differential equations.

People had been measuring mass with balance scales for thousands of years. One need not belabor that point.

Measuring acceleration can obviously be done with rulers and clocks. So Newton's second law can be seen as a physical statement about measurable quantities.

In modern physics the correctness of Newton's second law is taken for granted and we actually define the standard unit of force in terms of the standard units for mass, distance and time. The reason we do this has more to do with what quantities we can measure precisely than with what quantities are "physical" or "fundamental" (whatever those terms might mean).

Having done this it might seem that Newton's second law is a matter of definition rather than a matter of experimental fact.

This is very similar to the way we take the constancy of the speed of light for granted and define the standard unit of length in terms of the standard unit for time. That's because we can measure length using atomic clocks and interferometry better than we can measure the distance between two scratches on a rod, not because time is more physical or fundamental than distance.

Having done this, it might seem that the speed of light is a defined constant rather than an experimentally determined value.

5. ### bigerst

57
ok i get how forces are interactions but just one more question, does a force cause motion?
the example i stumbled upon was the function x=e^-1/t^2 (from mary boas) where the derivative of all orders vanish at t=0, hence is it correct to say that interactions produce newton's second law, the differential equation, whereas forces are not enough to cause motion?

6. ### jbriggs444

1,935
Not sure what you are getting at with that formula for position as a function of time.

The position at time t=0 is zero. The velocity at time t=0 is zero. The acceleration at time t=0 is zero. By Newton's second law the force at time t=0 is zero.

None of that is at all problematic.

The position at time t != 0 is non-zero. The velocity at time t != 0 is non-zero. The acceleration at time t != 0 is non-zero. By Newton's second law the force at time t != 0 is non-zero.

None of that is at all problematic.

If you are a mad scientist in control of the scenario then you are free to modulate the applied force f(t) to make it proportional to the second derivitive of e^-(1/t2) and thereby have a force that causes an object to have the specified trajectory.

Are you working toward some argument based an idea that a function that is continuously differentiable to all orders must be exactly determined by its Taylor series expansion? That premise is incorrect.

7. ### rahulpark

10
Take into account that, this displacement function x=e^-1/t^2 was caused by an interaction changing with time (variable force). The question is in the reverse order. If there weren't any force this function doesn't explain the object in motion.

8. ### bigerst

57
im just confused, intuitively i'd think a force would cause motion, but at t=0 there is no force, but at the next instant the object starts moving, so what causes it to move? i think there might be something wrong with saying "next instant" since time is continuous? is that the key?

Thanks

9. ### Jano L.

1,271
I am sympathetic to your confusion. This stems from the proposition "force causes motion".
I'll try to explain.

Newton stated his 2nd law in words, as the second law of three, as he considered the force as a more or less understood concept, associated with contact of bodies or muscular tension. He also spoke of changes as "compelled", or "caused" by other bodies.

I think this is a matter of choice of the most clear formulation. In his days, when mechanics was still developing, he probably saw his choice of words as better, as it appealed to daily experience and thinking.

However, I think today, it is more useful to define the technical concept of total force more precisely by F=dp/dt, so the proposition called by him "2nd law" becomes rather an advanced operational form of definition of this total force.

Also, I believe that when stating the basic mechanical laws, it is better to refrain from using the notion of causality altogether, as it is usually not necessary to set up the mathematical description of the motion and is quite heavy philosophical notion on its own (much more complicated than mechanics in my opinion).

So today, in effort to state the laws of nature in the most accurate way, we may be putting the "2nd law" differently, but still, the laws of mechanics are not just a bunch of useful definitions, but rather some valuable statements how the things in world usually behave (->force is a often function of t,x,v and -> independence/superposition of forces).

The advantage of the mechanical description with forces is that any motion can be described. In your example, if the coordinate as a function of time is

$$x(0)=0;$$
$$x(t)=e^{-\frac{1}{t^2}},$$

it is possible to find the force from the definition ##F(t) = m \frac{d^2x}{dt^2}##, and is equal to

$$F(0) = 0;$$

$$F(t) = -2 m \frac{1}{t^4} e^{-\frac{1}{t^2}}.~~~(*)$$

if I did not made a mistake in the calculation. Of course, this force is just a description, not an explanation of such motion - such force is as strange as the original motion.

This cannot be found just from the information given. The explanation or cause of such motion can be found only in finding its relation to other bodies, or possibly in answering the question "why is the force given by that expression (*)?". To find that, you have to find physical situation where such motion would occur...

10. ### bigerst

57
so the idea is that force is as much a description to the motion as displacement itself (am i understanding this correctly?)
then the above formulation completely overthrows the typical problem solving strategy i have encountered: identify the forces that cause the motion, add them up and solve differential equations. i think the principle behind that strategy is answering the questions "when these forces are applied, what motion results". though the formalism hardly change the idea behind it is drastically different?

thanks
bigerst

11. ### Andrew Mason

6,895
If F ≠ ma then the laws of physics would be different in different inertial frames. Galilean relativity would not hold. F=ma is essential to the principle that all inertial reference frames are equivalent (assuming time and distance measurements are equal for all reference frames).

When you are dealing with forces and change in motion, keep in mind that motion depends on the time interval through which it is applied. A bullet applies a significant force to a rifle and the shooter when it is fired, but the bullet goes down the barrel in such a short time that the gun does not have time to move a material distance before the bullet leaves the muzzle so accuracy is not affected.

AM

12. ### jbriggs444

1,935
I would say that for this particular conundrum that is exactly the key.

For each instant when you can look at the position of the object and ask "why is it here now" you can take the double-integral of force over time up to that instant and answer the question correctly.

Because you have modeled position as a real-valued function of real-valued time there is no "first instant". There is no non-zero time without a history of leading up to it.

You may have been aiming toward a somewhat more insidious conundrum where force is a function of position F(x) = e-1/x2. If my hazy memory of differential equations is correct, this leads to a differential equation that has multiple solutions for position as a function of time that are all consistent with boundary condition of an object that is stationary at position zero at time zero.

If you made a hill with a profile matching this force function and put a ball at the top then you could well ask "what caused the trajectory that was seen". This suggests a non-deterministic universe. But limited measurement precision hides such a situation from us. We cannot do a reproducible experiment to find out what would really happen. Similarly, we can't balance a pencil on its point.

Last edited: Jan 24, 2013
13. ### Jano L.

1,271
The acceleration or force ##F(t)## describes the motion, but less exhaustively than the position ##x(t)##; one can always find force merely from ##x(t)##, but doing it vice versa requires additional information.

I do not think that the proposition "forces cause motion" is a necessary part of newtonian mechanics. That would be close to the old Aristotelian view of physics. The great contribution of Galileo and Newton was that they realized that the motion is a natural state of body and force is present only when there is change of its motion.

Instead, sometimes it is said that force causes the change of motion and it does help to set up the differential equations. However, in the end, there is no causality in the mathematical description. One can often use the differential equation to determine both past and future position of the body, everything is symmetric.

As mentioned above, there are also cases where the force is such function of position that the equations of motion have multiple solutions not determined by the initial conditions. Then all solutions are equally valid and it follows that force alone does not determine what motion will occur - to determine that one needs additional information, perhaps higher derivatives of ##x(t)##, or something else that causes nature to realize one particular motion.