Personal Wiki of Subhrajit Bhattacharya

Newton's Laws of motion form the foundation to all concepts of what we call 'Newtonian mechanics', or the rules that govern the motion of particles and bodies in our macroscopic world. However, very often they are poorly understood or the real implication of these laws are incorrectly interpreted. The philosophies behind Newton's Laws and their proper interpretation constitute many of the fundamental pillars in classical physics with far reaching consequences. More often than not, his three laws of motion are introduced in high-school level physic classes without elaborating on or explaining some of the underlying concepts and assumptions. This eventually becomes a severe bottleneck in understanding more advanced concepts in physics like energy, relativistic mechanics and quantum mechanics. Very often those lack of understandings are shoved under the carpet.

In this article I would thus like to start from the very beginning - the statements of the three laws - and would try to elucidate the fundamental concepts behind the laws of motion in as simple words as possible. We will do so by taking ourselves back to the time of Isaac Newton, and will start with contemporary understandings about the classical world. We will see how Newton's laws, for the first time, shone light on some of the most haunting concepts in physics, but were able to do so only with some questions left open. We will thus use those to motivate the development of concepts that are part of some of the more recent advancements in physics, including the fundamental forces, relativistic mechanics and theory of electromagnetism.

[:intro standard:] In case you don't remember the three famous laws of motion taught in every high scool, let me help you recapitulate them...
Newton's First Law of motion: Every body continues to be in it's state of rest or of uniform motion in a straight line unless and untill compelled by some external imposed force.
Newton's Second Law of motion: The rate of change of momentum of a body is directly proportional to the Force acting on the body and the change takes place in the direction in which the force acts.
Newton's Third Law of motion: To every action there is an opposite and equal reaction.

So having revised the statements of the Laws, let's take a little insight into their contents.
First of all it is to be understood that Newton's first two laws are more of definitions than 'laws' of nature. And hence they can never be proved wrongs or can never change unless we change the definitions! If you define the name of an apple to be 'apple', and everybody accepts it, how can it ever go wrong unless some fine morning everybody change mind and prefer calling it something else? However if you postulate that the diameter of an apple is always less than 10cm, then it is highly possible that an apple is found by some means which violates the postulate. So that's the difference between a definition and a postulate. In one sense it is right to call a definition by the name 'Law', since a definition will never ever change!

So now it is time to understand why exactly Newton's first two laws are definitions.
First of all, let's forget what we know to be motion, force and mass. As a matter of fact, before Newton, force and mass were not defined properly. We will define them on our way of exploring the laws. So remember, at this point we have no idea what is mass or what is force! [:/intro:]

[:motionmeasure PhantomCoolForm:]

Measuring motion - concept of a reference frame:

Let's assume we have defined the unit of length. That's easy to do! Take a rod and call that to be one meter. Then 2 meters will be just 2 such similar rods placed one after another, and so on (However here is a deeper notion behind this. Here we assume that the space in which we live is homogeneous and isotropic. Hence there is no preffered origin or direction. Anyway, lets leave that aside for the time being.). Moreover we assume that we know what time is and we know how to measure it, and that time is an absolute quantity (that though does not hold true in light of Einstein's Special relativity. But we will not bother much about that for the time being.). Then one can measure speed of a body or particle relative to oneself just by measuring the distance that it moves and dividing it by the time taken to do so. Add to it the direction, and define it as velocity vector. So now we defined 'motion' by the velocity of a body. Note that here we are speaking of a 'relative velocity', that means the velocity measured by some person. The velocity measured by someone else moving relative to this person will be totally different. [:/motionmeasure:]

[:firstlaw standard:]

The First Law - concept of Force and Inertial Frame:

Before Newton there had been several attempts to understand what we call today 'force'. However force being such an abstract quantity couldn't be understood for a long time. In fact we'll see that Newton's first law gives the definition of force for the very first time.

Before Newton, many people used to believe that bodies in motion tend to stop after some time. This was very intuitive from the simple day-to-day experiences. When you roll a ball/cart it'll stop by itself after going a certain distance. Newton was first to view the thing from a different perspective. He proposed that the rolling cart stops because of something called a 'force' which acts on it due to its interaction with the ground and the air. And in fact it was found that if the ground was made smoother and the experiment was performed in presence of less air, a ball or a cart rolls upto a much longer distance before it stops. Hence it is intutive that if we can make the floor absolutely smooth and the place was made complete vacuum, the cart would have rolled on forever!
However it must be remembered that by adopting Newton's proposal we make a great hypothesis at the very first step... that is, we say that something called 'force' acts on the cart to slow it down. This is important because before this we didn't know precisely what force is. Thus Newton said that when an 'entity' called 'force' acts on the cart in motion, it's velocity tends to change. Previously we did not have the concept of a force. However we did know what motion is. Hence here we define force... That entity which tends to change the state of rest or of uniform motion of a body is defined as Force. So we see that Newton's first law basically gives the definition of Force.

It would have been simple if the story for the First Law have ended here. But there is still a lot more into it! We first notice that the First Law explicitly uses the term 'motion'. But we know motion is something relative (i.e., frame-dependent). So now the question arises that whether or not this definition of Force depends on the choice of reference frame as well. That means if I measure the change in motion of the rolling cart from another person's reference frame, can I conclude from this person's point of view that a force (as we just defined it in First Law) is still acting on the cart? The answer, unfortunately, turns out to be no! Let's consider the problem from the perspective of someone sitting on the rolling cart that is being slowed down due to the friction acting on it (i.e. we consider the reference frame of the cart). From that reference frame the cart is obviously at rest. Hence by Newton's first Law we should concluse that no Force is acting on the cart from reference frame of the cart. So force turns out to be frame dependent. But this leads to a problem since we desire a definition of force which will be indepencent of any reference frame. Hence comes the concept of Inertial Frame. Instead of re-defining force to make it frame independent, we state that Newton's First Law won't be valid in a frame like that of the cart, but will be valid from the reference frame of the ground. For now we shrug under the carpet the reason for this particular choice. We define an inertial frame, by choice, to be such a reference frame where Newton's First Law is valid for all bodies. Eventually we define the reference frame of the cart to be Non-Inertial and that of the ground to be Inertial. Hence our conclusion of no force acting on the cart was wrong from the reference frame of the cart, since it is a non-inertial frame, and hence the definition of force according to Newton's first law will not be valid in it. Hence in a way, Newton's First Law also gives a definition for Inertial Frame.

Note that upon choosing a particular inertial frame, every other frame that is in a state of rest or of uniform motion with respect to that chosen frame, becomes a valid inertial frame. The reason is obvious: If the cart's motion is changing with respect to the ground, its motion should also be changing with respect to a chair sitting on the ground or a car moving at a constant velocity on the ground. So by choosing one particular frame to be inertial, we essentially end up choosing a class/family of frame to be inertial.

We note that since Inertial frame is a necessity for stating Newton's First Law and defining force, its definition should have come before the First Law (i.e. the definition of force). So is there any way that we can conclude whether or not a frame is inertial before visiting the First Law? So let's investigate what actually made us conclude that the reference frame of the cart is non-inertial and that of the ground is inertial when we don't know the definition of Force. We could easily have defined the inertial and non-inertial frames other way! That is, we define inertial frame to be that of the cart. Then we would have concluded by the statement of the First Law that no force is acting on the cart since it is at rest, whereas some force is acting on the other objects lying on the ground (say the carpet on which the cart is rolling) since they are decelerating with respect to the frame of the cart (note that since the cart was decelerating with respect to ground, the ground should also be decelerating with respect to the cart, i.e. when seen from the cart). That conclusion would have been a direct consequence of the definition of Force from First Law. So what's the problem in that? Apparently there isn't any! So we fail to define/choose an inertial frame before defining force. [:/firstlaw:]

[:fallacy :]

The fallacy and its resolution - A more precise concept of Force:

Hence, to summerize, the outcomes of the discussion so far:

1. Newton's First Law is valid in certain reference frames called Inertial Frames (i.e., the definition of force, as prescribed by the First Law, should be obtained in an inertial frame),
2. Inertial Frames, by choice, are those where Newton's First Law is valid (i.e., knowing the definition of force, we call all those reference frames 'inertial' where this definition is valid).

This evidently leads to a great fallacy! We are using the concept of Inertial Frame in defining Force, and again we are using the concept of Force in defining an Inertial Frame!! It's a sort of a chicken-egg problem!

Image courtesy of classroomclipart.com

So now we need to do something to resolve this. We can

The first way is the easiest. Say if we perform experiments sitting on the ground we can simply resolve everything by setting our Inertial frame on the ground. But this is not an elegant solution, since logically why should we give any preference to the ground on which we are performing our experiments?

So, basically the second way is the one that gives the proper solution. We redefine forces to be ones that are originating only because of a) Gravitational, b) Electro-magnetic/contact, c) Strong or d) Weak interactions. That means these four can be the only cause of change of state of rest or of uniform motion of a body. If we find that there are some forces (i.e. the cause of change in state of rest or of uniform motion of a body) in a reference frame originating out of some cause other than these four, we conclude that the frame is non-inertial (in fact such forces are called 'fictitious forces' in a non-inertial frame). The fact that we included these 4 types in the list is a rather experimental deduction than anything else.
We however note that this is just an 'intermediate' definition of force introduced to resolve a fallacy and define 'inertial frames', and does not invalidate the definition of force by the First Law. In fact this intermediate definition of forces is solely used to define inertial frames. Thus we call a frame to be inertial if and only if the causes behind the change of state of rest or of uniform motion of bodies in that frame are composed of only a) Gravitational, b) Electro-magnetic/contact, c) Strong and d) Weak, interactions.

Since we have previously accepted the First Law as a definition for Forces, here we make a slight re-statement of what we just mentioned. We say that Inertial frames are those where the forces (i.e. the cause of change of state of rest or of uniform motion) of interaction are composed of only the 4 types of forces, namely, Gravitational, Electromagnetic, Strong & Weak. (Though we omit Gravitation when talking about General Relativity) Hence we define force from Newton's first law as an 'entity' that causes change in state of rest or of uniform motion when seen from such an Inertial frame.
So that more or less ends the discussion on Newton's First Law. [:/fallacy:]

[:equalforce standard:]

Concept of equal forces:

We'll need this concept before we can proceed to the Second Law.
So far what we have learnt to do is to understand when a force is acting on a body, and when it isn't. We said that seeing things from an inertial frame, if there is change in the state of rest or uniform motion of a body we say force is acting on it, else we say it isn't. Thus we have learnt to identify two states of force... zero force and non-zero force. Now we'll try to understand when we can call two forces to be equal.
The concept of equality of forces is rather involved! We can conclude from simple experiments whether or not two forces are equal. Say we take two identical springs and apply forces to cause extention in axial direction on both of them. Quite intuitively we'll say that if the extentions are equal then the forces are also possibly equal. However, with our current description of force, it is non-trivial to define equality of forces in a proper way when the systems are not identical.

Lets look at things from an inertial reference frame. Hence in this frame there are just 4 causes of forces, namely, Gravitational, Electro-magnetic/contact, Strong and Weak interactions. Say on a body, two of these forces (may or may not be of the same kind) are acting in such a way that the body is in a state of rest or of uniform motion -- that is the net force acting on the body is zero. Then we say that the two forces acting on the body are equal (and of course opposite). However there is a fallacy in this concept. We have defined force in such a way that it exists only when there is change in motion. So if we observe no change in motion then how can we conclude that forces are acting on the body? Moreover this particular way of identifying equal forces work if the forces are acting on the same body. How would you say if or not the forces acting on two different bodies are equal? So once again we need to resort to our intermediate definition for forces, that is, forces originate from Gravitational, Electromagnetic /contact, Strong and Weak interactions only. Hence we don't look for change in state of motion/rest to identify forces, rather we look for presence of any of these four causes.
It is rather difficult to identify the causes of forces intuitively since we don't yet know from what those four types of forces originate. It is a whole new subject to study each of these four types of forces and their causes. So for the time being we'll take for granted that we know whether or not each type of these forces are present at a point in the space and we can identify their causes.
If we try to investigate the spring that we talked about at the beginning of this paragraph, we'll find that when it is in an extended and equilibrium state because of no net force acting on any part of it, the extention and deformations in it is the proof of electromagnetic/contact force acting within it. If there is the gravitational field, then there is also a gravitational force. Hence even though there is no visible change in motion at any part of the spring, all these forces due to gravitational & electromagnetic/contact interactions (due to electrons of the atoms present inside the spring and the earth pulling the mass of the spring) are still present and acting on every part of the spring.

Since we assumed that we have a way of identifying the source/cause of all the 4 types of forces, we will now define equality of forces as follows: We say that if relative to two point in space, at some specified instants of time, the distribution of causes/sources of forces in the vicinity of each of those points have the exactly same configuration (i.e., they impart similar influence at the two points in space-time) then the forces acting on identical objects placed at those two points of space-time are equal. This is still quite ambiguous since we haven't quite discussed how to identify or quantify the causes/sources of the 4 types of forces. But at this point we can't do any better.

Image courtesy of clker.com

Let's try to understand the equality of forces, as we just defined, by a simplified example. Say you plan to push a tri-cycle and a truck with equal forces. So, first you push the truck. You get a 'feel' of how hard you have pushed. But the feel cannot be quantified since you don't yet know how to measure force. If we try to investigate the cause of force in this case, we'll find that while pushing the truck, the electron cloud on the surface of your palm maintained a certain distance with the electron cloud of the back on the truck where the push was being applied (the electron clouds are the source/causes of the force in this case). Had you pushed harder, the distance between the electron clouds would have just been smaller, and that would have resulted in a 'greater' force. So, the cause of the force in this case is a particular relative configuration of electron clouds on your palm and the truck giving rise to electrostatic (that's a special class of electromagnetic) interaction. If you manage to maintain the distance between the electron cloud of your hand and that of the back of the tri-cycle the exact same while pushing the tri-cycle, only then you can say that the forces applied on the tri-cycle is same as that applied to the back of the truck.
(However in this example we still make a great approximation that the forces due to the electron clouds only exist in a very localized region. Although the electron cloud configuration of the hand and the body of the person pushing the vehicles may remain the same in both the cases, the electron cloud configuration for the tri-cycle is definitely different from that of the truck. But with the assumption of only local effect, we can ignore the forces caused by electron clouds present further away from the place where the push is being applied.) [:/equalforce:]

[:secondlaw :]

The Second Law - measure of Force and the concept of Inertial Mass:

So now that we have got the definition of force, we need to find a way to measure it. As we have seen from the First Law, we define force to be non-zero only when there is a change in state of rest or of uniform motion of a body (and remember, change in motion of a body in an inertial frame can be caused only by Gravitational, Electromagnetic/contact, Weak or Strong interactions -- by the definition of inertial frame). And by motion we meant velocity. So it is quite obvious that Force will be, by some means, related to the change in velocity (i.e., acceleration). This relationship could have been developed in various ways. However Newton's Second Law gives an elaborate prescription to relate this change of velocity with the newly defined quantity called Force.

First we note that we have a good way of measuring length and time (as discussed earlier), hence it is easy to measure the rate of change of motion. We call it acceleration, . Where is the velocity.

We next assign a scaler quantity (i.e. a number) to each material body and call that 'inertial mass'. For the time being we choose to keep this 'inertial mass' a fixed value for a particular body which doesn't change with time or motion of the body. (However once we introduce the Third Law and adhire to the rigorous statement of the Second Law, we'll need to relax this in the light of Special Relativity). The product of this inertial mass and the velocity of the body will be called the 'linear momentum'. However we don't yet know how to find the 'inertial mass' of a body. So now our first target is to understand this number called inertial mass associated with each body.
We start off by denoting the 'inertial mass' of a specific body of interest by the letter (which, of course, we don't yet have any idea how to measure or assign numerical value to. All that we know is is that it is a number which have definite value for the particular body - the body of current interest - and does not vary with speed of the body or with time). Thus, is the linear momentum. Then by the statement of Newton's Second Law, we should measure force by the expression (here we conveniently chose the proportionality constant mentioned in the statement of the law to be 1. We could have chosen something else, but since this constant does not vary across different bodies, its exact value is immaterial.). This essentially is the prescription that the Second Law gives for measuring force. Since we assumed to be invariant with time, . So, upon measuring the value of (which is easy to measure using a scale and a clock as discussed earlier) and knowing the values of , we get the value of force, , as prescribed by Second Law. But here we again encounter another chicken-egg problem. We don't yet know how to assign a value for to a body. We have a single equation, namely , but two unknowns: and (strictly speaking there are 3 equations and 4 unknowns). So how do we resolve this problem? The concept of equal forces developed in the previous section comes to the rescue.

To determine for different bodies we apply equal forces, say , on each of them ( is chosen arbitrarily. We don't yet know how to measure - it's just a notation. But we can say whether or not another arbitrary force, , is equal to . Also, we can apply a force equal to . For example, maintaining the same distance between the electron clouds of the pushing device and the bodies being pushed, as discussed earlier, will impart equal forces on the bodies). So now we can easily measure/observe the rate of change of motion (i.e. ) of each of those bodies due to application of the force . Let's choose one of the bodies and assign it an unit value of inertial mass (For S.I. units this is done for a piece of platinum cylindrical piece called The U.S. National Prototype Kilogram). We apply the force to it and let the measured value of acceleration for that body with unit inertial mass be . Next, for the body of interest, whose inertial mass, , is to be determined, let the measured value of rate of change of motion upon application of be . Then we have, by the Second Law, . Thus, by simple concepts of vector algebra, . Thus, by performing this experiment on each and every body, we get a measure of inertial mass, , for each of the bodies (note that the unit is important here since we have fixed the mass of a standard body to unity).

Thus we now have a way of determining of a body. Also, we can experimentally obtain the acceleration, of any body using a scale and a clock. Thus, we can now measure (in an inertial frame) any force using the prescription of the Second law, i.e. .

A Convention: From now onwards whenever we use a term like "force acting on a particle A due to particle B", we'll mean that the presence of B is creating some sort of Gravitational, Electromagnetic, Strong or Weak fields, which result in some force on the particle A. [:/secondlaw:]

[#-addforces]

Vector addition of forces

Once we have a way of measuring forces (which we just saw are vector quantities), the immediate question that arises is whether or not we can add forces. The notion of addition of forces is more involved than it appears. It may appear very intuitive and convenient to vectorially add forces when multiple forces (originating out of the 4 causes, namely Gravitational, Electromagnetic, Strong & Weak) act on a particle. However often what appear to be too intuitive are things that are most difficult to justify. In fact there is no theoretical justification behind adding forces vectorially. Vector addition of forces is completely an experimental fact. In this discussion we won't go into much details of the consequences of this experimental result, rather just take for granted that we can vectorially add the forces (originating due to the different causes/sources) acting on a particle. [-#]

[:thirdlaw :]

The Third Law:

First of all the Third Law is not a definition, neither is it universally true. It is a common experience that whenever a body is acted upon by some force, some other body (most often that is the cause/source of the force) also feels a force equal to its magnitude and opposite in direction. This was just an experimental observation. In fact this is indeed true (till the accuracy that the experiments could reach) in case of forces like those due to Gravitation and Electrostatic interactions. However this law fails miserably in case of Electrodynamics or Electromagnetism.

[:/thirdlaw:]

[:linearmom :]

Combining Second Law and Third Law - Conservation of Linear Momentum:

If we however assume that the Third Law is true, we can couple it with the Second Law to get a rather interesting result. It can be proved that if some particles or bodies interact with each other by Forces such that the Third Law holds (i.e. if the force on one particle is due to presence of another particle, then the force on the second particle will be due to presence of the first), and if there is no external force acting on the system, then the net linear momentum of all the particles/bodies (the algebric sum of the components of the linear momentums) remain constant with time (of course from an inertial frame). [Refer to any high school physics book for the proof].

Now, this is a very interesting and useful result, since physicists always try to seek entities that remain constant in a system (often they call it Hamiltonian of the system). Such entities turn out to be very handy. So linear momentum and its conservation is much lucrative to us. Hence, even though the Third Law is not valid in many cases, we desire that linear momentum still remains conserved for any system without external Forces. Hence at places where linear momentum wouldn't have been conserved because of the failure of the Third law, we still forcefully keep it conserved by introducing some other conventions and concepts about linear momentum. Say in the case of Electromagnetism, we attach a concept of field momentum and say that whenever there is a decrease of linear momentum in the system (due to violation of Third law in Electromagnetism) there is a corresponding increase in the field momentum and vise-versa. Basically that's the reason we have momentum associated with photons even though they don't have any mass -- and most interestingly, such adjustments do fall in right place very nicely to create a consistent description. [Read the topic 'Electromagnetism' for further details or refer to any good book on Electrodynamics.] [:/linearmom:]

[#-conclusion] So that completes the Second and Third Laws in light of Newtonian Mechanics. For furthur discussions read the topics on 'Special Relativity', 'General Relitivity', 'Electromagnetism'. [-#]