Newton’s Second Law (law or definition?)

Question: State Newton’s second law of motion in words. Explain the meaning of Newton’s second law.

In Principia, Newton states the second law as “[t]he 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.” Simply phrased, this means that the force is proportional to the rate of change of momentum, but we do not find the equation F = ma in the original Newton’s second law. Depending on the grading criteria, students could be penalized when they state Newton’s second law that is related to F = ma instead of F = dp/dt. 

On the other hand, one may explain that Newton’s second law is a definition of force. Currently, there is no agreement on whether Newton’s second law is simply an empirical law or merely a definition. Furthermore, one may debate to what degree Newton’s second law is a definition or a law. However, physicists could be more precise by specifying their definition of definition. For instance, what they have in mind may be a theoretical definition or an operational definition. Alternatively, Newton’s second law can be considered to be a physical model or simply a mathematical relationship between force and motion.

Below are examples of Newton’s second law of motion that are stated in textbooks from different countries. A Russian textbook: A force acting on a body is equal to the product of the mass of the body and the acceleration produced by this force, the directions of the force and the accelerations coinciding. (Landsberg, 1971). A UK textbook: The rate of change of momentum of an object is proportional to the resultant force which acts on the object (Breithaupt, 2000). A US textbook: An object of mass m subjected to forces F1, F2, F3, … will undergo an acceleration a given by a = Fnet/m where the net force Fnet = F1 + F2 + F3 +…  is the vector sum of the individual forces. The acceleration vector a points in the same direction as the net force vector Fnet (Knight, 2004).

How would Feynman answer?

In The Feynman Lectures on Physics, we can find Newton’s second law of motion that is based on the equation F = ma and F = dp/dt. Additionally, Feynman disagrees that Newton’s second law is simply a definition. We will discuss possible answers of Feynman from the perspective of F = ma, F = dp/dt, and problems of defining force.

1. F = ma 

Although Feynman often makes fun of philosophers, he is interested in the meaning of knowledge, and thus opines that it is always important to ask, “What does it mean?” In Feynman’s words, “‘What is the meaning of the physical laws of Newton, which we write as F = ma? What is the meaning of force, mass, and acceleration?’ Well, we can intuitively sense the meaning of mass, and we can define acceleration if we know the meaning of position and time. We shall not discuss those meanings, but shall concentrate on the new concept of force. The answer is equally simple: ‘If a body is accelerating, then there is a force on it.’ That is what Newton’s laws say, so the most precise and beautiful definition of force imaginable might simply be to say that force is the mass of an object times the acceleration (Feynman et al., 1963, section 12–1 What is a force?) Importantly, he emphasizes that the force is supposed to have some independent properties, for example, it has a material origin, and thus, it is not just a definition.

In addition, Feynman explains that “the acceleration a is the rate of change of the velocity, and Newton’s Second Law says more than that the effect of a given force varies inversely as the mass; it says also that the direction of the change in the velocity and the direction of the force are the same (Feynman et al., 1963, section 9–1 Momentum and force).” Similarly, according to Feynman, “we see that Newton’s Second Law, in saying that the force is in the same direction as the acceleration, is really three laws, in the sense that the component of the force in the x-, y-, or z-direction is equal to the mass times the rate of change of the corresponding component of velocity: F_x = m(dv_x/dt) = m(d²x/dt²) =ma_x, F_y = m(dv_y/dt) = m(d²y/dt²) = ma_y, F_z = m(dv_z/dt) = m(d²z/dt²) = ma_z (Feynman et al., 1963, section 9–3 Components of velocity, acceleration, and force).” Essentially, physicists define the force in terms of F = ma in Euclidean space. Thus, we can visualize the force as a vector such that the directions of the force and acceleration are the same.

Interestingly, Feynman disagrees that F = ma is a definition because it is not exactly true. Firstly, Feynman mentions that “[o]ne might sit in an armchair all day long and define words at will, but to find out what happens when two balls push against each other, or when a weight is hung on a spring, is another matter altogether, because the way the bodies behave is something completely outside any choice of definitions (Feynman et al., 1963, section 12–1 What is a force?).” Note that we have idealized the equation F = ma and prediction cannot be simply made from a mathematical definition. Secondly, Feynman clarifies that “[t]he forces on a single thing already involve approximation, and if we have a system of discourse about the real world, then that system, at least for the present day, must involve approximations of some kind. (Feynman et al., 1963, section 12–1 What is a force?) That is, Newton’s second law is not exact and it is important to understand that this physical law involves idealizations and approximations.

2. F = dp/dt

Historically speaking, Newton proposes the second law of motion as the rate of change of motion instead of the product of a mass of an object and its acceleration. However, Feynman states that “the motion of an object is changed by forces in this way: the time-rate-of-change of a quantity called momentum is proportional to the force (Feynman et al., 1963, section 9–1 Momentum and force).” To be precise, Feynman uses the term momentum instead of motion that is adopted by Newton. Furthermore, he specifies force as the time-rate-of-change of momentum. This is more precise because the rate of change of momentum could be with respect to displacement instead of time. However, Feynman’s statement can be further improved. First, we can be more precise by using the term linear momentum that distinguishes from angular momentum. Better still, the word proportional can be replaced by directly proportional.

Feynman also mentions that “Newton’s Second Law may be written mathematically this way: F = d(mv)/dt. Now there are several points to be considered. In writing down any law such as this, we use many intuitive ideas, implications, and assumptions which are at first combined approximately into our ‘law.’ … First, that the mass of an object is constant; it isn’t really, but we shall start out with the Newtonian approximation that mass is constant, the same all the time, and that, further, when we put two objects together, their masses add. These ideas were of course implied by Newton when he wrote his equation, for otherwise it is meaningless. For example, suppose the mass varied inversely as the velocity; then the momentum would never change in any circumstance, so the law means nothing unless you know how the mass changes with velocity (Feynman et al., 1963, section 9–1 Momentum and force).” However, particle physicists prefer Newton’s second law to be written as F = d(γmv)/dt in which the Lorentz factor, γ, equals to 1/(1 – v²/c²)^1/2 and c is the speed of light. This is related to the concept of invariant mass that is velocity-independent.

Moreover, Feynman explains that “there is another interesting consequence of Newton’s Second Law, to be proved later, but merely stated now. This principle is that the laws of physics will look the same whether we are standing still or moving with a uniform speed in a straight line. For example, a child bouncing a ball in an airplane finds that the ball bounces the same as though he were bouncing it on the ground. Even though the airplane is moving with a very high velocity, unless it changes its velocity, the laws look the same to the child as they do when the airplane is standing still. This is the so-called relativity principle. As we use it here we shall call it ‘Galilean relativity’  to distinguish it from the more careful analysis made by Einstein, which we shall study later (Feynman et al., 1963, section 10–2 Conservation of momentum).” In other words, Newton’s second law is valid in an inertial frame of reference in which every free particle moves with a constant velocity.

3. Problems of defining force

Newton’s second law of motion is commonly known as a law of force or a definition of force. Feynman would discuss problems of defining force such as context, precision, and circularity as shown below: 

Context: Feynman mentions that “[m]omentum is not the same as velocity. A lot of words are used in physics, and they all have precise meanings in physics, although they may not have such precise meanings in everyday language (Feynman et al., 1963, section 9–1 Momentum and force).” Similarly, the term force has alternative definitions in the everyday context and technical context. For example, a definition of force in a dictionary or everyday language is “energy.” Moreover, Feynman clarifies that “[t]he first term is the mass times acceleration, and the second is the derivative of the potential energy, which is the force (Feynman et al., 1964, section 19–1 A special lecture—almost verbatim).” Depending on the context, force may be defined as “mass times acceleration,” “time rate of change of linear momentum,” or “derivative of the potential energy”, and thus, the term force could be confusing to students.

Precision: Feynman explains that “[t]he student may object, ‘I do not like this imprecision, I should like to have everything defined exactly; in fact, it says in some books that any science is an exact subject, in which everything is defined.’ If you insist upon a precise definition of force, you will never get it! First, because Newton's Second Law is not exact, and second, because in order to understand physical laws you must understand that they are all some kind of approximation (Feynman et al., 1963, section 12–1 What is a force?).” To illustrate this fact, Feynman gives the example in which the mass of a chair can be defined only approximately. He argues that it is difficult to distinguish the atoms that are chair, air, dirt, or paint.

Circularity: Feynman provides the following insights: “[w]e could also define force to mean that a moving object with no force acting on it continues to move with constant velocity in a straight line. If we then observe an object not moving in a straight line with a constant velocity, we might say that there is a force on it. Now such things certainly cannot be the content of physics, because they are definitions going in a circle (Feynman et al., 1963, section 12–1 What is a force?).” In a sense, it suggests that Newton’s first law and second law are circular: the first law states that zero force does not result in a change in velocity, whereas second law states that a force results in a change in velocity. Thus, both statements are essentially similar and the first law may be considered as a special case of second law. However, this is different from another circularity in which force and mass are defined based on Newton’s second law. That is, one should not define force by using the equation F = ma, and then define mass by using the equation m = F/a. (Some prefer to define mass using E/c².)

To conclude, Newton’s second law of motion can be stated based on the equation F = ma or F = dp/dt. To be more accurate, the concept of force should be defined as the time rate of change of linear momentum instead of simply the product of mass and acceleration. Importantly, Feynman disagrees that Newton’s second law is simply a definition because it is not exactly correct and it can be falsified by experiment. Furthermore, there are idealization and approximations in this physical law of force as well as it is valid in an inertial frame of reference. However, we should be cognizant of problems in defining force. 

References:

1. Breithaupt, J. (2000). Understanding Physics for Advanced Level (4th ed). Cheltenham: Stanley Thorne. 

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley. 

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley. 

4. Knight, R. D. (2004). Physics for Scientists and Engineers with Modern Physics. California: Addison-Wesley. 

5. Landsberg, G. S. (1971/2000). Textbook of Elementary Physics, Volume I. (A. Troitsky, Transl.) Honolulu, Hawaii: University Press of the Pacific.