Philosophical Discussion of Newton's Second Law

Question

I was thinking a bit more deeply about Newton's Second Law the other day, having obtained a copy of The Principia. Newton originally describes his Second Law as such:

"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."

In one of the earlier pages of his text, Newton defines the motive quantity of a force as a measure of the force that is proportional to the motion it generates in a given time. (Interestingly, we use the term 'momentum' to describe what Newton referred to as 'motion'.)

Lastly, Newton defines motion as a quantity of a body jointly arising from its mass and velocity combined.

Now, I was thinking about the extent to which this law is a numerical definition of a force. For it to be a numerical definition of a force, we would need to independently define mass. (Velocity and acceleration are easily defined as the first and second time derivatives of position, and position can be defined through the usage of an arbitrary coordinate system.) When working with non-relativistic masses, one can safely assume that the mass of a body is an intrinsic property, and we simply define a block at The International Bureau of Weights and Measures to have a mass of one kilogram. Now, we need a way to find the mass of an arbitrary body given this definition of one kilogram, so that we can treat Newton's Second Law as a definition of force.

The answer I was given was to use a spring. Extending a spring (with a block attached to it) by a certain length and releasing it results in motion of the block. We can analyze this motion to find the initial acceleration of the block. After experimenting a bit more, we realize that extending a spring by the same distance results in the block having an identical acceleration. (Now here's the bit I do not entirely follow along with.) The rationale then provided to me was that since the acceleration of the block is identical when displaced from the equilibrium by the same distance, the force exerted on the block must be the same when the displacement from the equilibrium is the same. Hence, we realize that the force exerted by a spring must be a function of its displacement from its equilibrium position. We arrive at the conclusion that releasing a block twice as massive from the same position results in half the acceleration. My question is: How does one arrive at this conclusion without using the Second Law, and therefore employing means of circular reasoning?

Answer 1

I'll try to write a short answer, focusing directly on your main question that I recast in the correct form as

How does one conclude that releasing a block twice as massive from the same position results in half the acceleration without using the Second Law, therefore employing circular reasoning?

Such a question may have circular or non-circular answers depending on the exact statement and interpretation of the principles of Mechanics.

Everything depends on the exact definition of the two key concepts of mass and force. We also need to be aware that in the history of mechanics, different, non-equivalent solutions have been proposed for such a central step.

In particular, the point of view taking the Second Law as the definition of the force (which was not Newton's point of view) requires an independent definition of the mass. That can be done in Mach's approach by defining the ratio of masses of two interacting forces as the inverse ratio of the modulus of the corresponding accelerations. After defining masses in this way, it is possible to say that Newton's Second Law defines the force without circular reasoning.

However, we must note that Mach's approach is not the only possibility. We can equally well introduce forces as primitive quantities, thus defining the mass as the ratio between the modulus of the force and of acceleration. What should not be allowed is to mix different approaches.

Answer 2

Newton's original philosophy is not particularly relevant to modern physics. He had the first word on the theory that bears his name, not the last word. There is no need and no particular benefit to following his historic reasoning.

For it to be a numerical definition of a force, we would need to independently define mass. ... we simply define a block at The International Bureau of Weights and Measures to have a mass of one kilogram. Now, we need a way to find the mass of an arbitrary body given this definition of one kilogram

The kilogram is no longer defined by the block at the BIPM. It is now defined in terms of Planck's constant. The BIPM also provides a Mise En Pratique for measuring mass accurately in kilograms. A mass standard can be measured using e.g. a Kibble balance or by counting particles using e.g. x-ray crystallography. An arbitrary mass can then be compared to a mass standard using e.g. a balance scale.

Whether or not this process is "independent" of Newton's 2nd law is not important. What is important is that it is consistent with Newton's laws. Also, the important thing about Newton's 2nd law is not the mass, but the net force. In other words, the important thing is that forces add like vectors rather than combining in some other way.

Answer 3

So that there can be no mistake, I will list the relevant definitions exactly as they appear in my edition of the Principia, which is the translation by Bernard Cohen and Anne Whitman, and provide some observations that take advantage of modern mathematical notation. (We could cite the original Latin, but it no longer common for academics to know how to read Latin anymore.) Much clarification is needed to address your question.

Definitions

1. Quantity of matter is a measure of matter that arises from its density and volume jointly.

Let's call this quantity of matter "mass" and denote it by $m$. Observe that Newton defines $m := \rho V$, where of course $\rho$ is density and $V$ is volume. He does not define $\rho := m/V$, as is common in introductory physics textbooks. That is, $\rho$ and $V$ have priority for Newton as directly measurable quantities. How so? We take as self-evident that identical materials have identical densities. We can define two objects to have equal volume if they displace equal volumes of water. Then, for two objects having equal volume, we define one to be more dense than the other if the one tips a balance scale in its favor. These considerations show that Newton's definition here is inspired by Archimedes and reminds us (as the struggles of every intro physics student show) that mass is not necessarily an intuitive concept (in fact, it is often confused with weight).

2. Quantity of motion is a measure of motion that arises from the velocity and the quantity of motion jointly.

We call this quantity of motion momentum and denote it as $\vec p$. You are right to observe that Newton considers momentum to be the true quantity of motion. This contrasts with Galileo and Leibniz who believed that the true quantity of motion was vis viva or "living force", which is defined as $mv^2$ or twice the kinetic energy as it is known today (cf. "virial theorem", where the mean potential energy is twice the negative of the mean kinetic energy). More interestingly, the Lagrangian and Hamiltonian formulations of classical mechanics (insofar as they are derived from the Newtonian framework) would seem to suggest that energy is the true quantity of motion. Thus, modern physics seem to side with Leibniz in more than just notation. Of course, it is fair to ask why there should be only one "true" quantity of motion. However, the question nowadays is moot since momentum and energy are unified in the modern relativistic framework via 4-vectors. Even more interestingly, the principle of relativity hearkens all the way back to Galileo, if I am not mistaken, in his work Dialogue Concerning the Two Chief World Systems.

3. Inherent force of matter is the power of resisting by which every body, so far as it is able, perseveres in its state either of resting or of moving uniformly straight forward.

We call this inherent force of matter inertia. This definition is more or less the adoption of a claim from Avicenna (who was known to the Scholastics with whom Galileo frequently quarreled) that a body moving in vacuum would not stop unless acted upon. Although Newton undoubtedly had in mind Euclidean geometry, if we interpret "straight forward" to mean along its geodesic, then we have a perfect statement of general relativity. The only thing really missing is a decision whether the speed of information transmission should be finite or infinite. Unfortunately, Newton implicitly chose an infinite speed of information transmission when he elaborated his conception of absolute time, but we can hardly fault him considering what a breakthrough it was to present his laws of motion and universal law of gravitation in the first place.

4. Impressed force is the action exerted on a body to change its state either of resting or of moving uniformly straight forward.

We call the impressed force impulse. Newton here defines impressed force as the cause of a change in an object's quantity of motion, i.e., momentum. It is a philosophical principle that a thing cannot cause its own change except insofar as one part of a thing acts on another part. Thus, change requires some cause, and Newton here simply singles out those causes of change in a body's quantity of motion and gives them the name force.

Newton goes on to define central forces, but that is superfluous to your question.

Laws of Motion

1. Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.

Newton's first law defines inertial frames, in which his remaining laws hold.

2. A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.

Using modern notation $$\text{impressed force} = \int \mathrm dt \, \vec F \propto \vec p$$ taking the derivative of both sides with respect to time yields Newton's second law in the usual differential form.

Now to answer your question:

I can restate what you say at the end of your penultimate paragraph in the following manner: "If we define 1 kg as the mass of a certain bar of platinum, how can we determine the mass of an arbitrary body?" The simplest answer is not to use a spring but rather to use a balance scale, as I mentioned in my remarks on Newton's first definition. Given the 1 kg mass, we can make a 1/2 kg mass and a 1/3 kg mass and so on. (I leave it as an exercise to the reader to figure out how to make these.) Given these, we can determine the mass of an arbitrary body to an arbitrary degree of precision.

There are some misconceptions in your last paragraph concerning using a spring. The point of using a spring is that, for small enough displacement from equilibrium, the resulting force (i.e., cause of change in quantity of motion) on an attached body is approximately linear in $x$ and thus obeys Hooke's law $F=-kx$. That a device (the spring) should exert a linear force is a matter of empirical observation (look at the change in velocity, i.e., the acceleration). This linear force is related to mass via Newton's second law by definition.

There is no circularity in the logic. $ma = F = -kx$. The first equality holds by definition, the second by empirical fact. Solving for $m$ is a viable way to calculate the mass, assuming you already know $k$. How do you determine $k$? Well, that is a topic for another question, but it shouldn't be too hard to figure out ;)

Also, you might want to rework your algebra. $m \to 2m$ does not mean you have twice the acceleration.

Answer 4

We arrive at the conclusion that releasing a block twice as massive from the same position results in half the acceleration. My question is: How does one arrive at this conclusion without using the Second Law, and therefore employing means of circular reasoning?

This is a statement that was obtained either by some observations, experiments, or by a logical/mathematical argument based on some assumption (Newton's 2nd law).

Logic and mathematics can derive a lot of things, but only if we put some assumptions in. No assumption at all? Nothing can be derived. The more you assume, the more you can derive. 2nd law is such an assumption that allows us to derive a lot. It is a generalization of those observations and experiments we do not want to repeat everytime we think of questions in mechanics.

If you do not want to use 2nd law as an assumption to find inertial mass, because you want to use $m\mathbf a$, with already known mass $m$, to define the concept of force, you are in a bad situation, because this was never successfully done, and not for lack of trying. The problem with this is that thinking of 2nd law as definition of force can be used to determine net force on the body from its mass and acceleration, but it can't be used to determine all different real impressed forces that act on the body. The fact net force is a vector sum of individual acting forces does not follow from this tentative definition.

For example, $\mathbf F_{net}=m\mathbf a$ does not define what all the acting forces are when acceleration of the body is zero, like in statics, which is an important part of mechanics. The concept of force is richer than what 2nd law can imply, already in non-relativistic mechanics.

And in relativistic mechanics, we've realized that $\mathbf F_{net}=m\mathbf a$ is only an approximate law to the more correct $\mathbf F_{net} = \frac{d\mathbf p}{dt}$.

So using $\mathbf F_{net} = m\mathbf a$ to define the concept of force has problems; these perhaps can be overcome by adding more statements defining all the other properties of the concept of force (it is a vector, more forces add as vectors to create net force), but then 2nd law alone, whether $\mathbf F_{net} = m\mathbf a$, or $\mathbf F_{net} = m\frac{d(\gamma \mathbf v)}{dt}$, is not a complete definition of all forces in general.

The traditional viewpoint is much better, forces are primary, they are vectors that add up as vectors to create net force, by definition. Then 2nd law is an approximate law for this net force, that comes as generalization of observations. And this approximate law can be then used to define inertial mass of a body.