# Newton's Third Law As Discussed in Den Hartog's Mechanics

A question, but some preliminaries in order to avoid answers involving concepts not yet introduced in the text I'm reading--Den Hartog's Mechanics.

Near the very beginning of his book, Den Hartog tries to articulate the intuitive notion of (i.e., some might say 'defines') 'mechanical contact force' in his introductory comments on statics (page 3):

"If there is any doubt as to whether there is a mechanical contact force between two bodies, we may imagine them to be separated by a small distance and a small mechanical spring to be inserted between them, with the ends of the spring attached to the bodies. If this spring were to be elongated or shortened...there would be a direct contact force."

In relation to springs, he notes that a 'dynamometer", "force meter" or what we might call a 'scale' is used for the practical measurement of force:

"In a spring the elongation (or compression) is definitely related to the force on it so that a spring can be calibrated against..."

He then states (p.4) the first axiom of statics AKA Newtown's third law. He gives about the simplest possible illustration of the axiom: a book at rest on a table, stating

"the push down on the table by a book is equal to the push up on the book by the table."

(a still simpler case, I suppose, would be book resting on earth)

He continues:

"Thinking about our imaginary experiment of inserting a thin dynamometer spring between the book and the table, the proposition looks to be quite obvious: there is only one force in the spring."

I'm puzzled by this last (bolded) comment. Let me try to be more specific. I've seen this experiment (both in thought and laboratory) described a little differently: In what I recall, two (not one, but two) spring-operated scales are hooked to each other, back-to-back -- i.e., one connected to table, the other connected to the book, and the two scales connected to each other in the middle. In this type of arrangement, one scale measures the push of the book, and the other measures the push of the table, and we find (consistent with the Third Law), that they come out equal.

Each of those two equal forces acts on the connected spring system in opposite directions and everything remains at rest as expected.

It's also clear that you can think of this connected double-spring system as a single spring, essentially no different than Hartog's set up, with the exception that this arrangement has two force measuring pointers, one at each end.

Again, it is really his conclusion that the third law in this case is obvious or makes sense because

"there is only one force in the spring,"

that I find as odd. Is it even a true statement? What I think is true is that the spring measures the force acting on it by the book (from above) and the opposite and equal force of the table (from below) and is therefore at rest. There are two forces (from two different objects) acting on one object (the spring).

Any thoughts appreciated!

• Here is a link to Den Hartog's book. Click on the 'page 4' hyperlink.books.google.com/… – pi kappa Jan 24 '16 at 23:07
• A spring has two ends and there is a force on each end. If there was only one force the spring would move. – Peter R Jan 25 '16 at 0:12
• Hmm strange wording indeed. Does he mean that it's quite obvious that a spring is exerting one value of a force, e.g. 10 Newtons, in opposite directions, such that Newton's 3rd law obviously holds? – innisfree Jan 25 '16 at 4:33
• To me, the phrase "force in the spring" is meaningless. What does it mean for a force to be in something? I think your understanding is correct; Hartog's strange language is throwing you off the track. – garyp Mar 14 '18 at 12:48

## 4 Answers

Newton’s third law is logically independent from the rest. It’s completely impossible to prove it using the other two laws and “logic” unless you slip it in as an assumption somehow. That’s exactly what Hartog is doing, by using vague phrases like "the force in the spring", or a mathematically ill-defined dynamometer.

I think it's important to distinguish between physical and logical validity. Newton's third law really is physically intuitive, if you've lived in this world, and maybe Hartog's examples can help show that. But they are not logically valid, and indeed cannot be -- Newton's third law is our starting point, not our ending point.

• Yes, that was my reaction as well-- the reasoning is circular, but the circularity is well concealed so it's easy to miss. Perhaps we should say Hartog is making the third law seem like a plausible way to build a universe, not that the universe is logically required to work that way. At some level, that probably suggests it doesn't exactly work that way! – Ken G Apr 14 '18 at 2:59

Even though im not completely sure, where your exact problem is, ill give it a try:

Lets first think about a book lying directly on earth's surface:

First of all theres a gravitational force $F_G=\frac{m_1\cdot m_2}{r^2}$ This force does not specify which object is "the pulling one" and which object is "the pulled one" - its just a force, acting between two objects, pulling both together. Of course since the earth's mass is "sligthly" greater than the books mass, the actual acceleration $a = \frac{F}{m}$ will be "slightly" different, but thats not the point here.
What we see is, that the acceleration seems to be $a = 0$ here for both objects - so we need some force which adds up together with gravitation to $F=0$ and call it contact force.

Thats exactly the point where this little imaginary experiment comes in:
We assume theres a spring between the two objects and ask about "how much it is compressed".
From our perspective we can say it acts as a scale, measuring the weight of the book on earth. But we could as well say its a scale, measuring the weight of the earth on the book. Or we could say its just a spring, pushing the two objects away from each other with exactly $F_G$.
All descriptions are the same, we are just used to talking about a scale if one object has a relative high mass (e.g. earth) and the other has a relative low mass (e.g. book) while we are used to talk about a spring if they have comparable masses (e.g. two books).

You could as well think about the two connection points of the spring.
Which forces act on the connection point between the spring and the book?
Obviously what we are used to refer to as "gravitational force on the book" and the force of the spring, caused by its compression. Since the point is not moving, both must be equal in strength and opposite in direction.
Which forces act on the connection point of the spring with the earth?
Obviously the force of the spring, cause by its compression. Since we are talking about the same spring, it needs to be the same force as before (just the opposite direction). And since this point is not moving as well, there needs to be a second force counteracting it - you might be tempted to call this a contact force, keeping the spring (and the book) from moving "into earths surface", which i would not call wrong, but lets think for a moment where this force comes from. It's basically the gravitational force from the book on the earth.

Lets think about a last experiment: Think of a stack of massless scales on earth's surface with a book on top.
What weight would the top scale measure? Of course, the weight of the book.
What weight would the lowest scale measure? Since all scales are massless, it would still measure the weight of the book, no matter how many scales are in between.

All of this boils down to one conclusion: If you have a book lying on earth's surface, there are two forces: The gravitation and some sort of "contact force" counteracting the gravitation. If we model this contact force as a spring, only one force is left to compress this spring: gravitation. From what i've read, thats the point Den Hartog is trying to make.
Of course this gravitational force acts on two objects: The earth and the book. And thats the point the experiment with two springs is trying to make.
The same applies to the contact force: Theres only one contact force keeping the objects from moving into another. But it also acts on both objects.

At the end, all of the descriptions are the same, its just a different view.

• Disclaimer: English is a foreign language for me, feel free to edit any spelling/grammar-mistakes – Anedar Jan 25 '16 at 0:26
• Thanks. I agree with much of what you've written. But Is it correct to say, (paraphrasing Den Hartog) that there is only one force acting in the at-rest spring ? As to there being only one contact force keeping the objects from moving into each other, but it also acts on both objects, that's incoherent unless it's short hand for "magnitude." What is true is that the contact force of the spring on the book has the same magnitude but opposite direction of the contact force of the spring on the earth/table etc. Of course, I could be missing some obvious point. – pi kappa Jan 25 '16 at 2:25
• To be more specific about my objection, see Hartog's comment on page 4: "Contact forces are always exerted by one body on another body." I see the action (force) of the spring on the book which has the same magnitude but opposite direction of action of the spring on the earth/table etc. But that is two forces. Of course, I still think I'm missing some obvious point of Den Hartog's (or yours) but don't see it. – pi kappa Jan 25 '16 at 2:35
• Hmmm, thats a question about the definition of "1 force". If you define force as wikipedia does ("any interaction that, when unopposed, will change the motion of an object") then you always need "2 forces". So i think what Den Hartog is trying to tell you is this: Whether you refer to it as "1 interaction resulting in 2 forces" or "1 force resulting in 2 objects beeing affected" is the same. And from my experience its common to refer to a force as "an interaction between two objects" - you just dont have the case where a force only affects one object. – Anedar Jan 25 '16 at 2:49
• I agree. There is an interaction between two objects. And if we're going to be sufficiently specific in accord with the third law we need to specify how each object invloved is affected. – pi kappa Jan 26 '16 at 23:47

Talking strictly about Newtonian mechanics, if an object has no acceleration, it means that no force is applied on it. Once you know the result force applied is zero, you can deconstruct it any way you want, as long as the sum still stays zero. Deconstructing non-accelerating forces is like doing: 0 = 1 - 1 = 2(1 - 1) etc.

On a more advanced notion though, forces were invented by Newton because of his lack of knowledge, at the time, about potentials. You see, forces are a though construction made to soften things a bit. What is really important in mechanics is energy and it's different forms.

Den Hartog is only trying to explain things. He might have used the term "force" in an informal way. Since the spring is not moving, the two forces on the ends of the spring must be equal in magnitude and directed in the opposite ways. This one magnitude of force is "one force" he is talking about.

If you think about it, when you use a dynamometer you are already assuming the third law, because a dynamometer measures only the force acting on it, not the force it is imposing on other bodies.