Mr. Newton’s journey to the gravitational formula
Hello Readers,
Today’s journey will be, as the title suggests, “Mr. Newton’s journey to the gravitational law” — an imaginary journey with Mr. Newton as our guide. So you must be wanting to know what all we will face in this little adventure of ours. Hence, first some travel plans for you.
When in high-school, physics text books stated that prior to Newton there were Kepler’s laws of planetary motion. Based on these laws Newton discovered his celebrated “Universal theory of gravitation”. And one very common exercise of the text books was to derive Kepler’s laws out of Newton’s law of gravitation (and of course Newton’s three laws of motion). The problem lay in the fact that some teachers claimed this to be the proof of Newton’s Gravitational Formula. That is like saying Newton went to sleep, the entire formula just clicked in his mind, he got up and derived Kepler’s laws out of this one formula and lo and behold! He had his grand theory. That story is as good as saying scientists do science accidentally, not intentionally. Thus although the exercise of deriving Kepler’s laws out of Newton’s law of gravitation is a good exercise in itself, that’s surely not the way Newton arrived at the formula.
So in context of the above, our aim would be to try to imagine how Newton might have reached his theory standing on “Kepler’s shoulders”. I am sure most of you already know the gravitational force formula -
Thus the formula has four aspects: G (the gravitational constant), m1, m2 and r. We will investigate how Newton reached at each of these four elements entering the formula in the way they do.
1/r²:
The force is proportional to 1/r², r being the distance between the two objects (point objects). This is a standard derivation based on Kepler’s laws and uses a lot of Calculus and some formula of ellipses. I studied this as part of an exercise while going through “Differential Equations with Applications and Historical Notes” by George F. Simmons. This probably is a part of undergraduate or graduate physics curriculum. Probably this was also the hardest portion for Newton to arrive at because he did not have such an advanced calculus theory at hand (rather he had to develop it himself). But we will not be proving that here. It is all mathematical gymnastics, beautiful but rigorous (and dry).
m1:
We are studying the gravitational force between two point objects. First we study the gravitational force of object 1 over object 2, or mass 1 (m1)over mass 2 (m2). We will call object 1 the central body and object 2 the peripheral body. This terminology will be very useful down the line. So gravitational force is proportional to the mass of the central body m1. Why so?
For that please note the below figure:
Here m1, m2 and m3 are three masses. F_31 (bold denotes they are vectors) is the force by m1 on m3 and F_32 is the force on m3 by m2.
Now the net force on m3 is F_net = F_31 + F_32. This is the superposition principle in play. The gravitational force on one body due to various other other bodies are all independent of each other. Hence the net gravitational force is just a summation of the various independent forces, not some crazy net force produced out of the intermingling of the individual forces. Thus F_net does not equal say m2.F_31 + m1.F_32. This superposition principle, as many of you might know, is a fact displayed by nature and is thus based on our day to day observations.
Now had F_31 and F_32 been directed along the same line the forces would have simply added up. Keep this in mind. It will be used next.
Now suppose you have the central mass m1 attracting peripheral mass m2 with a force F. On top of m1 you place another similar point mass m1. What is the force on m2 now? F + F = 2.F (since the forces are similarly directed, rather similar). The force is twice. Great.
What is the mass of this couplet of two m1 s that we now have as the central body? m1 + m1 = 2.m1. This says something that is easy to miss given the obviousness of its nature. It says that masses, when superimposed on each other, add up like scalars. This is again a fact of nature. If we tie up two m1 s together, whether the combination behaves like m1² or 3.m1 or 2.m1 is not in our hands. That is something nature displays.
Owing to the logic presented in the above two paragraphs we see that as the mass of the central body is doubled or tripled the force doubles or triples respectively. Thus the force is proportional to the mass of the central body m1. And remember this is an outcome of two facts of nature — the superposition of forces and the scalar addition of mass (when defined as the proportionality constant between force and acceleration) on superimposition.
So have reached that the gravitational force is proportional to m1/r². We move on.
m2:
How come the peripheral mass enters the formula in the same way that the central mass does? The gravity of the question becomes clearer if you ask yourselves the following (just as I did):
Why is it m1.m2 in the formula? Why not m1.m2² or just m1?
I personally feel the answer to this question leads us to the profoundest of truths about nature. However, before going for the answer we need to take a look at Newton’s third law of motion once.
Every action has an equal and opposite reaction.
The force on mass m2 by m1 F_21 (here m1 and m2 are just two masses, not necessarily our central and peripheral masses) is equal and oppositely directed to the force on m1 by m2 F_12. This happens for every pair of bodies in the universe. That’s all it says.
Now we return to our original “why m1.m2?” question regarding gravitational force.
The gravitational pull on peripheral body m2 by central body m1 is K(m1/r²). Now let us study the pull on the central body m1 by m2. That is, now our “central” and “peripheral” bodies get exchanged. We hope nature shows symmetry in the sense that when our bodies get exchanged the nature of gravitational force (and hence its defining formula) remains the same. After all, why should nature and hence gravitation (being a child of nature) distinguish one body from another. Thus the pull on m1 by m2 should be K(m2/r²).
But, according to Newton’s third law of motion, is not the pull on m1 by m2 “equal” in magnitude to the pull on m2 by m1. Thus, we should have -
K(m2/r²) = K(m1/r²) where m1 and m2 are not necessarily equal and definitely not 0.
This would be lead to a contradiction and the only way out of the contradiction is that the force depends on the “peripheral mass” and the “central mass” in a similar way. That is, if the force is proportional to the mass of the central body it is also proportional to the mass of the peripheral body. Hence “m1.m2”.
Thus proportionality to the peripheral mass is an outcome of the “symmetrical nature of gravitational force equation between a pair bodies” and “Newton’s third law of motion” trying to be in sync with each other.
Now the final aspect G.
G:
This one is based more on the history of gravitation, as I have read. First, Newton took the apple and earth; earth is the central body and apple the peripheral one. Apple is pulled towards the center of the earth. By what force? One that is proportional to (m_Earth.m_Apple/r_EarthApple²). Thus
F_EarthApple = G_EarthApple.(m_Earth.m_Apple/r_EarthApple²)
Here G_EarthApple is some constant that is peculiar to the system containing earth and apple.
Now Newton observes the moon and identifies that while going round the earth what the moon is actually doing is constantly accelerating towards the center of the earth, just like the apple. So maybe moon is attracted by the earth through a similar force of gravitation. If it be so, then-
F_EarthMoon = G_EarthMoon.(m_Earth.m_Moon/r_EarthMoon²)
Now according to Newton’s second law of motion F=ma -
a_Apple/a_Moon [accelerations of apple and moon]
= (F_EarthApple/m_Apple)/(F_EarthMoon/m_Moon)
= (G_EarthApple/G_EarthMoon) x (r_EarthMoon/r_EarthApple)²
At this point, Newton observed (through astronomical observations) that the ratio of the acceleration of apples (a_Apple) and acceleration of moon (a_Moon) was the same as (r_EarthMoon/r_EarthApple)². This led to the fact that-
G_EarthApple = G_EarthMoon
Thus the proportionality constant that previously was thought to be peculiar to the system involving a particular pair of bodies is actually the same for any pair of bodies throughout the universe. Hence it is G throughout the universe. Herein lies the “universality” of the theory of gravitation; herein lies the generalization to a “universal” theory of gravitation.
Thus the climb up the mountain is complete, the summit point being -
F = G(m1m2/r²)
I hope this gave a flavour of the immense nature of Newton’s work. But does it not also show that truth can be reached through deeper and deeper levels of investigation? Science, being a pursuit of truth, is supposed to follow a similar pattern. As I set out to show at the very beginning -
Science is intentional not accidental, progressing through ever deepening inquiries in a systematic way.
Goodbye! Until next time.
