# How Did Newton Find Acceleration due to Gravity? (If he had, indeed)

Well, this is my first post here. Hoping for fun times.

So, straight to topic, my title describes what I want to know. I tried searching online, but I did not come upon anything that explains how Newton came to find the value 9.8m/s^2, or even whether he did.

Also, if you happen to know, how did Newton come up with his law of universal gravitation? [G=m1*m2/r^2]

All replies will be appreciated. Thank you!

tiny-tim
Homework Helper
welcome to pf!

hi damnmad! welcome to pf! (try using the X2 icon just above the Reply box )
I tried searching online, but I did not come upon anything that explains how Newton came to find the value 9.8m/s^2, or even whether he did.

Galileo found that (before Newton, of course) … he did very accurate experiments with balls rolling down inclined calibrated pieces of wood.

(do remember that Newton didn't discover gravity, he only discovered that the same gravity applies to the moon and planets )
Also, if you happen to know, how did Newton come up with his law of universal gravitation? [G=m1*m2/r^2]

He got it from Kepler's laws and centripetal acceleration …

see http://en.wikipedia.org/wiki/Newton's_law_of_universal_gravitation#History" for some details Last edited by a moderator:

hi damnmad! welcome to pf! (try using the X2 icon just above the Reply box )

Galileo found that (before Newton, of course) … he did very accurate experiments with balls rolling down inclined calibrated pieces of wood.

[...]

Why thank you, Tim, for a most informative reply. Cleared up everything Now I know that after my death, I can rest in peace, after I try to replicate Galilio's experiments, of course.

Again, thanks. I owe you one, and I will forever :tongue:

Cleonis
Gold Member

Galileo [...] did very accurate experiments with balls rolling down inclined calibrated pieces of wood.

Elaborating:
The setups with balls rolling down inclines were used to investigate whether the Earth's gravity gives uniform acceleration. If acceleration is uniform then covered distance is proportional to the square of time.

With balls rolling down an incline not all work done goes into linear velocity. Some of the work done goes to spinning of the ball. Galileo didn't have the means to assess how much force went into the spinning. It was only later that the mechanics of spinning solids was developed.

In Newton's time the most accurate method available for measuring the Earth's gravitational force was to find the period of a pendulum. The period of a pendulum is a function of it's length and the gravitational force.

Most likely Huygens had obtained the magnitude of the Earth's gravity that way, and no doubt Newton redid those measurements on his own. It would be interesting to find out how accurate Newton's measurements were.

Cleonis
Gold Member
Also, if you happen to know, how did Newton come up with his law of universal gravitation? [G=m1*m2/r^2]

In the years before the Principia was published a number of scientists were aware that if the Sun exerts an force that falls off with the square of distance, then the resulting planetary orbits are a close match for Kepler's third law. Among them were Robert Hooke, Christopher Wren, Halley.

But the only case for which they could prove that was for perfectly circular orbits. The actual orbits of the planets are very close to circular, but not perfectly. Kepler's first law states that the planets are ellipses, and Hooke, Wren and Halley didn't have the mathematical means to handle that.

Newton's achievement was that he provided mathematical proof. To do that he had to develop a new branch of mathematics, which today we call 'differential calculus'.

Newton did not share his new mathematics with others. He kept his 'method of fluxions' to himself. In the Principia Newton used reasoning with geometrical constructions to establish the same results. One can say that in the Principia Newton was doing things the hard way. What Newton's contemporaries learned from the Principia was that it could be done. When Leibniz published his mathematics of differention others started reproducing Newton's results with differential calculus.

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K^2

He got it from Kepler's laws and centripetal acceleration …
You also need Galileo's result and Newton's 3rd to get the m1m2 bit of it. Kepler's + Centripetal only gives you the 1/r² dependence.

tiny-tim
Homework Helper
With balls rolling down an incline not all work done goes into linear velocity. Some of the work done goes to spinning of the ball. Galileo didn't have the means to assess how much force went into the spinning. It was only later that the mechanics of spinning solids was developed.

In Newton's time the most accurate method available for measuring the Earth's gravitational force was to find the period of a pendulum.…

ooh yes, you're right of course, Cleonis! if he'd used that experiment to measure g, he'd have got 5/7 of the correct result …

the force is mgsinθ, but the effective mass is the actual mass plus the "rolling mass", m + I/r2, or 7m/5. Must have been the pendulum! arildno
Homework Helper
Gold Member
Dearly Missed

Deleted due to merely repeating what Cleonis actually said, rather than what I thought he said.

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K^2
Right. So long as he used identical shapes of homogeneous solids, he should have gotten identical rolling speeds regardless of mass and size of the sphere. It's a far easier experiment to carry out than actually dropping things and hoping to time them.

AlephZero
Homework Helper
Well, this is my first post here. Hoping for fun times.

So, straight to topic, my title describes what I want to know. I tried searching online, but I did not come upon anything that explains how Newton came to find the value 9.8m/s^2, or even whether he did.
See "Principia", Book 3, Proposition 20. Specifically, this is about determining the shape of the earth from the difference in surface gravity at different latitudes.

The variation of the period of a pendulum was well known to mariners who used pendulum clocks on long sea voyages, but in Newton's time some claimed the that the gravitational force was temperature dependent and therefore varied between the poles and the equator. They did not have a well defined scale of temperature and liquid-in-glass thermometers had not yet been invented so experimenting on this was difficult. Newton included the effect of the earth's rotation on surface gravity and proposed that the rest of the variation was caused by the earth not being a perfect sphere, which COULD be studied experimentally by accurate surveying.

As far as I know Newton did not do any experiments on this himself, but Principia Book 3 lists the experimental data he had collected from many sources (and includes some amusing rants on the incompetence of French scientists!)

Also, if you happen to know, how did Newton come up with his law of universal gravitation? [G=m1*m2/r^2]

The short answer is "read the rest of Principia book 3" (and the references back to book 1, for the details of the math). The best evidence for this available to Newton was the motion under gravity of systems of multiple bodies, in particular the effect of the sun's gravitational attraction on the moon's orbit round the earth.

Cleonis
Gold Member
Also, if you happen to know, how did Newton come up with his law of universal gravitation? [G=m1*m2/r^2]

Here is how knowing required centripetal force combined with Kepler's third law suggests an inverse square law for gravity.

Huygens was the first to show that for circumnavigating motion the required centripetal force is proportional to the radial distance r and to the square of the angular velocity. Since the period of revolution P is the inverse of the angular velocity we have the following rule of proportion:

$$F_c \propto \frac{r}{P^2}$$

Kepler's third law: the square of the Period is in proportion to the cube of the radial distance:

$$P^2 \propto r^3$$

Which can be rearranged as follows:

$$\frac{1}{r^2} \propto \frac{r}{P^2}$$

Arriving at an inverse square law is pretty much inevitable.

The problem was to handle the actual orbits, which are ellipses. Will an inverse square law give Kepler orbits?

Incidentally, Newton, in the course of his investigations, had derived the expression for required centripetal force independently. There is an article by http://msx4.pha.jhu.edu/rch.html" [Broken] (PDF-file). Newton's derivation is efficient and elegant.

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I apologize for taking long to answer; school's been keeping me busy.

I have to admit this is the first time I encounter Kepler's laws and centripetal acceleration, so this was a double-enlightenment moment for me.

K^2, thanks for your comment. Tim, good thing that you agree . AlephZero, thanks for clarifying.
You guys are awesome Thanks again.