I Why Did Newton Use d² for Gravitation Law?

[bazer43]

When Newton developed his law of universal gravitation, why would he use distance squared d², instead of 4/3πr³ as the field would expand in a sphere around the body?

 

[vanhees71]

I'm not sure what you are referring to. Newton figured out that the gravitational force between two bodies that can be considered pointlike is attractive with the direction along the line connecting the positions of the two bodies and its magnitude is proportional to the product of the two masses and the inverse squared distance of the bodies, i.e., for any two bodies there's a universal constant $G$ such that
$$\vec{F}_{12}=-G \frac{m_1 m_2 (\vec{x}_1-\vec{x}_2)}{|\vec{x}_1-\vec{x}_2|^3}.$$
That's an empirical fact, following from Kepler's three laws of planetary motion. Nowadays of course we use Newton's universal law to derive Kepler's laws :-).

 

[weirdoguy]

why would he use distance squared d², instead of 4/3πr³

Because the latter would violate Keplers' laws.

 

[anuttarasammyak]

When Newton developed his law of universal gravitation, why would he use distance squared d², instead of 4/3πr³ as the field would expand in a sphere around the body?

Your observation in other words is that area A=$4\pi r^2$ or $4\pi d^2$ following your reference, is used. Same amount of Force is distributed on sphere surface area of any $r$ or $d$.

\int \mathbf{a}\cdot d \mathbf{A}=-4\pi G m
or
\nabla \cdot \mathbf{a} = -4\pi G \rho
where a is acceleration caused by mass m around it. $\rho$ is mass density.

 

[Lord Jestocost]

When Newton developed his law of universal gravitation, why would he use distance squared d², instead of 4/3πr³ as the field would expand in a sphere around the body?

"Any point source which spreads its influence equally in all directions without a limit to its range will obey the inverse square law."
see: http://hyperphysics.phy-astr.gsu.edu/hbase/Forces/isq.html

 

[russ_watters]

Because "what is the volume containing the field?" was not the question he was trying to answer.

 

[Herman Trivilino]

Because the area of a sphere is proportional to the square of its radius.

 

[sophiecentaur]

When Newton developed his law of universal gravitation, why would he use distance squared d², instead of 4/3πr³ as the field would expand in a sphere around the body?

No one has dealt exactly with why the inverse cube law is not appropriate. The inverse square law is followed because there is nothing to decrease the effect of the attraction in the free space between the masses and the only reduction is because of the spreading out of the 'lines of gravitational force' over an increasing area as the distance increases.
The inverse square law is followed in several other places in Physics. Light spreads out in empty space and the reduction in intensity follows the ISL. But, if the space in between contains an absorbent material (say dust) the rate of additional fall-off can be much higher. It can be proportional to the distance and that sometimes 'beats' the ISL. It's the same for sound energy which can be attenuated at different rates, according to humidity, for instance.

The ISL only even works for gravity if the mass m can be treated as being at a point or a body with spherical symmetry. If you take points below the Earth's surface (or strictly a sphere of uniform density) the gravity gets greater, proportionally with the radius. So you have to check that the ISL actually applies whenever you want to use it.

 

[Arjan82]

Fun fact: It seems that it was actually Hooke who suggested the inverse square law to Newton...
https://en.wikipedia.org/wiki/Inverse-square_law#Gravitation

 

[sophiecentaur]

Newton was a real swine about anyone he saw as competition. He did some dirty deeds, they say, to suppress Hooke's career and work. At least we all remember Hooke's Law for springs, even if the rest of his stuff has been neglected.

 

[anuttarasammyak]

We know gravity of infinite long bar is proportional to $r^{-1}$ where r is distance from the bar.
We may interpret gravity proportional to $r^{-2}$ from the "point" matter is actually from "infinite long bar" in 4-dimension space one dimension along the bar of which we cannot conceive. In this way , playfully defending OP, we may be able to imagine that gravity in 4-dimension space follows $r^{-3}$ law.