Is Newton's Equation for Gravity Accurate for All Situations?

[Justin Hunt]

I have several questions about the Newtonian equation for Gravity Gm1m2/r^2. First, I want to point out this equation is only valid for point masses or when the two objects are sufficiently far apart.This is because the r squared term on bottom means that the relationship between force and distance is exponential rather than linear. The curve flattens out as the objects become further away, which is why at sufficient distances this approximation is valid. The basic premise of Newton was that all matter attracts all other matter, using the center of gravity was just a simplification in order to create an equation. This is why his equation was unable to determine the orbit of Mercury for instance.

Examples are why tidal forces would pull you apart when entering a black hole or why if you were in a hollow sphere you would NOT be pulled towards the center of it.

My question is one, was Newton aware of this limitation?

Two, did Newton or other people since then try to create a more accurate equation for certain situations such as an equation for determining the force of gravity between a point mass and a uniform sphere of radius r that is distance d away? (integrating the force over the volume rather than using the center of mass).

 

[Dale]

First, I want to point out this equation is only valid for point masses or when the two objects are sufficiently far apart.

This is not quite correct. It is valid outside of spherically symmetric masses regardless of the distance. For example, the Earth is approximately spherical so it is approximately valid for us on the Earth despite us being close to it.

Furthermore, it can be easily generalized to cover non spherical distributions and gravity inside a mass:

$$F=\int_M -\frac{Gm}{r^2}\hat r dM$$

Examples are why tidal forces would pull you apart when entering a black hole or why if you were in a hollow sphere you would NOT be pulled towards the center of it.

Tidal forces can easily be calculated with the simplified formula and the hollow sphere can easily be calculated with the generalized formula. Newton was well aware of both and published the hollow sphere calculation as Newton’s shell theorem.

 

[Ibix]

First, I want to point out this equation is only valid for point masses or when the two objects are sufficiently far apart.

The extension to extended bodies is trivial (at least in principle - the maths gets horrible fairly quickly, particularly if the bodies are non-rigid). You just replace the masses with densities and integrate over the volume.

This is because the r squared term on bottom means that the relationship between force and distance is exponential rather than linear.

No - the relationship is an inverse square.

The basic premise of Newton was that all matter attracts all other matter, using the center of gravity was just a simplification in order to create an equation. This is why his equation was unable to determine the orbit of Mercury for instance.

No. The final 43 seconds of arc per century in the precession of Mercury is not explicable by Newtonian gravity because (loosely speaking) Newtonian gravity doesn't include curvature of spacetime. There are effects due to the non-sphericity of the Sun, but these had been accounted for using Newtonian gravity prior to the development of general relativity.

Examples are why tidal forces would pull you apart when entering a black hole or why if you were in a hollow sphere you would NOT be pulled towards the center of it.

I'm not sure what you are saying these are examples of. Both tidal forces and the shell theorem were known and understood in Newtonian gravity prior to general relativity.

My question is one, was Newton aware of this limitation?

He derived the shell theorem in the Principia, as far as I am aware. Which, incidentally, contradicts your "using the center of gravity was just a simplification in order to create an equation". Newton in fact proved rigorously that the gravitational field outside a spherically symmetric mass was the same as the field from a point of the same mass. And since most gravitating bodies are very close to spherical you seldom need anything more sophisticated.

 

[Justin Hunt]

Thank you for pointing out the shell theorem. This address my question for why spheres are being treated as point masses. I will need to do the math myself though, because my back of the envelope calculations don't agree with the theorem (Will probably take a few days at least to do a thorough analysis). I will post my work after i finish.

 

[Nugatory]

Thank you for pointing out the shell theorem. This address my question for why spheres are being treated as point masses. I will need to do the math myself though, because my back of the envelope calculations don't agree with the theorem (Will probably take a few days at least to do a thorough analysis). I will post my work after i finish.

Google should find you a number of good derivations, as this is standard fare in a first-year physics class.

 

[Justin Hunt]

Well, I wasn't able to prove it mathematically, but i did write a C++ program to test the theorem. I have attached the source code as a text document. the program creates a 3D array using the subscripts as XYZ locations on a 3D grid. I first used a formula to assign the mass of points inside the sphere to 1 and points of the cube outside the sphere to 0. I then calculated the force to each point and broke it into XYZ components and stored that info in 3 additional arrays. at the end i compare the sum sum of the XYZ components to the scenario of all the mass at the center of the sphere. I found some inconsistencies when choosing a point very close to the sphere, but I account that to imperfections in the model. Any point chosen about 50 units away will result in identical results using both methods.

My program modeled the sphere with a uniform mass distribution, but i plan on trying different scenarios later and seeing how it affects the results.

 

[Dale]

I wasn't able to prove it mathematically, but i did write a C++ program to test the theorem.

Good approach! You do have to pay attention to numerical issues in such an approach, but if you do it right then you can get some intuition about situations where you cannot make a rigorous proof

 

[PeroK]

Well, I wasn't able to prove it mathematically,

The key to the mathematical proof is to exploit the symmetry of the sphere. There's a proof here, for example:

http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/sphshell.html#wtls