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

• Justin Hunt
In summary, the Newtonian equation for gravity, Gm1m2/r^2, is only valid for point masses or when the two objects are sufficiently far apart. This is because the relationship between force and distance is inverse square, not exponential. However, it can be easily generalized to cover non-spherical distributions and gravity inside a mass. Newton was aware of this limitation and published his shell theorem to account for it. Tidal forces and the shell theorem were known and understood in Newtonian gravity prior to the development of general relativity.
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).

Justin Hunt said:
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:

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

Justin Hunt said:
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.

vanhees71
Justin Hunt said:
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.
Justin Hunt said:
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.
Justin Hunt said:
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.
Justin Hunt said:
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.
Justin Hunt said:
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.

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.

Dale
Justin Hunt said:
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.

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.

#### Attachments

• shell test.txt
2.5 KB · Views: 286
vanhees71, PeroK and Dale
Justin Hunt said:
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

vanhees71
vanhees71

## 1. What is Newtonian physics and how does it relate to gravity?

Newtonian physics, also known as classical mechanics, is a branch of physics that describes the motion of objects under the influence of forces. It was developed by Sir Isaac Newton in the 17th century and is based on three laws of motion. One of these laws, the law of universal gravitation, explains the force of gravity between two objects and how it relates to their masses and distance.

## 2. How does gravity affect the motion of objects?

Gravity is a force that pulls objects towards each other. It affects the motion of objects by causing them to accelerate towards the center of mass of the larger object. For example, the Earth's gravity pulls objects towards its center, causing them to fall towards the ground.

## 3. What is the difference between mass and weight in Newtonian physics?

Mass is a measure of the amount of matter in an object, while weight is a measure of the force of gravity acting on an object. In Newtonian physics, mass is constant and does not change, while weight can vary depending on the strength of the gravitational force.

## 4. Can gravity be shielded or cancelled out?

In Newtonian physics, gravity cannot be shielded or cancelled out. It is a fundamental force that acts between all objects with mass. However, the effects of gravity can be counteracted by other forces, such as the normal force or the force of air resistance.

## 5. How does the force of gravity change with distance?

According to the law of universal gravitation, the force of gravity decreases as the distance between two objects increases. This means that the farther apart two objects are, the weaker the force of gravity between them will be. This relationship is described by the inverse-square law, which states that the force is inversely proportional to the square of the distance between the objects.

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