# How Does the Force Between Two Spheres Relate to Their Potential Energy?

• lark
In summary: Yes, this is exactly what Meir said. Newton was able to prove this using Euclidean geometry, but this discovery is actually based on the idea of spherical symmetry.
lark
Something I realized the other day - trying to figure out what the gravitational (or electrostatic) force would be between two spheres -
the force ON a sphere from another sphere, is the same as if the other sphere's mass were concentrated at its center.

So what is this force? It turns out that the force on a sphere FROM a point mass is the same as the force a sphere exerts ON a point mass - that is, the sphere is attracted to another point as if the sphere were a point itself! (from Newton's third law or whatever - action = reaction)

So the force between two spheres is the same as if the mass in the two spheres were concentrated at their centers.

In general if potential energy is 0 at infinity, the potential energy of object 1 from the gravitational field of object 2 is the same as the potential energy of object 2 in the grav. field of object 1 ...

laura

lark said:
Something I realized the other day - trying to figure out what the gravitational (or electrostatic) force would be between two spheres -
the force ON a sphere from another sphere, is the same as if the other sphere's mass were concentrated at its center.

So what is this force? It turns out that the force on a sphere FROM a point mass is the same as the force a sphere exerts ON a point mass - that is, the sphere is attracted to another point as if the sphere were a point itself! (from Newton's third law or whatever - action = reaction)

So the force between two spheres is the same as if the mass in the two spheres were concentrated at their centers.

laura
Congratulations!
You have just shown one of the most important discoveries Sir Isaac Newton made with his theory of gravitation.
this was not meant ironic, it is indeed a pleasing and elegant result.

Poor Newton didn't know Gauss's law (He didn't even know Gauss), so he had to work that out in a very complicated derivation.

arildno said:
You have just shown one of the most important discoveries Sir Isaac Newton made with his theory of gravitation.

just want to reiterate to the OP what Meir said, that this can be shown pretty clearly using spherical symmetry and Gauss's Law.

Is it Newton's fault he didn't know Gauss, and essentially proved this, if I recall correctly, within the context of Euclidean geometry?

## 1. What is potential energy?

Potential energy is the stored energy an object has due to its position or state. It is the energy that an object possesses because of its potential to do work.

## 2. How is potential energy calculated for a sphere?

The potential energy of a sphere is calculated by multiplying its mass by the acceleration due to gravity and its height from the reference point.

## 3. What is the reference point for calculating potential energy of a sphere?

The reference point for calculating potential energy of a sphere is usually the surface or ground level. This is where the gravitational potential energy is zero.

## 4. How does potential energy change for a sphere as it moves to a different height?

As a sphere moves to a higher position, its potential energy increases. This is because the sphere now has more potential to do work due to its increased height from the reference point. Conversely, as the sphere moves to a lower position, its potential energy decreases.

## 5. Can potential energy of a sphere be converted into other forms of energy?

Yes, potential energy can be converted into other forms of energy. For example, if a sphere is dropped from a certain height, its potential energy is converted into kinetic energy as it falls. This can also be seen in a bouncing ball, where the potential energy is converted into kinetic energy as the ball bounces back up.

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