# Unraveling Newton’s Shell Theorem: A 20-Year Journey

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## Chapter 1: The Enigma of Gravity

For two decades, Newton grappled with a profound question that had perplexed humanity since the days of ancient Greece: why do objects fall? His explorations led him to insights based on the pioneering work of Tycho de Brahe and Johannes Kepler, which informed his understanding of gravitational mechanics. However, while he made significant strides in explaining how gravity operates, the deeper question of why gravity exists would be left for future thinkers, including Einstein.

The core challenge for Newton was conceptualizing the mass of a planet from the perspective of an external observer. He had an intuitive sense that treating a planet as a point mass might be the right approach. Yet, intuition alone was insufficient; rigorous proofs were essential for a theory to gain scientific credence. After two decades of exploration, he achieved this breakthrough, a moment likely filled with immense satisfaction akin to the relief of surfacing after a deep dive.

In educational settings, we learn about gravity through the formula that relates two masses with the gravitational constant, divided by the square of their distance apart:

When discussing planets, we often consider the distance from the center of the planet to a specified point in space without fully appreciating the intricate interplay of forces at work. The gravitational forces (dF) exerted by individual mass points (dm) effectively combine to create a net force as if the entire mass were concentrated at the planet's center. This simplification holds true under the condition of uniform density throughout the planet's spherical shape.

To understand this better, imagine a planet as composed of infinitely thin shells, from the outermost layer down to the innermost. Each shell is centered at the planet's core, maintaining an approximately uniform density.

Now, let's delve into the proof that demonstrates the gravitational force of a planet acts as if all its mass were concentrated at its center.

### The Proof

First, it's important to set the scene for our analysis.

Here, we have a circular shell representing a planet with radius r, density ?, and mass M. Each shell can be subdivided into an infinite number of rings, each with an infinitesimal width (dw), a fixed distance (d) from point P, and a mass denoted as dM. Each shell's center is located at point O, which is at a distance R from point P.

To analyze the forces, we will sum the forces (dF) exerted by these rings on point P, starting from the topmost ring. It's crucial to note that due to symmetry, only the component of force along the line R will contribute. The vertical component (y) of each force from any point mass in a ring has a corresponding counterpart that cancels out.

Thus, we can express the net force from the ring as follows:

Here, dM represents the mass of the ring, which can be calculated as its area multiplied by its density. The area of the ring is the circumference (2?y) times its width (dw), leading to the following expression:

Recognizing that r ? sin? equals y and r ? d? equals dw, we can rewrite the equation accordingly:

Next, by expressing the density of the shell as M/4?r²:

We can substitute dM back into the force equation.

In this scenario, we have three independent variables: d, ?, and ?. These need to be simplified into a single variable. Using the law of cosines will aid in this transformation. Thus, we can derive the following equations:

and

However, we need to differentiate to switch from cos? to sin? ? d?. This results in:

Now, d has become our sole independent variable, and the gravitational force equation transforms into:

This expression represents the force exerted by a single ring.

The final step involves summing the forces from all rings of the shell, which translates to performing an integration from the rightmost ring to the leftmost. The limits of integration will range from d=R-r to d=R+r. Evaluating this integral yields the desired result.

As we anticipated, the cumulative gravitational forces from the infinitesimal mass points on the entire shell act as if all their masses were located at the center. Each time I encounter such elegant results from calculations, I am increasingly struck by the symmetrical nature of our universe. Perhaps this realization is a form of appreciation. Until next time!

The first video titled "A QUICK Proof of Shell Theorem in Under 3 Minutes!" provides a brief yet insightful explanation of the Shell Theorem, showcasing its fundamental principles in a concise format.

The second video, "Newton's Shell Theorem Part 1 (conceptual)," delves into the conceptual underpinnings of the Shell Theorem, offering a deeper understanding of its implications in gravitational theory.