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Understanding how mass is distributed within a planet is crucial when assessing gravitational properties. In our planetary model, the planet consists of two distinct parts: the core and the shell.

Let's break it down:

• The core, the inner part, has a radius of \(R\) and a mass of \(M\). It forms the central mass concentration of the planet.
• The outer shell surrounds the core, with a mass of \(4M\). It stretches from an inner radius \(R\) to an outer radius of \(2R\).

This setup means the total mass of the planet is \(M + 4M = 5M\). The mass distribution affects how gravitational forces act both inside and outside the shell. Specifically, within the shell, only the mass at smaller radii influences gravitational pull, while outside, the entire mass seems to pull from the center.

Think of it as if you're inside and just influenced by the inner part. Once outside, the whole planet seems like one single mass.

The inverse square law gives us a fundamental understanding of how gravitational force operates. It states that gravitational force diminishes with the square of the distance from the mass center.

Mathematically, it's written as:

• Gravitational force \(F\) is inversely proportional to the square of the distance \(r\): \( F \propto \frac{1}{r^2} \).

This means that if you double the distance from the center of mass, the gravitational pull is one-fourth of its original strength.
This law is vital in calculating gravitational acceleration at different points around a planet. For instance, as seen in the solution, moving from a radius of \(R\) to \(3R\) decreases the gravitational acceleration, because you've tripled the distance from the center, reducing gravitational forces significantly.

Calculating gravitational acceleration at different radii is key in understanding a planet's gravity's variation. In our scenario, gravitational acceleration or "g" is calculated at two points: at radius \(R\) and \(3R\).
At radius \(R\), the mass enclosed (only the core) affects gravity:

• Using the formula: \( g(R) = \frac{GM}{R^2} \)

Where \(G\) is the gravitational constant. At \(R\), only the core's mass \(M\) influences gravity, giving us a stronger pull.

At radius \(3R\), the entire mass \(5M\) is considered. The calculation changes to:

• \( g(3R) = \frac{G(5M)}{(3R)^2} \)

Here, since the distance is greater, gravitational acceleration diminishes due to the inverse square law.
This decrease illustrates how being farther from mass centers weakens gravitational effects.

Gravitational force is what keeps planets, moons, and all celestial bodies in motion within the universe. This force is the attraction between two masses.
Key aspects include:

• It's directly proportional to the product of the two masses (e.g., planet mass and a particle mass).
• The force is inversely proportional to the square of the distance between their centers.

The gravitational force can be expressed as \(F = \frac{G imes m_1 imes m_2}{r^2}\), where \(m_1\) and \(m_2\) are masses, and \(r\) is the distance between them.

In our exercise, the gravitational force created by the planet's core and shell influences a particle at different radii differently. At smaller radii, gravitational forces are stronger. As you move outward, as seen from \(R\) to \(3R\), the force reduces due to increased distance.
This core concept helps in understanding how objects within a gravitational field experience different pulls depending on their location, making orbits and motions predictable.