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Newton's Law of Gravitation is crucial in understanding how planets and other celestial bodies interact. According to this law, any two objects with mass attract each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This can be expressed mathematically as:\[F = \frac{G \cdot M_1 \cdot M_2}{r^2}\]Where:

• \( F \) is the gravitational force between the objects.
• \( G \) is the gravitational constant, approximately \(6.674 \times 10^{-11} \, \text{m}^3\text{kg}^{-1}\text{s}^{-2}\).
• \( M_1 \) and \( M_2 \) are the masses of the two objects.
• \( r \) is the distance between the centers of the two masses.

This concept helps us understand gravitational acceleration, which is the gravitational force per unit mass applied to an object near a celestial body like a planet.

Understanding planetary mass distribution is paramount when calculating gravitational fields, as the mass distribution affects how gravity acts at different points around the planet. In our exercise, the planet consists of a core and an outer shell:

• The core has a radius \(R\) and a mass \(M\).
• The outer shell extends from radius \(R\) to \(2R\) and has a mass of \(4M\).

In this setup, the gravitational effects at different radii vary due to the presence or absence of mass. For instance, inside the first radius \(R\), only the core influences the gravitational acceleration because the shell does not exert a gravitational force inside its own radius. Beyond the surface of the core, the total mass affecting gravitational forces changes, accounting for both the core and the shell mass. Recognizing how mass is arranged within a planet allows us to better predict and calculate the gravitational forces experienced at different distances.

Calculating the gravitational field at various points outside and inside a planet requires careful consideration of both the mass and the distance involved. This is done using the formula for gravitational acceleration:\[g = \frac{G \cdot M}{r^2}\]Here:

• \( g \) is the gravitational acceleration experienced at a distance \( r \).
• \( M \) is the mass causing the gravitational field (can be the core mass or the total mass depending on the location).
• \( r \) is the distance from the center of the mass.

In our case:- At distance \( R \), only the core mass \( M \) influences \( g \), as the shell contributes nothing internally.- At distance \( 3R \), the gravitational force comes from the entire planet, which includes both the core and shell, totaling a mass of \( 5M \).These calculations demonstrate how gravitational acceleration changes with both varying mass amounts and distance, providing insights into planetary gravitational fields.