91Ó°ÊÓ

The concept of gravitational force is key to understanding many physical movements and phenomena. In the exercise, it's given that inside the Earth, the force of gravity is proportional to the distance from the center. This can be expressed by the equation \( F = -k \times r \), where \( F \) is the gravitational force, \( k \) is a proportionality constant, and \( r \) is the distance from the center of the Earth. The negative sign simply indicates that the force is directed towards the center. Gravitational force acts as the pull that reinvigorates Newton's compelling idea that objects are attracted to each other, based on their masses and the distance between them. This fundamental force is what keeps the planets in orbit around the sun and makes objects fall to the ground on Earth.

The challenge in this exercise lies in the understanding that gravitational force changes according to location within the Earth. Unlike the surface where gravity is constant, inside the Earth, it changes linearly, which means it varies directly with distance. This requires a different approach than you might be used to on the surface.

Newton's Second Law establishes the relationship between an object's mass, its acceleration, and the applied force acting on it. It's famously summarized as \( F = m \times a \), where \( F \) represents the force applied, \( m \) is the mass of the object, and \( a \) is its acceleration.

In the context of our problem, this law helps us connect gravitational force to the resulting acceleration of the rock. Given \( F = -k \times r \) and applying Newton's second law, we interpret this as \( m \times a = -k \times r \), leading to \( a = -(k/m) \times r \). Such expression indicates that the acceleration is proportional to the distance from the center, following a behavior characterized by simple harmonic motion (SHM).

This elegant revelation that acceleration is proportional to distance opens up the beautiful symmetry in our understanding of physical systems, linking kinematic behavior to influencing forces.

Acceleration explains how quickly an object changes its velocity over time. In this case, we are interested in how the acceleration of the rock is influenced by its distance from the Earth's center. By noting the expression \( a = -(k/m) \times r \), we see that acceleration is directly proportional to the distance \( r \), but in the opposite direction.

Imagine the rock moving in the drilled hole: as it approaches the center, the acceleration reduces, reaching zero at the very midpoint—this midpoint being where forces balance out in the simple harmonic motion. Then, just as it passes the center, the acceleration direction reverses, slowing down the rock and pulling it back, in repetitive oscillations.

This situation is reminiscent of the plausible explanations given by harmonic motions, seen in objects like oscillating springs or pendulums.

The proportionality constant \( k \) is crucial for understanding how the gravitational force changes inside the Earth. In our problem context, \( k \) relates the gravitational force to the distance from Earth's center, given by \( F = -k \times r \).

To find \( k \), we equate the gravitational force experienced at the Earth's surface, \( F = G M m/R^2 \), to \( k \times R \) (since \( r = R \) at the surface). Through this, we derive that \( k = G M/R^3 \), where \( G \) is the gravitational constant and \( M \) is the mass of the Earth.

This relationship gives insight into how force diminishes as you descend to the Earth's core, aligning perfectly with the idea that the force acting on the rock is directly changed by how far it is from the core. The proportionality constant thus serves as a bridge between gravitational behavior and practical motion predictions.