If you could drill a hole through the earth and drop an object into the hole, how would Earth's gravity affect the object and why?

If you could drill a hole through the Earth and drop an object into it, the behavior of the object as it falls would be influenced by gravity in a unique way.

1. **Initial Fall**: When the object is first dropped, it would experience the full force of Earth's gravity. This means that it would accelerate downward toward the center of the Earth at a rate of approximately \(9.81 \, \text{m/s}^2\).

2. **Gravity Variation**: As the object falls deeper into the hole, the gravitational force acting on it would change. According to the shell theorem, the gravitational force inside a spherical shell of uniform density is zero. Therefore, as the object moves toward the center of the Earth, the effective gravitational force acting on it decreases. Specifically, if we assume the Earth has a uniform density, the gravitational force \(F\) acting on the object is proportional to its distance \(r\) from the center of the Earth. Mathematically, this can be described by:

\[
F = \frac{G M(r) m}{r^2}
\]

where \(M(r)\) is the mass enclosed within radius \(r\). As \(r\) decreases, \(M(r)\) does not increase proportionally to \(r\) (it decreases), which results in less gravitational pull on the object.

3. **At the Center**: When the object reaches the very center of the Earth, the gravitational forces from all directions would perfectly cancel each other out resulting in a net gravitational force of zero. The object would be weightless at this point.

4. **Beyond the Center**: As the object continues to fall past the center and toward the other side, it would once again feel a gravitational attraction pulling it back toward the center. However, now the force would be directed opposite to its motion. Consequently, the object would decelerate until it comes momentarily to rest at the surface on the opposite side of the Earth.

5. **Oscillation**: If we disregard air resistance and other forces, the object would oscillate back and forth through the Earth, taking the same time to travel from one side of the Earth to the other, and it would repeat this indefinitely as a simple harmonic oscillator. The time period \(T\) for this oscillation can be approximated by:

\[
T \approx 2\pi \sqrt{\frac{R}{g}}
\]

where \(R\) is the radius of the Earth and \(g\) is the acceleration due to gravity at the surface.

Thus, the overall effect of gravity on the object dropped into a hole through the Earth would be to cause it to fall, decelerate, and oscillate back and forth between the two ends of the hole, experiencing variations in gravitational force as it nears the center.