Imagine that two tunnels are bored completely through the Earth, passing through the center. Tunnel 1 is along the Earth's axis of rotation, and tunnel 2 is in the equatorial plane, with both ends at the Equator. Two identical balls, each with a mass of \(5.00 \mathrm{~kg}\), are simultaneously dropped into both tunnels. Neglect air resistance and friction from the tunnel walls. Do the balls reach the center of the Earth (point \(C\) ) at the same time? If not, which ball reaches the center of the Earth first?

Understanding the gravitational force calculation is crucial when delving into physics problems involving two masses attracting each other. According to Newton's law of gravitation, the force of gravity between two objects can be calculated using the equation:

\( F = G \frac{m_1 m_2}{r^2} \)

In this equation, \(F\) represents the gravitational force, \(G\) is the gravitational constant (approximately \(6.674 \times 10^{-11} N(m/kg)^2\)), \(m_1\) and \(m_2\) are the masses of the objects, and \(r\) is the distance between their centers. For our exercise, one mass is the Earth and the other is the ball. The simplification yielded from
\( F = G \frac{mM}{x^2} \)
shows a direct proportionality between the force and Earth's mass (\(M\)) and an inverse-square relation to the distance from Earth's center (\(x\)). This becomes foundationally important when considering the motion of objects under gravity alone, like the balls in the tunnels.

When objects are influenced only by gravity, the acceleration due to gravity serves as a uniform field simplifying the analysis of their motions. Newton's second law (\(F = ma\)) provides a link between force (
\(F\)) and acceleration (
\(a\)), with \(m\) being the object's mass. Plugging in the gravitational force while cancelling out the mass of the object yields:

\(a = G \frac{M}{x^2}\)

This equation reveals that an object's acceleration towards Earth only depends on Earth's mass and the object's distance from Earth's center, not on the object's mass itself. Therefore, in our scenario with the two balls, despite the orientation of the tunnels, the balls' radial accelerations are identically affected by gravity as they travel towards the center of Earth, leading to their synchronized arrival.

The Coriolis effect is a deflection pattern experienced by objects moving within a rotating frame of reference, such as Earth. It's essential when understanding weather patterns, ocean currents, and any motion over a large scale on Earth's surface. However, in the context of our exercise, the effect is still present but doesn’t affect the balls' travel time in the tunnels.

For the ball in Tunnel 2, its motion is subject to the Coriolis force due to Earth's rotation. This force acts perpendicular to the movement towards Earth's center, resulting in a curved path, but it does not change the speed of the object in the direction towards the center. Since we are only concerned with the time it takes for the balls to reach the center of Earth, the Coriolis force can be ignored in solving this problem—the radial acceleration (towards the Earth's center) remains unchanged, which assures the time for both balls to reach point C is equal.