Savchenko Problem 3.2.13
Physics · mechanics · electromagnetism
Problem
A straight tunnel is dug in the Earth that does not pass through its center. Determine the time of movement of a train with engines turned off through such a tunnel if the influence of the Earth’s rotation on the movement of the train and friction are neglected.Intuitive explanation
The key insight is that inside a uniform-density sphere, gravity acts like a spring force—only the mass within your current radius pulls you down. For any straight tunnel through Earth, you can decompose the gravitational force into a component along the tunnel that's proportional to your displacement from the tunnel's midpoint, creating simple harmonic motion. The remarkable result is that the period depends only on Earth's density, not on the tunnel's length or position, so the transit time is the same for all such tunnels.
Intuitive explanation
The key insight is that inside a uniform-density sphere, gravity acts like a spring force—only the mass within your current radius pulls you down. For any straight tunnel through Earth, you can decompose the gravitational force into a component along the tunnel that's proportional to your displacement from the tunnel's midpoint, creating simple harmonic motion. The remarkable result is that the period depends only on Earth's density, not on the tunnel's length or position, so the transit time is the same for all such tunnels.