Planetary Tunnels with Coriolis Effect
If you were to dig a tunnel into a planet, what shape would the tunnel have to be so that when you drop an object into it, the object's fall is unobstructed? The object will not necessarily fall in a straight line because of Coriolis forces. This Demonstration shows the path needed for the tunnel under various planetary conditions, including planet radius, surface gravity, rotational period, and initial starting latitude.
The following equation gives the general form for the relative motion of a particle between an inertial reference frame and a rotating one:
Here is the acceleration of the particle with respect to the inertial frame, is the acceleration of the rotating frame's origin with respect to the inertial frame, is the angular velocity of the rotating frame about its origin, and is the position vector of the particle in the rotating frame.
Using Gauss's law applied to gravity, it can be shown that for , where is the gravitational acceleration at the surface of the planet and is the radius of the planet. In detail, , where , the effective attractive mass of the planet inside a sphere of radius (). Also, note that . Given that position is a function of time, is zero, and is constant, the appropriate substitutions can be made into the above equation to give
Solving this differential equation for gives the path for the falling object with respect to the rotating reference frame of the planet.