Planetary Tunnels with Coriolis Effect

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

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.

Contributed by: Ariel Krasik-Geiger and Ramona Barber (May 2012)
Open content licensed under CC BY-NC-SA


Snapshots


Details

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.



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send